Operatorial approach to the non-Archimedean stability of a Pexider
The non-self-adjointness of the radial momentum operator in n dimensions
a r X i v :m a t h -p h /0009016v 2 24 A p r 2002The non-self-adjointness of the radial momentumoperator in n dimensionsGil PazPhysics Department,Technion-Israel Institute of Technology,3200Haifa,IsraelAbstractThe non self-adjointness of the radial momentum operator has been noted before by several authors,but the various proofs are incorrect.We give a rigorous proof that the n -dimensional radial momentum operator is not self-adjoint and has no self-adjoint extensions.The main idea of the proof is to show that this operator is unitarily equivalent to the momentum operator on L 2[(0,∞),dr ]which is not self-adjoint and has no self-adjoint extensions.I INTRODUCTIONThe radial momentum operator was the subject of long discussions since the early days of Quantum Mechanics.Its exact form and relation to the Hamiltonian were considered by many authors[1,2,3,4].Unlike the classical radial momentum,the connection between the radial mo-mentum operator and the Hamiltonian of a free particle is not trivial[5,6].In fact,the connection between the radial momentum and the Hamiltonian in n dimensions is[6]:ˆH=ˆP2r2mr2+¯h24r2.(at least formally,in principle one has to define the self-adjoint extension ofˆP2r[7,8],which is not self-adjoint.)Another important question that was raised is whether the radial momentum operator is an ob-servable.Although Dirac claimed in“The Principles of Quantum Mechanics”that the radial momentum operator is“real”[2],many authors realized that the radial momentum operator is not self-adjoint[4,9,10,11].Unfortunately none of these proofs is correct.In order to be an observable the radial momentum operator should be self-adjoint.Simply check-ing that the eigenvalues are real(like[4,9]do)is not sufficient(or necessary),one has to pay attention to the domain on which the operator is defined.Perhaps the most appealing(but in-correct)argument appears in[10,11](we use units where¯h=1):“Since ˆr,ˆP r =i the unitary transformation e−iaˆP r shifts the operatorˆr by a(because e iaˆP rˆr e−iaˆP r=ˆr+a)while leaving its spectrum invariant(being a unitary transformation).Therefore the spec-trum ofˆr must be(−∞,∞).Since the spectrum ofˆr is(0,∞)the operatorˆP r cannot be self-adjoint.”This statement,had it been true,would have prevented any operator,which has canonical com-mutation relation withˆr,from being self-adjoint.Unfortunately,this statement cannot be true as we can see from the following counter example.Consider the space:L2[(0,1),dx],the momentum operatorˆP=−i dThere’s seem to be almost a consensus in the literature that the operator which correspond to the radial momentum in n dimensions is the operator−i ∂2r [3,5].We shall now show that this operator is not self-adjoint and has no self-adjoint extensions.3II THE PROOFIt is well known that the momentum operator in L2[(0,∞),dr]is not self-adjoint and has no self-adjoint extensions[12].When we say the“momentum operator”we mean the operatorˆP=−i dr n−1r n−12ϕ∈L2[(0,∞),dr].This transformation(with n=3)is well known from elementary textbooks on quantum mechanics,where it is used to solve the Schr¨o dinger equation for the hydrogen atom[13].U−1is defined by:U−1:L2[(0,∞),r n−1dr]→L2[(0,∞),dr](U−1ϕ)(r):=r n−12 −i d2=−i d2r ,(5)which is,formally,the n-dimensional radial momentum operator.Ifϕ∈D(ˆP r)then U−1ϕ∈D(ˆP).Therefore ifϕ∈D(ˆP r)then: a. r n−12ϕis absolutely continuous in(0,∞)c.(r n−1drr n−12 2r n−1dr⇒ r n−1and we have(a).ˆPis(at least)symmetric,that is ifϕ,χ∈D(ˆP r)then the following equality should hold:r∞0 ˆP rχ ϕr n−1dr.(7)However,∞0χr1−n dr r n−1r n−1dr r n−1r n−12ϕ|∞0+i ∞0dr r n−12ϕdr=(−i)2χ r n−1 ˆP rχ ϕr n−1dr,(8)where we were forced to assume(b)in order to use integration by parts.(b)also ensures that the boundary term in(8)is zero at infinity.In order that(7)will hold,we have to assume that the boundary term also vanishes at the origin,i.e.assume(c).Thus we have definedˆP r with its proper domain and we see that it is symmetric.Furthermore it is also closed since it is unitarily equivalent toˆP,which is closed[12].As we have said earlier this operator is the radial momentum operator.In order tofind out whether this operator is self-adjoint or at least have self-adjoint extensions we have to check the dimensionality of the two subspaces:K−=ker(i+ˆP∗r)and K+=ker(i−ˆP∗r).If they do not have the same dimensionality the operator is not self-adjoint and has no self-adjoint extensions[7].ˆP∗is easy tofind since[12]ˆP∗r=UˆP∗U−1and we have:rϕ∈ker i±ˆP∗r ⇒ i±ˆP∗r ϕ=0⇒i±UˆP∗U−1 ϕ=0⇒U i±ˆP∗ U−1ϕ=0⇒i±ˆP∗ U−1ϕ=0⇒U−1ϕ∈ker i±ˆP∗ .(9) In a similar way we can show that:ψ∈ker i±ˆP∗ ⇒Uψ∈ker i±ˆP∗r (10) and we conclude that:dim ker i±ˆP∗r =dim ker i±ˆP∗ .(11)ˆP∗has the following property[12]:dim ker i+ˆP∗ =0dim ker i−ˆP∗ =1,(12)5becauseker i+ˆP∗ ={ce r|c∈C}⊆L2[(0,∞),dr]ker i−ˆP∗ ={ce−r|c∈C}⊆L2[(0,∞),dr].(13) Thereforedim ker i+ˆP∗r =dim ker i−ˆP∗r (14) andˆP r is not self-adjoint and does not have self-adjoint extensions.Using this equivalence we can understand the non self-adjoint nature of the radial momentum operator on a more intuitive level,by transferring the problem into a one dimensional problem. In one dimension we can,in general,consider three types of interval:infinite interval,finite in-terval(“particle in a box”)and semi-infinite interval.In thefirst case the momentum operator is self-adjoint because we can translate a wave packet to both sides.In the case of a“particle in a box”we can translate a wave packet and whatever“comes out”at one end we can enter at the other end(possibly with a different phase which corresponds to a specific self-adjoint extension [7]).Therefore the momentum operator,which is not self-adjoint,has self-adjoint extensions.In the case of a semi-infinite interval,we can move a wave packet to the right but if we try to move it to the left what“comes out”at the origin cannot be entered at the other end since the“other end”is infinity.Therefore the translation operator is not unitary and the momentum operator is not self-adjoint and has no self-adjoint extensions.To summarize,we have seen that the radial momentum operator is unitarily equivalent to the momentum operator on the half line(0,∞).Since this operator is not self-adjoint(and has no self-adjoint extensions),the radial momentum operator is not self-adjoint(and has no self-adjoint extensions).ACKNOWLEDGMENTSThis work was supported by the Technion Graduate School.I thank Aviv Censor and Amnon Harel for useful discussions.6References[1]B.Podolsky,Phys.Rev.32812(1928).[2]P.A.M Dirac,The Principles of Quantum Mechanics4th ed.(Revised)(Oxford University Press,Hong Kong,1995)p.152-153.[3]B.S.deWitt,Phys.Rev.85,653(1952).[4]A.Messiah,Quantum Mechanics,(North-Holland,Amsterdam,1965).[5]H.Ess´e n,Am.J.Phys.46,983(1978).[6]G.Paz,Eur.J.Phys.22337(2001).[7]M.Reed,B.Simon,Methods of Modern Mathematical Physics Vol.II,(Academic Press,NY, 1975).[8]S.Albeverio,F.Gesztesy,R.Høegh-Krohn,H.Holden,Solvable Models in Quantum Mechan-ics,(Springer-Verlag,NY,1988).[9]R.L.Liboff,I.Nebenzahl,H.H.Fleischmann,Am.J.Phys.41,976(1973).[10]O.Levin,A.Peres J.Phys.A:Math.Gen.27,L143(1994).[11]J.Twamley J.Phys.A:Math.Gen.31,4811(1998)..[12]J.Blank,P.Exner,M.Havliˇc ek,Hilbert Space Operators in Quantum Physics,(AIP Press, NY,1994).[13]C.Cohen-Tannoudji,B.Diu loe,Quantum Mechanics,(Wiley-Interscience,NY,1977).7。
M,
uz ¯)u = ¯ = (∂/∂ z
We are mostly concerned with contractible solutions, u : T2 → M is contractible. (2)
Equation (1) linearized about a conFredholm operator. Essentially for this reason for a typical f the equation has a finite number of solutions. Our goal is to estimate this number from below by investigating the manifold formed by all pairs (u, f ) where u solves (1), (2) with f = f .
Improved $alpha^4$ Term of the Muon Anomalous Magnetic Moment
This result provides the most stringent test of the Standard Model. Unfortunately, such a test must wait for further improvement in the uncertainty of the hadronic corrections to aµ [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The lowest-order hadronic vacuum-polarization effect has thus far been determined from two sources, (i) e+ e− annihilation cross section, and (ii) hadronic τ decays. Several recent evaluations are listed in Table I. Their differences (except for the one obtained from the τ decay data) are due to different interpretations and treatments of basically identical data. However, they all agree that the measurement of the e+ e− annihilation cross section, in particular in the region below ρ − ω resonances, must be improved substantially in order to reduce the experimental uncertainty significantly. Such efforts are underway at several laboratories. Particularly interesting and promising is new radiative-return measurements [17]. On the other hand, it is not clear at present whether the value from the τ -decay data can be improved much further because of the difficulty in evaluating more precisely the effect of isospin breaking [6, 7]. A new theoretical development is an attempt to calculate the hadronic vacuumpolarization effect on muon g − 2 in lattice QCD [18]. The NLO hadronic contribution has been evaluated by two groups [8, 19]: aµ (had.NLO ) = −10.1 (0.6) × 10−10 , aµ (had.NLO ) = −9.8 (0.1)exp (0.0)rad × 10−10 . (3)
Nonnegatively and Positively curved manifolds
a rXiv:077.391v1[mat h.D G]2J u l27NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS BURKHARD WILKING The aim of this paper is to survey some results on nonnegatively and positively curved Riemannian manifolds.One of the important features of lower curvature bounds in general is the invariance under taking Gromov Hausdorfflimits.Cele-brated structure and finiteness results provide a partial understanding of the phe-nomena that occur while taking limits.These results however are not the subject of this survey since they are treated in other surveys of this volume.In this survey we take the more classical approach and focus on ”effective”re-sults.There are relatively few general ”effective”structure results in the subject.By Gromov’s Betti number theorem the total Betti number of a nonnegatively curved manifold is bounded above by an explicit constant which only depends on the dimension.The Gromoll Meyer theorem says that a positively curved open manifold is diffeomorphic to the Euclidean space.In the case of nonnegatively curved open manifolds,the soul theorem of Cheeger and Gromoll and Perelman’s solution of the soul conjecture clearly belong to the greatest structure results in the subject,as well.Also relatively good is the understanding of fundamental groups of nonnega-tively curved manifolds.A theorem of Synge asserts that an even dimensional orientable compact manifold of positive sectional curvature is simply connected.An odd dimensional positively curved manifold is known to be orientable (Synge),and its fundamental group is finite by the classical theorem of Bonnet and Myers.The fundamental groups of nonnegatively curved manifolds are virtually abelian,as a consequence of Toponogov’s splitting theorem.However,one of the ”effective”conjectures in this context,the so called Chern conjecture,was refuted:Shankar [1998]constructed a positively curved manifold with a non cyclic abelian funda-mental group.As we will discuss in the last section the known methods for constructing non-negatively curved manifolds are somewhat limited.The most important tools are the O’Neill formulas which imply that the base of a Riemannian submersion has nonnegative (positive)sectional curvature if the total space has.We recall that a smooth surjective map σ:M →B between two Riemannian manifolds is called a Riemannian submersion if the dual σad∗:T σ(p )B →T p M of the differential of σislength preserving for all p ∈M .Apart from taking products,the only other method is a special glueing technique,which was used by Cheeger,and more recently by Grove and Ziller to construct quite a few interesting examples of nonnegatively curved manifolds.By comparing with the class of known positively curved manifolds,the nonnega-tively curved manifolds form a huge class.In fact in dimensions above 24all known simply connected compact positively curved manifolds are diffeomorphic to rank 1symmetric spaces.Due to work of the author the situation is somewhat better in12BURKHARD WILKINGthe class of known examples of manifolds with positive curvature on open dense sets,see section4.Given the drastic difference in the number of known examples,it is somewhat painful that the only known obstructions on positively curved compact manifolds, which do not remain valid for the nonnegatively curved manifolds,are the above quoted results of Synge and Bonnet Myers on the fundamental groups.Since the list of general structure results is not far from being complete by now, the reader might ask why a survey on such a subject is necessary.The reason is that there are a lot of other beautiful theorems in the subject including structure results,but they usually need additional assumptions.We have subdivided the paper infive sections.Section1is on sphere theorems and related rigidity results,some notes on very recent significant developments were added in proof and can be found in section6.In section2,we survey results on compact nonnegatively curved manifolds,and in section3,results on open non-negatively curved manifolds.Then follows a section on compact positively curved manifolds with symmetry,since this was a particularly active area in recent years. Although we pose problems and conjectures throughout the paper we close the paper with a section on open problems.We do not have the ambition to be complete or to sketch all the significant historical developments that eventually led to the stated results.Instead we will usually only quote a few things according to personal taste.1.Sphere theorems and related rigidity results.A lot of techniques in the subject were developed or used in connection with proving sphere theorems.In this section we survey some of these results.We recall Toponogov’s triangle comparison theorem.Let M be a complete manifold with sectional curvature K≥κand consider a geodesic triangle∆in M consisting of minimal geodesics with length a,b,c∈R.Then there exists a triangle in the2-dimensional complete surface M2κof constant curvatureκwith side length a,b,c and the angles in the comparison triangle bound the corresponding angles in∆from below.1.1.Topological sphere theorems.We start with the classical sphere theorem of Berger and Klingenberg.Theorem1.1(Quarter pinched sphere theorem).Let M be a complete simply con-nected manifold with sectional curvature1/4<K≤1.Then M is homeomorphic to the sphere.The proof has two parts.Thefirst part is to show that the injectivity radius of M is at leastπ/2.This is elementary in even dimensions.In fact by Synge’s Theorem any even dimensional oriented manifold with curvature0<K≤1has injectivity radius≥π.In odd dimensions the result is due to Klingenberg and relies on a more delicate Morse theory argument on the loop space.The second part of the proof is due to Berger.He showed that any manifold with injectivity radius≥π/2and curvature>1is homeomorphic to a sphere.In fact by applying Toponogov’s theorem to two points of maximal distance,he showed that the manifold can be covered by two balls,which are via the exponential map diffeomorphic to balls in the Euclidean space.NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS3 Grove and Shiohama[1977]gave a significant improvement of Berger’s theorem, by replacing the lower injectivity radius bound by a lower diameter bound.Theorem1.2(Diameter sphere theorem).Any manifold with sectional curvature ≥1and diameter>π/2is homeomorphic to a sphere.More important than the theorem was the fact the proof introduced a new con-cept:critical points of distance functions.A point q is critical with respect to the distance function d(p,·)if the set of initial vectors of minimal geodesics from q to p intersect each closed half space of T q M.If the point q is not critical it is not hard to see that there is a gradient like vectorfield X in a neighborhood of q.A vectorfield is said to be gradient like if for each integral curve c of X the map t→d(p,c(t)) is a monotonously increasing bilipschitz map onto its image.An elementary yet important observation is that local gradient like vectorfields can be glued together using a partition of unity.Proof of the diameter sphere theorem.We may scale the manifold such that its di-ameter isπ/2and the curvature is strictly>1.Choose two points p,q of maximal distanceπ/2,and let z be an arbitrary third point.Consider the spherical compar-ison triangle(˜p,˜q,˜z).We do know that the side length of(˜p,˜z)and(˜q,˜z)are less or equal toπ/2whereas d S2(˜p,˜q)=π/2.This implies that the angle of the triangle at˜z is≥π/2.By Toponogov’s theorem any minimal geodesic triangle with corners p,q,z in M has an angle strictly larger thanπ/2based at z.This in turn implies that the distance function d(p,·)has no critical points in M\{p,q}.Thus there is a gradient like vectorfield X on M\{p,q}.Furthermore without loss of generality Xis given on B r(p)\{p}by the actual gradient of the distance function d(p,·),where r is smaller than the injectivity radius.We may also assume X(z) ≤d(q,z)2for all z∈M\{p,q}.Then theflowΦof X exists for all future times and we can define a diffeomorphismψ:T p M→M\{q}as follows:for a unit vector v∈T p M and a nonnegative number t putψ(t·v)= exp(tv)if t∈[0,r]andψ(t·v)=Φt−r(exp(rv))if t≥r.Clearly this implies that M is homeomorphic to a sphere.There is another generalization of the sphere theorem of Berger and Klingenberg.A manifold is said to have positive isotropic curvature if for all orthonormal vectors e1,e2,e3,e4∈T p M the curvature operator satisfiesR(e1∧e2+e3∧e4,e1∧e2+e3∧e4)+R(e1∧e3+e4∧e2,e1∧e3+e4∧e2)>0 By estimating the indices of minimal2spheres in a manifold of positive isotropic curvature,Micallef and Moore[1988]were able to show thatTheorem1.3.A simply connected compact Riemannian manifold of positive isotropic curvature is a homotopy sphere.A simple computation shows that pointwise strictly quarter pinched manifolds have positive isotropic curvature.Thus the theorem of Micallef and Moore is a generalization of the quarter pinched sphere theorem.A more direct improvement of the quarter pinched sphere theorem is due to Abresch and Meyer[1996]. Theorem1.4.Let M be a compact simply connected manifold with sectional cur-vature14BURKHARD WILKING•M is homeomorphic to a sphere.•n is even and the cohomology ring H∗(M,Z2)is generated by one element.It is a well known result in topology that the Z2cohomology rings of spaces which are generated by one element are precisely given by the Z2-cohomology rings of rank1symmetric spaces RP n,CP n,HP n,Ca P2and S n,cf.[Zhizhou,2002].The proof of Theorem1.4has again two parts.Abresch and Meyerfirst establish that the injectivity radius of M is bounded below by the conjugate radius which in turn is bounded below byπ.From the diameter sphere theorem it is clear that without loss of generality diam(M,g)≤π(1+10−6).They then establish the horse shoe inequality,which was conjectured by Berger:for p∈M and any unit vector v∈T p M one hasd(exp(πv),exp(−πv))<π.In particular exp(πv)and exp(−πv)can be joined by a unique minimal geodesic. Once the horse shoe inequality is established it is easy to see that there is a smooth map f:RP n→M n such that in odd dimensions the integral degree is1and in even dimensions the Z2-degree is1.The theorem then follows by a straightforward cohomology computation.The horse shoe inequality relies on a mixed Jacobifield estimate.We only state the problem here in a very rough form.Let c be a normal geodesic in M and J a Jacobifield with J(0)=0.Suppose that at time t0=2πNONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS5 The work of Weiss[1993]goes in a different direction.He uses the fact that a quarter pinched sphere M n has Morse perfection n.A topological sphere M n is said that to have Morse perfection≥k if there is a smooth mapΨ:S k→C∞(M,R) satisfyingΨ(−p)=−Ψ(p),and for each p∈S k the functionΨ(p)is a Morse function with precisely two critical points.It is not hard to see that a quarter pinched sphere has Morse perfection n.Weiss used this to rule out quite a few of the exotic spheres by showing that their Morse perfection is<n.He showed that in dimensions n=4m−1any exotic sphere bounding a parallelizable manifold has odd order in the group of exotic spheres unless the Morse perfection≤n−2.By Hitchin there are also exotic spheres with a non-vanishingα-invariant,and thus these spheres do not even admit metrics with positive scalar curvature,see the survey of Jonathan Rosenberg.Similar to the quarter pinched sphere theorem,one can also strengthen the assumptions in the diameter sphere theorem in order to get a differentiable sphere theorem.This was carried out by Grove and Wilhelm[1997].Theorem1.6.Let M be an n-manifold with sectional curvature≥1containing (n−2)-points with pairwise distance>π/2.Then M is diffeomorphic to a sphere.If one has only k points with pairwise distance>π/2,then Grove and Wilhelm obtain restrictions on the differentiable structure of M.With a slight variation of the proof of Grove and Wilhelm one can actually get a slightly better result.Let M be an inner metric space.We say that M has a weak 2-nd packing radius≥r if diam(M)≥r.We say it has a weak k-th packing radius ≥r if there is a point p∈M such that∂B r(p)is connected and endowed with its inner metric has weak(k−1)-th packing radius≥r.Theorem1.7.Let(M,g)be an n-manifold with sectional curvature≥1and weak (k+1)-th packing radius>π/2.Then there is a family of metrics g t(t∈[0,1) with sectional curvature≥1and g0=g such that(M,g t)converges for t→1to an n-dimensional Alexandrov space A satisfying:If k≥n,then A is isometric to the standard sphere.If k<n,then A is given by the k-th iterated suspensionΣk A′of an n−k-dimensional Alexandrov space A′.Corollary1.8.Letε>0.A manifold with sectional curvature≥1and diameter >π/2also admits a metric with sectional curvature≥1and diameter>π−ε.As in the paper of Grove and Wilhelm,one can show in the situation of Theo-rem1.7that there is a sequence of positively metrics˜g i on the standard sphere with curvature≥1such that(S n,˜g i)i∈N converges to A as well.In particular,Grove and Wilhelm showed that an affirmative answer to the following question would imply the differentiable diameter sphere theorem.Question1.9(Smooth stability conjecture).Suppose a sequence of compact n-manifolds(M k,g k)with curvature≥−1converges in the Gromov Hausdorfftopol-ogy to an n-dimensional compact Alexandrov space A.Does this imply that for alllarge k1and k2the manifolds M k1and M k2are diffeomorphic?By Perelman’s stability theorem it is known that M k1and M k2are homeomor-phic for all large k1and k2,see the article of Vitali Kapovitch in this volume. Sketch of the proof of Theorem1.7.Let p,q∈M be points such that d(p,q)>π/2+εfor someε>0.We claim that we canfind a continuous family of metrics6BURKHARD WILKINGwith g0=g and K t≥1such that(M,g t)converges for t→1to the suspension of ∂Bπ/2(p).We consider the suspension X of M,i.e.,X=[−π/2,π/2]×M/∼where the equivalence classes of∼are given by p+:={π/2}×M,p−:={−π/2}×M and the one point sets{(t,p)}for|t|=π/2.Recall that X endowed with the usual warped product metric is an Alexandrov space with curvature≥1.We consider the curve c(t)=((1−t)π/2,p)as a curve in X,r(t):=π/2+ε(1−t) and the ball B r(t)(c(t))⊂X.Put N t:=∂B r(t)(c(t)).Since X\B r(t)(c(t))is strictly convex and N t is contained in the Riemannian manifold X\{p±}for all t=1,it follows that N t is an Alexandrov space with curvature≥1for all t∈[0,1].Clearly N0is up to a small scaling factor isometric to M.Moreover N1is isometric to the suspension of∂Bπ/2(p)⊂M.Using that N t is strictly convex in the Riemannian manifold X\{p±}for t∈[0,1),it follows that the family N t can be approximated by a family of strictly convex smooth submanifolds˜N t⊂X\{p±},t∈[0,1).Furthermore,one may assume that lim t→1N t=N1=lim t→1˜N t.We found a family of metrics g t of curvature>1such that(M,g t)converges to the suspension of∂Bπ/2(p).We may assume that∂Bπ/2(p)has weak k-th packing radius>π/2and k≥2.We now choose a curve of points q t∈M converging for t→1to a point on the equator q1∈∂Bπ/2(p)of the limit space such that there is a point q2in∂Bπ/2(p) whose intrinsic distance to q1is>π/2.We now repeat the above construction for all t∈(0,1)with(M,g,p)replaced by (M,g t,q t).This way we get for each t an one parameter family of smooth metrics g(t,s)with K≥1which converges for s→1to the suspension of the boundary of Bπ/2(q t)⊂(M,g t).It is then easy to see that one can choose the metrics such that they depend smoothly on s and t.Moreover,after a possible reparameterization of g(s,t)the one parameter family t→g(t,t)converges to the double suspension of the boundary of Bπ/2(q1)⊂∂Bπ/2(p).Clearly the theorem follows by iterating this process.We recall that to each Riemannian manifold(M,g)and each point p∈M one can assign a curvature operator R:Λ2T p M→Λ2T p M.We call the operator2-positive if the sum of the smallest two eigenvalues is positive.It is known that manifolds with2-positive curvature operator have positive isotropic curvature. Theorem1.10.Let(M,g)be a compact manifold with2-positive curvature opera-tor.Then the normalized Ricciflow evolves g to a limit metric of constant sectional curvature.In dimension3the theorem is due to Hamilton[1982].Hamilton[1986]also showed that the theorem holds for4-manifolds with positive curvature operator. This was extended by Chen to4-manifolds with2-positive curvature operator.In dimension2it was shown by Hamilton and Chow that for any surface the normalized Ricciflow converges to limit metric of constant curvature.In dimensions above4 the theorem is due to B¨o hm and Wilking[2006].For n≥3the proof solely relies on the maximum principle and works more generally in the category of orbifolds.We recall that a family of metrics g t on M is said to be a solution of the Ricci flow if∂NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS7 Hamilton showed that if one represents the curvature operator R with respect to suitable moving orthonormal frames,then∂R=R2+R#.dtSketch of the proof of Theorem1.10.We let S2B so(n) denote the vectorspace of algebraic curvature operators satisfying the Bianchi identity.We call a continuous family C(s)s∈[0,1)⊂S2B(so(n))of closed convex O(n)-invariant cones of full dimension a pinching family,if(1)each R∈C(s)\{0}has positive scalar curvature,(2)R2+R#is contained in the interior of the tangent cone of C(s)at R forall R∈C(s)\{0}and all s∈(0,1),(3)C(s)converges in the pointed Hausdorfftopology to the one-dimensionalcone R+I as s→1.The argument in[B¨o hm and Wilking,2006]has two parts.One part is a general argument showing for any pinching family C(s)(s∈[0,1))that on any compact manifold(M,g)for which the curvature operator is contained in the interior of C(0)at every point the normalized Ricciflow evolves g to a constant curvature limit metric.In the proof of this result onefirst constructs to such a pinching family a pinching set in the sense Hamilton which in turn gives the convergence result.The harder problem is actually to construct a pinching family with C(0)being the cone of2-nonnegative curvature operators.Here a new tool is established.It is a formula that describes how this ordinary differential equation R′=R2+R# changes under O(n)-equivariant linear transformations.By chance the transforma-tion law is a lot simpler than for a generic O(n)-invariant quadratic expression.The transformation law often allows to construct new ODE-invariant curvature cones as the image of a given invariant curvature cone under suitable equivariant linear transformation l:S2B so(n) →S2B so(n) .This in turn is used to establish the existence of a pinching family.1.3.Related rigidity results.Wefirst mention the diameter rigidity theorem of Gromoll and Grove[1987]Theorem1.11(Diameter rigidity).Let(M,g)be a compact manifold with sec-tional curvature K≥1and diameter≥π/2.Then one of the following holds:a)M is homeomorphic to a sphere.b)M is locally isometric to a rank one symmetric space.The original theorem allowed a potential exceptional case•M has the cohomology ring of the Cayley plane,but is not isometric to the Cayley plane.8BURKHARD WILKINGThis case was ruled out much later by the author,see[Wilking,2001].The proof of the diameter rigidity theorem is closely linked to the rigidity of Hopffibrations which was established by Gromoll and Grove[1988]as well Theorem1.12(Rigidity of Hopffibrations).Letσ:S n→B be a Riemannian sub-mersion with connectedfibers.Thenσis metrically congruent to a Hopffibration. In particular thefibers are totally geodesic and B is rank one symmetric space.Similarly to the previous theorem,the original theorem allowed for a possible exception,Grove and Gromoll assumed in addition(n,dim B)=(15,8).Using very different methods,the rigidity of this special case was proved by the author in [Wilking,2001].This in turn ruled out the exceptional case in the diameter rigidity theorem as well.Sketch of the proof of the diameter rigidity theorem.The proof of the diameter rigid-ity theorem is the most beautiful rigidity argument in positive curvature.One as-sumes that the manifold is not homeomorphic to a sphere.Let p be a point with N2:=∂Bπ/2(p)=∅.One defines N1=∂Bπ/2(N2)as the boundary of the distance tube Bπ/2(N2)around N2.It then requires some work to see that N1and N2are totally geodesic submanifolds without boundary satisfying N2=∂Bπ/2(N1).Not both manifolds can be points,since otherwise one can show that M is homeomorphic to a sphere.If one endows the unit normal bundleν1(N i)with its natural connection metric,then Grove and Gromoll show in a next step that the mapσi:ν1(N i)→N j,v→exp(π/2v)is a Riemannian submersion,{i,j}={1,2}. Furthermoreσi restricts to a Riemannian submersionν1q(N i)→N j for all q∈N i.In the simply connected case one shows that N i is simply connected as well,i= 1,2.By the rigidity of submersions defined on Euclidean spheres(Theorem1.12) we deduce that N i is either a point or a rank one symmetric space with diameter π/2.Going back to the definition of N1,it is then easy to see that N1={p}.Using thatσ1:S n−1→N2is submersion with totally geodesicfibers,one can show that the pull back metric exp∗p g on Bπ/2(0)⊂T p M is determined byσ1.Thus M is isometric to a rank one symmetric space.In the non simply connected case one can show that either the universal cover is not a sphere and thereby symmetric or dim(N1)+dim(N2)=n−1.In the latter case it is not hard to verify that M has constant curvature one.Since the proof of the differentiable sphere theorem for manifolds with2-positive curvature follows from a Ricciflow argument it is of course not surprising that it has a rigidity version as well.Theorem1.13.A simply connected compact manifold with2-nonnegative curva-ture operator satisfies one of the following statements.•The normalized Ricciflow evolves the metric to a limit metric which is upto scaling is isometric to S n or CP n/2.•M is isometric to an irreducible symmetric space.•M is isometric to nontrivial Riemannian product.Of course in the last case the factors of M have nonnegative curvature operators. By Theorem2.2(M,g)admits a possibly different metric g1such that(M,g1)is locally isometric to(M,g)and(M,g1)isfinitely covered by a Riemannian product T d×M′where M′is simply connected and compact.This effectively gives a reduction to the simply connected case.NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS9 The theorem has many names attached to it.Of course Theorem1.10(Hamilton [1982,1986],B¨o hm and Wilking[2006])enters as the’generic’case.This in turn was used by Ni and Wu[2006]to reduce the problem from2-nonnegative curvature operators to nonnegative curvature operators.One has to mention Gallot and Meyer’s[1975]investigation of manifolds with nonnegative curvature operator using the Bochner technique.Berger’s classification of holonomy groups,as well as Mori’s [1979],Siu and Yau’s[1980]solution of the Frankel conjecture are key tools.Based on this Chen and Tian[2006]proved convergence of the Ricciflow for compact K¨a hler manifolds with positive bisectional curvature.Sketch of a proof of Theorem1.13.Considerfirst the case that the curvature op-erator of M is not nonnegative.We claim that then the Ricciflow immediately evolves g to a metric with2-positive curvature operator.We consider a short time solution g(t)of the Ricciflow and let f:[0,ε)×M→R, denote the function which assigns to(t,p)the sum of the lowest two eigenvalues of the curvature operator of(M,g(t))at p.Wefirst want to show that f(t,·)is positive somewhere for small t>0.We may assume that f(0,p)=0for all p.It is straightforward to check that f satisfies∂f(λ1+λ2) R+t(R2+R#) .∂t|t=0+From the invariance of2-nonnegative curvature operators it is known that q(R)≥0. In fact a detailed analysis of the proof shows that q(R)≥2(λ1(R))2.In the present situation we deduce by afirst order argument that f(t,p)becomes positive somewhere for small t>0.Now it is not hard to establish a strong maximum principle that shows that f(t,·)is everywhere positive for small t>0,see Ni and Wu[2006].In other words(M,g t)has2-positive curvature operator for t>0and the result follows from Theorem1.10.We are left with the case that the curvature operator of(M,g)is nonnegative. Essentially this case was already treated by Gallot and Meyer using the Bochner technique,see[Petersen,2006].We present a slightly different argument following Chow and Yang(1989).Using Hamilton’s[1986]strong maximum principle one deduces that for t>0the curvature operator of(M,g t)has constant rank and that the kernel is parallel.Thus either R t is positive or the holonomy is non generic.We may assume that M does not split as a product.Hence without loss of generality M is irreducible with non generic holonomy.Since(M,g t)clearly has positive scalar curvature Berger’s classification of holonomy groups implies that Hol(M)∼=U(n/2),Sp(1)Sp(n/4)unless(M,g)is a symmetric space.In the case of Hol(M)∼=Sp(1)Sp(n)we can employ another theorem of Berger[1966]to see that M is up to scaling isometric to HP n/4,since in our case the sectional curvature of (M,g t)is positive.In the remaining case Hol(M)=U(n/2)it follows that M is K¨a hler and(M,g t)has positive(bi-)sectional curvature.By Mori[1979]and Siu and Yau’s[1980]solution of the Frankel conjecture M is biholomorphic to CP n/2. In particular,M admits a K¨a hler Einstein metric.Due to work of Chen and Tian [2006]it follows,that the normalized Ricciflow on M converges to the Fubini study metric which completes the proof.pact nonnegatively curved manifoldsThe most fundamental obstruction to this date is Gromov’s Betti number theo-rem.10BURKHARD WILKINGTheorem2.1(Gromov,1981).Let M n be an n-dimensional complete manifold with nonnegative sectional curvature,and let F be afield.Then the total Betti number satisfiesb(M,F):=ni=0b i(M,F)≤1010n4.Gromov’s original bound on the total Betti number was depending double expo-nentially on the dimension.The improvement is due to Abresch[1987].However, this bound is not optimal either.In fact Gromov posed the problem whether the best possible bound is2n,the total Betti number of the n-dimensional torus.The statement is particularly striking since the nonnegatively curved manifolds in a fixed dimension≥7have infinitely many homology types with respect to integer coefficients.More generally Gromov gave explicit estimates for the total Betti num-bers of compact n-manifolds with curvature≥−1and diameter≤D.The proof is an ingenious combination of Toponogov’s theorem and critical point theory. Sketch of the proof of Theorem2.1.The most surprising part in the proof is a def-inition:Gromov assigns to every ball B r(p)⊂M afinite number called the corank of the ball.It is defined as the maximum over all k such that for all q∈B2r(p) there are points q1,...,q k withd(q,q1)≥2n+3r,d(q,q i+1)≥2n d(p,q i)and q i is a critical point of the distance function of q in the sense of Grove and Shiohama.One can show as follows that the corank of a ball is at most2n:Choose a minimal geodesic c ij from q i to q j,i<j and minimal geodesic c i from q to q i, i=1,...,k.Since q i is a critical point we canfind a possibly different minimal geodesic˜c i from q to q i such that the angle of the triangle(˜c i,c j,c ij)based at q i is ≤π/2.Therefore L(c j)2≤L(c ij)2+L(c i)2.Applying Toponogov’s theorem to the triangle(c i,c j,c ij)gives that the angleϕij between c i and c j satisfies tan(ϕij)≥2n. Thusϕij≥π/2−2−n.The upper bound on k now follows from an Euclidean sphere packing argument in T q M.By reverse induction on the corank,one establishes an estimate for the content of a ball cont(B r(p))which is defined as the dimension of the image of H∗(B r(p)) in H∗(B5r(p)).A ball B r(p)with maximal corank is necessarily contractible in B5r(p)since for some q∈B2r(p)the distance function of q has no critical points in B8r(q)\{q}.This establishes the induction base.It is immediate from the definition that corank(Bρ(q))≥corank(B r(p))for all q∈B3r/2(p)and allρ≤r/4. In the induction step one distinguishes between two cases.In thefirst case,one assumes that corank(Bρ(q))>corank(B r(p))for all q∈B r(p)andρ:=r4n .Thus for some point x∈B2ρ(q)there is no criticalpoint of the distance function of x in B8r(x)\B2−n+3r(x).This implies that one can homotop B r(p)to a subset of B r/4(x)in B5r(p).From this it is not hard to deduce that cont(B r/4(x))≥cont(B r(p)).We have seen above corank(B r/4(x))≥corank(B r(p)).One can now apply the same argument again with B r(p)replaced。
IATA AHM目录
Airport Handling ManualEffective 1 January—31 December 201838NOTICEDISCLAIMER. The information contained in thispublication is subject to constant review in the lightof changing government requirements and regula-tions. No subscriber or other reader should act onthe basis of any such information without referringto applicable laws and regulations and/or withouttak ing appropriate professional advice. Althoughevery effort has been made to ensure accuracy, theInternational Air Transport Association shall not beheld responsible for any loss or damage caused byerrors, omissions, misprints or misinterpretation ofthe contents hereof. 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No part of this publication maybe reproduced, recast, reformatted or trans-mitted in any form by any means, electronic ormechanical, including photocopying, record-ing or any information storage and retrieval sys-tem, without the prior written permission from:Senior Vice PresidentAirport, Passenger, Cargo and SecurityInternational Air Transport Association800 Place VictoriaP.O. Box 113Montreal, QuebecCANADA H4Z 1M1Airport Handling ManualMaterial No.: 9343-38ISBN 978-92-9229-505-9© 2017 International Air Transport Association. All rights reserved.TABLE OF CONTENTSPage Preface (xv)Introduction (xvii)General (1)AHM001Chapter0—Record of Revisions (1)AHM011Standard Classification and Numbering for Members Airport Handling Manuals (2)AHM012Office Function Designators for Airport Passenger and Baggage Handling (30)AHM020Guidelines for the Establishment of Airline Operators Committees (31)AHM021Guidelines for Establishing Aircraft Ground Times (34)AHM050Aircraft Emergency Procedures (35)AHM070E-Invoicing Standards (53)Chapter1—PASSENGER HANDLING (91)AHM100Chapter1—Record of Revisions (91)AHM110Involuntary Change of Carrier,Routing,Class or Type of Fare (92)AHM112Denied Boarding Compensation (98)AHM120Inadmissible Passengers and Deportees (99)AHM140Items Removed from a Passenger's Possession by Security Personnel (101)AHM141Hold Loading of Duty-Free Goods (102)AHM170Dangerous Goods in Passenger Baggage (103)AHM176Recommendations for the Handling of Passengers with Reduced Mobility(PRM) (105)AHM176A Acceptance and Carriage of Passengers with Reduced Mobility(PRM) (106)AHM180Carriage of Passengers with Communicable Diseases (114)AHM181General Guidelines for Passenger Agents in Case of SuspectedCommunicable Disease (115)Chapter2—BAGGAGE HANDLING (117)AHM200Chapter2—Record of Revisions (117)AHM210Local Baggage Committees (118)AHM211Airport Operating Rules (124)Airport Handling ManualPageChapter2—BAGGAGE HANDLING(continued)AHM212Interline Connecting Time Intervals—Passenger and Checked Baggage (126)AHM213Form of Interline Baggage Tags (128)AHM214Use of the10Digit Licence Plate (135)AHM215Found and Unclaimed Checked Baggage (136)AHM216On-Hand Baggage Summary Tag (138)AHM217Forwarding Mishandled Baggage (139)AHM218Dangerous Goods in Passengers'Baggage (141)AHM219Acceptance of Firearms and Other Weapons and Small Calibre Ammunition (142)AHM221Acceptance of Power Driven Wheelchairs or Other Battery Powered Mobility Aidsas Checked Baggage (143)AHM222Passenger/Baggage Reconciliation Procedures (144)AHM223Licence Plate Fallback Sortation Tags (151)AHM224Baggage Taken in Error (154)AHM225Baggage Irregularity Report (156)AHM226Tracing Unchecked Baggage and Handling Damage to Checked and UncheckedBaggage (159)AHM230Baggage Theft and Pilferage Prevention (161)AHM231Carriage of Carry-On Baggage (164)AHM232Handling of Security Removed Items (168)AHM240Baggage Codes for Identifying ULD Contents and/or Bulk-Loaded Baggage (169)Chapter3—CARGO/MAIL HANDLING (171)AHM300Chapter3—Record of Revisions (171)AHM310Preparation for Loading of Cargo (172)AHM311Securing of Load (174)AHM312Collection Sacks and Bags (177)AHM320Handling of Damaged Cargo (178)AHM321Handling of Pilfered Cargo (179)AHM322Handling Wet Cargo (180)AHM330Handling Perishable Cargo (182)AHM331Handling and Protection of Valuable Cargo (184)AHM332Handling and Stowage of Live Animals (188)AHM333Handling of Human Remains (190)Table of ContentsPageChapter3—CARGO/MAIL HANDLING(continued)AHM340Acceptance Standards for the Interchange of Transferred Unit Load Devices (191)AHM345Handling of Battery Operated Wheelchairs/Mobility AIDS as Checked Baggage (197)AHM350Mail Handling (199)AHM351Mail Documents (203)AHM353Handling of Found Mail (218)AHM354Handling of Damaged Mail (219)AHM355Mail Security (220)AHM356Mail Safety (221)AHM357Mail Irregularity Message (222)AHM360Company Mail (224)AHM380Aircraft Documents Stowage (225)AHM381Special Load—Notification to Captain(General) (226)AHM382Special Load—Notification to Captain(EDP Format and NOTOC Service) (231)AHM383Special Load—Notification to Captain(EDP NOTOC Summary) (243)AHM384NOTOC Message(NTM) (246)Chapter4—AIRCRAFT HANDLING AND LOADING (251)AHM400Chapter4—Record of Revisions (251)AHM411Provision and Carriage of Loading Accessories (252)AHM420Tagging of Unit Load Devices (253)AHM421Storage of Unit Load Devices (263)AHM422Control of Transferred Unit Load Devices (268)AHM423Unit Load Device Stock Check Message (273)AHM424Unit Load Device Control Message (275)AHM425Continued Airworthiness of Unit Load Devices (279)AHM426ULD Buildup and Breakdown (283)AHM427ULD Transportation (292)AHM430Operating of Aircraft Doors (295)AHM431Aircraft Ground Stability—Tipping (296)AHM440Potable Water Servicing (297)AHM441Aircraft Toilet Servicing (309)Airport Handling ManualPageChapter4—AIRCRAFT HANDLING AND LOADING(continued)AHM450Standardisation of Gravity Forces against which Load must be Restrained (310)AHM451Technical Malfunctions Limiting Load on Aircraft (311)AHM453Handling/Bulk Loading of Heavy Items (312)AHM454Handling and Loading of Big Overhang Items (313)AHM455Non CLS Restrained ULD (316)AHM460Guidelines for Turnround Plan (323)AHM462Safe Operating Practices in Aircraft Handling (324)AHM463Safety Considerations for Aircraft Movement Operations (337)AHM465Foreign Object Damage(FOD)Prevention Program (340)Chapter5—LOAD CONTROL (343)AHM500Chapter5—Record of Revisions (343)AHM501Terms and Definitions (345)AHM503Recommended Requirements for a New Departure Control System (351)AHM504Departure Control System Evaluation Checklist (356)AHM505Designation of Aircraft Holds,Compartments,Bays and Cabin (362)AHM510Handling/Load Information Codes to be Used on Traffic Documents and Messages (368)AHM513Aircraft Structural Loading Limitations (377)AHM514EDP Loading Instruction/Report (388)AHM515Manual Loading Instruction/Report (404)AHM516Manual Loadsheet (416)AHM517EDP Loadsheet (430)AHM518ACARS Transmitted Loadsheet (439)AHM519Balance Calculation Methods (446)AHM520Aircraft Equipped with a CG Targeting System (451)AHM530Weights for Passengers and Baggage (452)AHM531Procedure for Establishing Standard Weights for Passengers and Baggage (453)AHM533Passengers Occupying Crew Seats (459)AHM534Weight Control of Load (460)AHM536Equipment in Compartments Procedure (461)AHM537Ballast (466)Table of ContentsPageChapter5—LOAD CONTROL(continued)AHM540Aircraft Unit Load Device—Weight and Balance Control (467)AHM550Pilot in Command's Approval of the Loadsheet (468)AHM551Last Minute Changes on Loadsheet (469)AHM561Departure Control System,Carrier's Approval Procedures (471)AHM562Semi-Permanent Data Exchange Message(DEM) (473)AHM564Migration from AHM560to AHM565 (480)AHM565EDP Semi-Permanent Data Exchange for New Generation Departure Control Systems (500)AHM570Automated Information Exchange between Check-in and Load Control Systems (602)AHM571Passenger and Baggage Details for Weight and Balance Report(PWR) (608)AHM580Unit Load Device/Bulk Load Weight Statement (613)AHM581Unit Load Device/Bulk Load Weight Signal (615)AHM583Loadmessage (619)AHM587Container/Pallet Distribution Message (623)AHM588Statistical Load Summary (628)AHM590Load Control Procedures and Loading Supervision Responsibilities (631)AHM591Weight and Balance Load Control and Loading Supervision Training and Qualifications (635)Chapter6—MANAGEMENT AND SAFETY (641)AHM600Chapter6—Record of Revisions (641)AHM610Guidelines for a Safety Management System (642)AHM611Airside Personnel:Responsibilities,Training and Qualifications (657)AHM612Airside Performance Evaluation Program (664)AHM615Quality Management System (683)AHM616Human Factors Program (715)AHM619Guidelines for Producing Emergency Response Plan(s) (731)AHM620Guidelines for an Emergency Management System (733)AHM621Security Management (736)AHM633Guidelines for the Handling of Emergencies Requiring the Evacuation of an Aircraft During Ground Handling (743)AHM650Ramp Incident/Accident Reporting (745)AHM652Recommendations for Airside Safety Investigations (750)AHM660Carrier Guidelines for Calculating Aircraft Ground Accident Costs (759)Airport Handling ManualChapter7—AIRCRAFT MOVEMENT CONTROL (761)AHM700Chapter7—Record of Revisions (761)AHM710Standards for Message Formats (762)AHM711Standards for Message Corrections (764)AHM730Codes to be Used in Aircraft Movement and Diversion Messages (765)AHM731Enhanced Reporting on ATFM Delays by the Use of Sub Codes (771)AHM780Aircraft Movement Message (774)AHM781Aircraft Diversion Message (786)AHM782Fuel Monitoring Message (790)AHM783Request Information Message (795)AHM784Gate Message (797)AHM785Aircraft Initiated Movement Message(MVA) (802)AHM790Operational Aircraft Registration(OAR)Message (807)Chapter8—GROUND HANDLING AGREEMENTS (811)AHM800Chapter8—Record of Revisions (811)AHM801Introduction to and Comments on IATA Standard Ground Handling Agreement(SGHA) (812)AHM803Service Level Agreement Example (817)AHM810IATA Standard Ground Handling Agreement (828)AHM811Yellow Pages (871)AHM813Truck Handling (872)AHM815Standard Transportation Documents Service Main Agreement (873)AHM817Standard Training Agreement (887)AHM830Ground Handling Charge Note (891)AHM840Model Agreement for Electronic Data Interchange(EDI) (894)Chapter9—AIRPORT HANDLING GROUND SUPPORT EQUIPMENT SPECIFICATIONS (911)AHM900Chapter9—Record of Revisions (911)AHM901Functional Specifications (914)AHM904Aircraft Servicing Points and System Requirements (915)AIRBUS A300B2320-/B4/C4 (917)A300F4-600/-600C4 (920)A310–200/200C/300 (926)A318 (930)A319 (933)Table of ContentsPageChapter9—AIRPORT HANDLING GROUND SUPPORT EQUIPMENT SPECIFICATIONS(continued) AHM904Aircraft Doors,Servicing Points and System Requirements for the Use of Ground Support Equipment(continued)A320 (936)A321 (940)A330-200F (943)A330-300 (948)A340-200 (951)A340-300 (955)A340-500 (959)A340-600 (962)Airbus350900passenger (965)AIRBUS A380-800/-800F (996)ATR42100/200 (999)ATR72 (1000)AVRO RJ70 (1001)AVRO RJ85 (1002)AVRO RJ100 (1003)B727-200 (1004)B737–200/200C (1008)B737-300,400,-500 (1010)B737-400 (1013)B737-500 (1015)B737-600,-700,-700C (1017)B737-700 (1020)B737-800 (1022)B737-900 (1026)B747–100SF/200C/200F (1028)B747–400/400C (1030)B757–200 (1038)B757–300 (1040)Airport Handling ManualPageChapter9—AIRPORT HANDLING GROUND SUPPORT EQUIPMENT SPECIFICATIONS(continued) AHM904Aircraft Doors,Servicing Points and System Requirements for the Use of Ground Support Equipment(continued)B767—200/200ER (1041)B767—300/300ER (1044)B767—400ER (1048)B777–200/200LR (1051)B777–300/300ER (1055)Boeing787800passenger (1059)BAe ATP(J61) (1067)Bombardier CS100 (1068)Bombardier CS300 (1072)CL-65(CRJ100/200) (1076)DC8–40/50F SERIES (1077)DC8–61/61F (1079)DC8–62/62F (1081)DC8–63/63F (1083)DC9–15/21 (1085)DC9–32 (1086)DC9–41 (1087)DC9–51 (1088)DC10–10/10CF (1089)DC10–30/40,30/40CF (1091)EMBRAER EMB-135Regional Models (1092)EMBRAER EMB-145Regional Models (1094)Embraer170 (1096)Embraer175 (1098)Embraer190 (1100)Embraer195 (1102)FOKKER50(F27Mk050) (1104)FOKKER50(F27Mk0502) (1106)Chapter9—AIRPORT HANDLING GROUND SUPPORT EQUIPMENT SPECIFICATIONS(continued) AHM904Aircraft Doors,Servicing Points and System Requirements for the Use of Ground Support Equipment(continued)FOKKER70(F28Mk0070) (1108)FOKKER100(F28Mk0100) (1110)FOKKER100(F28Mk0100) (1112)IL-76T (1114)MD-11 (1116)MD–80SERIES (1118)SAAB2000 (1119)SAAB SF-340 (1120)TU-204 (1122)AHM905Reference Material for Civil Aircraft Ground Support Equipment (1125)AHM905A Cross Reference of IATA Documents with SAE,CEN,and ISO (1129)AHM909Summary of Unit Load Device Capacity and Dimensions (1131)AHM910Basic Requirements for Aircraft Ground Support Equipment (1132)AHM911Ground Support Equipment Requirements for Compatibility with Aircraft Unit Load Devices (1136)AHM912Standard Forklift Pockets Dimensions and Characteristics for Forkliftable General Support Equipment (1138)AHM913Basic Safety Requirements for Aircraft Ground Support Equipment (1140)AHM914Compatibility of Ground Support Equipment with Aircraft Types (1145)AHM915Standard Controls (1147)AHM916Basic Requirements for Towing Vehicle Interface(HITCH) (1161)AHM917Basic Minimum Preventive Maintenance Program/Schedule (1162)AHM920Functional Specification for Self-Propelled Telescopic Passenger Stairs (1164)AHM920A Functional Specification for Towed Passenger Stairs (1167)AHM921Functional Specification for Boarding/De-Boarding Vehicle for Passengers withReduced Mobility(PRM) (1169)AHM922Basic Requirements for Passenger Boarding Bridge Aircraft Interface (1174)AHM923Functional Specification for Elevating Passenger Transfer Vehicle (1180)AHM924Functional Specification for Heavy Item Lift Platform (1183)AHM925Functional Specification for a Self-Propelled Conveyor-Belt Loader (1184)AHM925A Functional Specification for a Self-Propelled Ground Based in-Plane LoadingSystem for Bulk Cargo (1187)Chapter9—AIRPORT HANDLING GROUND SUPPORT EQUIPMENT SPECIFICATIONS(continued) AHM925B Functional Specification for a Towed Conveyor-Belt Loader (1190)AHM926Functional Specification for Upper Deck Catering Vehicle (1193)AHM927Functional Specification for Main Deck Catering Vehicle (1197)AHM930Functional Specification for an Upper Deck Container/Pallet Loader (1201)AHM931Functional Specification for Lower Deck Container/Pallet Loader (1203)AHM932Functional Specification for a Main Deck Container/Pallet Loader (1206)AHM933Functional Specification of a Powered Extension Platform to Lower Deck/Container/ Pallet Loader (1209)AHM934Functional Specification for a Narrow Body Lower Deck Single Platform Loader (1211)AHM934A Functional Specification for a Single Platform Slave Loader Bed for Lower DeckLoading Operations (1213)AHM936Functional Specification for a Container Loader Transporter (1215)AHM938Functional Specification for a Large Capacity Freighter and Combi Aircraft TailStanchion (1218)AHM939Functional Specification for a Transfer Platform Lift (1220)AHM941Functional Specification for Equipment Used for Establishing the Weight of aULD/BULK Load (1222)AHM942Functional Specification for Storage Equipment Used for Unit Load Devices (1224)AHM950Functional Specification for an Airport Passenger Bus (1225)AHM951Functional Specification for a Crew Transportation Vehicle (1227)AHM953Functional Specifications for a Valuable Cargo Vehicle (1229)AHM954Functional Specification for an Aircraft Washing Machine (1230)AHM955Functional Specification for an Aircraft Nose Gear Towbar Tractor (1232)AHM956Functional Specification for Main Gear Towbarless Tractor (1235)AHM957Functional Specification for Nose Gear Towbarless Tractor (1237)AHM958Functional Specification for an Aircraft Towbar (1240)AHM960Functional Specification for Unit Load Device Transport Vehicle (1242)AHM961Functional Specification for a Roller System for Unit Load Device Transportation on Trucks (1245)AHM962Functional Specification for a Rollerised Platform for the Transportation of Twenty Foot Unit Load Devices that Interfaces with Trucks Equipped to Accept Freight ContainersComplying with ISO668:1988 (1247)AHM963Functional Specification for a Baggage/Cargo Cart (1249)AHM965Functional Specification for a Lower Deck Container Turntable Dolly (1250)AHM966Functional Specification for a Pallet Dolly (1252)Chapter9—AIRPORT HANDLING GROUND SUPPORT EQUIPMENT SPECIFICATIONS(continued) AHM967Functional Specification for a Twenty Foot Unit Load Device Dolly (1254)AHM968Functional Specification for Ramp Equipment Tractors (1256)AHM969Functional Specification for a Pallet/Container Transporter (1257)AHM970Functional Specification for a Self-Propelled Potable Water Vehicle with Rear orFront Servicing (1259)AHM971Functional Specification for a Self-Propelled Lavatory Service Vehicle with Rear orFront Servicing (1262)AHM972Functional Specifications for a Ground Power Unit for Aircraft Electrical System (1265)AHM973Functional Specification for a Towed Aircraft Ground Heater (1269)AHM974Functional Specification for Aircraft Air Conditioning(Cooling)Unit (1272)AHM975Functional Specifications for Self-Propelled Aircraft De-Icing/Anti-Icing Unit (1274)AHM976Functional Specifications for an Air Start Unit (1278)AHM977Functional Specification for a Towed De-Icing/Anti-Icing Unit (1280)AHM978Functional Specification for a Towed Lavatory Service Cart (1283)AHM979Functional Specification for a Towed Boarding/De-Boarding Device for Passengers with Reduced Mobility(PRM)for Commuter-Type Aircraft (1285)AHM980Functional Specification for a Self-Propelled Petrol/Diesel Refueling Vehicle forGround Support Equipment (1287)AHM981Functional Specification for a Towed Potable Water Service Cart (1289)AHM990Guidelines for Preventative Maintenance of Aircraft Towbars (1291)AHM994Criteria for Consideration of the Investment in Ground Support Equipment (1292)AHM995Basic Unit Load Device Handling System Requirements (1296)AHM997Functional Specification for Sub-Freezing Aircraft Air Conditioning Unit (1298)Chapter10—ENVIRONMENTAL SPECIFICATIONS FOR GROUND HANDLING OPERATIONS (1301)AHM1000Chapter10—Record of Revisions (1301)AHM1001Environmental Specifications for Ground Handling Operations (1302)AHM1002Environmental Impact on the Use of Ground Support Equipment (1303)AHM1003GSE Environmental Quality Audit (1305)AHM1004Guidelines for Calculating GSE Exhaust Emissions (1307)AHM1005Guidelines for an Environmental Management System (1308)Chapter11—GROUND OPERATIONS TRAINING PROGRAM (1311)AHM1100Chapter11—Record of Revisions (1311)AHM1110Ground Operations Training Program (1312)Appendix A—References (1347)Appendix B—Glossary (1379)Alphabetical List of AHM Titles (1387)IATA Strategic Partners..............................................................................................................................SP–1。
A Numerical Method for Solution of Ordinary Differential Equations of Fractional Order
(6)
In the general case the Riemann-Liouville fractional derivatives do not commute
β α+β β α α (a Dt (a Dt y ))(t) = (a Dt )(t) = (a Dt (a Dt y ))(t) .
(7)
Extending our considerations, the integer operator commutes with the fractional operator
β α−β α (a Dt (a It y ))(t) = (a Dt )(t) ,
(9)
but they do not commute in the opposite way. We turn our attention to the composition rule in two following ways. First of all we found in literature [10,11] the fact, that authors neglect the general property of fractional derivatives given by eqn. (7). Regarding to solution of fractional differential equations we will apply above properties in the next sections.
α y )(t) = (a It
Scattering and Inverse Scattering on ACH Manifolds
dt2 + h(t) t2 ∞ ¯ The function ρ := ψ∗ t is where h(t), t ∈ [0, ǫ) is a C 1-parameter family of C ∞ metrics on ∂ X. ¯ ¯ ¯ Note that a boundary defining function in X near ∂ X, which can be extended smoothly to X. n the boundary represents the geometric infinity of X , as does the sphere S for the hyperbolic space Hn+1 the Laplacian of (X, g ) was studied in [27]; it consists a finite pure point spectrum σpp (∆), which is the set of L2 (X ) eigenvalues, and an absolutely continuous spectrum σac (∆) satisfying The resolvent σac (∆) = n2 /4, ∞ and σpp (∆) ⊂ 0, n2 /4 .
R(λ) = (∆g − λ(n − λ))−1 ,
2000 Mathematics Subject Classification. Primary 58J50, Secondary 35P25. Key words and phrases. scattering, pseudoconvex domains, asymptotically complex hyperbolic manifolds. The first author was partially supported by French ANR grants no. [JC05-52556] and [JC05-46063], and an Australian National University postdoctoral fellowship. Both authors were supported in part by the National Science Foundation Grant DMS 0500788.
Noncompact Heisenberg spin magnets from high-energy QCD II. Quantization conditions and ene
A(s, t) ∼ −is
(iα ¯ s )N
N =2
¯ s EN /4 s−α ξa,N (t)ξb,N (t) , (α ¯ s σN ln s)1/2
(1.1)
where α ¯ s = αs Nc /π is the QCD coupling constant and the sum goes over an arbitrary number of reggeized gluons exchanged in the t−channel, N = 2, 3, .... In the GLLA approximation, the reggeized gluons interact with each other elastically and form colour-singlet compound states [7]. These states can be defined as solutions to the Bartels-Kwiecinski-Praszalowicz (BKP) equation [3, 8] HN Ψ(z1 , z2 , ..., zN ) = EN Ψ(z1 , z2 , ..., zN ) . (1.2) The effective QCD Hamiltonian HN acts on the colour SU (Nc ) charges of N reggeized gluons and their two-dimensional transverse coordinates, zk (k = 1, ..., N ), which belong to the hyperplane orthogonal to the momenta of two scattered particles. The contribution to the scattering amplitude (1.1) of the compound states built from N reggeized gluons, Ψ(z1 , z2 , ..., zN ), has the ¯ s EN /4 standard Regge form ∼ s−α . It is dominated at large s by the contribution of the ground state EN for Eq. (1.2). Obviously, it increases (or decreases) with s if the energy of the ground state is negative (or positive). As we will show below, the spectrum of (1.2) is gapless (see Eq. (5.6) below) and, as a consequence, one has to retain in (1.1) the contribution of the excited states next to the ground state. This amounts to appearance of the additional factor (α ¯s σN ln s)−1/2 in the r.h.s. of (1.1). The residue factors ξa(b),N (t) measure the overlap of Ψ(z1 , z2 , ..., zN ) with the wave functions of the scattered particles and they depend, in general, on the momentum 2 transferred, t, and the colour factor, 1/Nc . Calculation of the spectrum of the compound states (1.2) for arbitrary number of reggeized gluons N and eventual resummation of their contribution to the scattering amplitude (1.1) is a longstanding problem in high-energy QCD [9, 10, 11, 12]. At N = 2, the solution to (1.2) has been found a long time ago – the well-known Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron [7]. At N = 3 the solution to (1.2) – the Odderon state in QCD [13], was formulated only a few years ago by Janik and Wosiek [14] by making use of the remarkable integrability properties of the effective QCD Hamiltonian [1, 2]. Their solution has been later verified in Refs. [15, 16]. The methods employed at N = 3 in [14] can not be generalized, however, to higher (N ≥ 4) reggeized gluon compound states and very little is known about the solutions to (1.2) for N ≥ 4. Recently, a significant progress has been made in solving the Schr¨ odinger equation (1.2) for higher N in the multi-colour limit, Nc → ∞ and α ¯ s = fixed [5, 16]. The first results of the calculation of the ground state energy EN for higher reggeized gluon compound states in multi-colour QCD 2
Non-Destructive Testing and Evaluation
Non-Destructive Testing and Evaluation Non-Destructive Testing and Evaluation Non-destructive testing (NDT) is awide group of analysis techniques used in science and industry to evaluate the properties of a material, component, or system without causing damage. This is in contrast to destructive testing, which is used to determine the properties of a material by causing damage to the material. NDT is commonly used in the aerospace, automotive, and construction industries to ensure the safety and reliability of products and structures. In this essay, we will discuss the importance of non-destructive testing and evaluation and its various techniques. First and foremost, non-destructive testing and evaluation play a crucial role in ensuring the safety and reliability of products and structures. By using NDT techniques, manufacturers and engineers can detect flaws, defects, and discontinuities in materials and components without causing any damage. This allows for the early detection and prevention of potential failures, which is essential for maintaining the integrity of critical infrastructure and ensuring the safety of end users. There areseveral commonly used NDT techniques, each with its own advantages and limitations. One of the most widely used techniques is ultrasonic testing, which uses high-frequency sound waves to detect internal and surface defects in materials. Another common technique is radiographic testing, which uses X-rays or gamma rays tocreate images of the internal structure of a material. Other techniques include magnetic particle testing, liquid penetrant testing, and eddy current testing. Each of these techniques has its own unique applications and is used depending on the specific requirements of the material or component being evaluated. Inaddition to ensuring the safety and reliability of products and structures, non-destructive testing and evaluation also play a critical role in quality controland process optimization. By using NDT techniques, manufacturers can identify and eliminate defects in the early stages of production, leading to cost savings and improved efficiency. Furthermore, NDT can be used to monitor the performance and integrity of materials and components over time, allowing for predictive maintenance and extending the lifespan of critical infrastructure. Despite its numerous advantages, non-destructive testing and evaluation also have some limitations. One of the main challenges is the need for highly trained and skilledpersonnel to perform NDT techniques accurately and reliably. Additionally, some NDT techniques may be time-consuming and require specialized equipment, which can increase the overall cost of testing. Furthermore, certain materials and components may be difficult to inspect using conventional NDT techniques,requiring the development of new and innovative methods. In conclusion, non-destructive testing and evaluation are essential tools for ensuring the safety, reliability, and quality of products and structures. By using a variety of NDT techniques, manufacturers and engineers can detect flaws and defects in materials and components without causing any damage, leading to cost savings, improved efficiency, and extended lifespan of critical infrastructure. While NDT has its limitations, ongoing research and development in this field continue to expand the capabilities of non-destructive testing and evaluation, making it an indispensable part of modern industry.。
Conformal collider physics Energy and charge correlations
symmetry. In this case the theory is not conformal, but at high enough energies we can approximate the process as a conformal one to the extent that we can ignore the running of the coupling and the details of the hadronization process. In this paper we will analyze similar processes but in conformal field theories.
Fig. 1: A localized excitation is produced in a conformal field theory and its decay products are measured by calorimeters sitting far away.
1
For early work on the applications of scale invariance to strong interactions and, in particular, collisions, see [1].
arXiv:0803.1467v3 [hep-th] 4 May 2008
Conformal collider physics: Energy and charge correlations
The $n$-component KP hierarchy and representation theory
on the wave function (0.1.4) w(x, z ) = (1 + w1 (x)z −1 + w2 (x)z −2 + . . . )ex1 z+x2 z
2
+...
.
*Supported in part by the NSF grant DMS-9103792. **Supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences. Typeset by AMS-TEX 1
−...
be the adjoint wave function. Then the fourth formulation of the KP hierarchy is the following bilinear identity (0.1.8) Resz=0 w(x, z )w∗ (x′ , z )dz = 0 for any x and x′ .
THE n-COMPONENT KP HIERARCHY AND REPRESENTATION THEORY.
arXiv:hep-th/9308137v2 1 Sep 1993
V.G. Kac and J.W. van de Leur
§0. Introduction. 0.1. The remarkable link between the soliton theory and the group GL∞ was discovered in the early 1980s by Sato [S] and developed, making use of the spinor formalism, by Date, Jimbo, Kashiwara and Miwa [DJKM1,2,3], [JM]. The basic object that they considered is the KP hierarchy of partial differential equations, which they study through a sequence of equivalent formulations that we describe below. The first formulation is a deformation (or Lax) equation of a formal pseudo-differential operator L = ∂ + u1 ∂ −1 + u2 ∂ −2 + . . . , introduced in [S] and [W1]: (0.1.1) ∂L = [ B n , L] , n = 1, 2 , . . . . ∂xn
A Virtual Work Approach to Modeling the Nonlinear
Barry T. Rosson
Department of Civil, Environmental and Geomatics Engineering, Florida Atlantic University, Boca Raton, FL 33431, USA
Abstract: The stiffness reduction is studied in detail of compact W-Shapes (wide-flange steel shapes) that results from yielding of the
Key words: Nonlinear analysis, steel beam-columns, stiffness reduction, material model, virtual work.
1. Ihavior of steel frames with compact doubly-symmetric beam-columns that are subjected to major or minor axis bending has been shown to exhibit significant differences in their response based on plastic hinge and plastic zone analyses [1, 2]. Frames of this type with little to no redundancy can be very sensitive to the refinement of the inelastic analysis procedure employed [3, 4]. Recent research has focused on developing improved empirical relationships to account for the reduction in stiffness that occurs due to yielding of the beam-column’s cross-section [5-7]). The objective of this paper is to present the findings from a detailed fiber element model investigation of the stiffness reduction that develops as a result of yielding in the flanges and web over the full range of moment and axial load combinations from initial yield to the fully plastic condition. Considering major axis or minor axis bending under axial compression conditions, analytical expressions are presented to determine the moment and axial load combinations at the initial onset of yielding
Trajectory tracking control for maneuverable nonholonomic systems
Sergei V. Gusev
arXiv:math/0507567v1 [math.OC] 27 Jul 2005
Department of Mathematics and Mechanics, St Petersburg State University 2 Bibliotechnaya sq., Peterhof, St.Petersburg, 198904, Russia E-mail address: gusev@
Abstract
The paper considers a motion control problem for kinematic models of nonholonomic wheeled systems. The class of maneuverable wheeled systems is defined consisting of systems that can follow any sufficiently smooth non-stop trajectory on the plane. A sufficient condition for maneuverability is obtained. The design of control law that stabilizes motion along the desired trajectory on the plane is performed in two steps. On the first step the trajectory on the configuration manifold of the system and the input function are constructed that ensure the exact reproduction of the desired trajectory on the plane. The second step is the stabilization of the constructed trajectory on the configuration manifold of the system. For this purpose a recursive procedure is used that is a version of backstepping algorithm meant for non-stationary systems nonlinearly depending on input. The procedure results in the continuous memoryless feedback that stabilizes the trajectory on the configuration manifold of the system. As an example the motion control problem for a truck with multiple trailers is considered. It is shown that the proposed control law stabilizes the desired trajectory of the vehicle on the plane for all initial states of the system from some open dense submanifold of the configuration manifold, i. e., almost globally. The statement takes place both for a truck pulling any number of trailers in a forward direction and for a truck pushing any number of trailers in a backward direction. The latter result is the solution of the intuitively hard problem of the road train reverse motion control. The effectiveness of the proposed control is demonstrated by simulation. Animated examples are presented at Sergei V. Gusev Web Page. Keywords: Nonholonomic systems, kinematics, nonlinear systems, trajectory tracking, backstepping. AMS subject classification: 93D15; 70Q05; 70B15.
Collisional decoherence reexamined
a rXiv:q uant-ph/03394v216M ay23Collisional decoherence reexamined Klaus Hornberger and John E.Sipe ∗Universit¨a t Wien,Institut f¨u r Experimentalphysik,Boltzmanngasse 5,1090Wien,Austria (Dated:Apil 17,2003)Abstract We re-derive the quantum master equation for the decoherence of a massive Brownian particle due to collisions with the lighter particles from a thermal environment.Our careful treatment avoids the occurrence of squares of Dirac delta functions.It leads to a decoherence rate which is smaller by a factor of 2πcompared to previous findings.This result,which is in agreement with recent experiments,is confirmed by both a physical analysis of the problem and by a perturbative calculation in the weak coupling limit.PACS numbers:03.65.Yz,03.65.Ta,03.75.-bI.INTRODUCTIONA classic result of decoherence theory is the rapid decay in the off-diagonal matrix el-ements in the coordinate representation of the density operatorρ(R1,R2;t)of a massive Brownian particle suffering collisions with the lighter particles of a thermal bath.Early cal-culations by Joos and Zeh[1]were improved by later authors,and the result of Gallis and Fleming[2]seems to be the most widely quoted[3].Theyfind,in the limit of an infinitely massive Brownian particle,that∂ρ(R1,R2;t)m dˆn1dˆn2equation derivations undertaken in,e.g.,quantum optics.In thefirst calculation wefind(2) withε=1.In the second wefind(2)withε=1and f(qˆn2,qˆn1)replaced by f B(qˆn2,qˆn1), thefirst Born approximation to that scattering amplitude.This is precisely what would be expected,since the second calculation requires the assumption of weak interaction;it thus serves to confirm theε=1result of thefirst.Neither of these is the most elegant or general calculation one could imagine;thefirst is rather cumbersome,and the second would be neater if generalized to second quantized form[5].But thefirst has the advantage of displaying the physics of decoherence in an almost pictorial way,while allowing a calculation involving the full scattering amplitude.And the second,in its simple form,establishes a clear connection with the usual approach to decoherence through the master equation approach common in quantum optics.Totally separate in their approaches,we feel that together they are a convincing demonstration thatε=1.These two calculations are presented in sections II and IV below.In section III we return to the traditional derivation and highlight its inherent shortcomings.We show how it should be modified by using a simple physical argument,which leads to a replacement rule for the occurring square of a Dirac delta function.This treatment then also yields the resultε=1. Our concluding remarks are presented in section V.II.SCATTERING CALCULATIONTo set our notation we begin with a review of the standard approach used to calculate collisional decoherence.However,we also wish to point out the difficulties that can arise in its application,so we begin in a more detailed way than is normally done.To apply scattering theory in a careful way one has to begin with an asymptotic-in state |φm |ψ ,a normalized ket that is the direct product of a Brownian particle ket|φm and abath particle ket|ψ .The asymptotic-in ket is the result of the evolution of a product ket |φ(−∞)m |ψ(−∞) at t=−∞to t=0under the Hamiltonian that describes the free evolution of both particles,without interaction.The effect of the two-particle scattering operator S on this asymptotic-in state,S(|φm |ψ ),then produces the asymptotic-out state.When evolved from t=0to t=∞by the non-interacting Hamiltonian,the asymptotic-out stateyields the actual state at t=∞that evolves from|φ(−∞)m |ψ(−∞) at t=−∞under the influence of the full Hamiltonian.FIG.1:Sketched are the wave packets associated with|φm and|ψ at t=0.In configuration(a) the state|φm |ψ could be taken as both an asymptotic-in state and an initial state at t=0;for configuration(b)that would not be possible.In general,of course,|φm |ψ does not describe the actual ket at t=0that evolves from |φ(−∞)m |ψ(−∞) at t=−∞,because the evolution of that actual ket involves the particle interaction.But if the kets|φm and|ψ are such that the(short-range)interaction between the particles has not yet had an effect(e.g.,Fig.1a but not Fig.1b),then|φm |ψ can be taken as the actual ket at t=0as well as the asymptotic-in ket.We only consider kets|φm and|ψ of this form below.We now turn to the impending collision of a bath particle characterized by|ψ and a Brownian particle described by a reduced density operator at t=0given by a convex sum of projectors|φm φm|,ρin= m p m|φm φm|= d R1d R2|R1 ρo(R1,R2) R2|,with probabilities p m>0, p m=1.Here the|R1,2 label position eigenkets of the Brownian particle,andρo(R1,R2)= m p m R1|φm φm|R2 (3) its position representation.Thenρtotalin=ρin⊗|ψ ψ|(4) can be considered both as the full initial(at t=0)density operator,and the full asymptotic-in density operator.The full asymptotic-out density operator is thenρtotal out =Sρtotalin S†= d R1d R2S(|R1 |ψ )ρo(R1,R2)( ψ| R2|)S†.To determine terms such as S(|R |ψ )it is useful tofirst consider the effect of the S operator on direct products|P |p of eigenkets|P of the Brownian particle momentum and eigenkets |p of the bath particle momentum.Since the total momentum commutes with the Soperator the scattering transformation can be reduced to a one-particle problem,with S(|P |p )= d q|P−q |p+q m∗M P+q|S o|m∗M P ,where the matrix element here is that of the one-particle scattering operator S o corre-sponding to the two-body interaction acting in the Hilbert space of the bath particle,and m∗=mM/(m+M)is the reduced mass.In the limit that the Brownian particle is much more massive than the bath particle,M≫m,this reduces toS(|P |p )→ d q|P−q |p+q p+q|S o|por,moving to a position representation for the Brownian particle,S(|R |p )= d q|R e−i q·R/ |p+q p+q|S o|p= d q|R |p+q p+q|e−i p·R/ S o e i p·R/ |p=|R e−i p·R/ S o e i p·R/ |p ,where p is the momentum operator for the bath particle,and so for general states|ψS(|R |ψ )=|R e−i p·R/ S o e i p·R/ |ψ≡|R ψR ,whereψR =e−i p·R/ S o e i p·R/ |ψ ,and thusρtotal= d R1d R2|R1 ψR1 ρo(R1,R2) ψR2 R2|.outis not thefinal density operator at t=∞,but only the asymptotic-out density Althoughρtotaloutoperator,it evolves to thefinal density operator through the non-interacting Hamiltonian, and overlaps of the form ψR2|ψR1 will be preserved during this free evolution.So thefinal reduced density operator for the Brownian particle at t=∞isρfinal= d R1d R2|R1 ψR2|ψR1 ρo(R1,R2) R2|≡ d R1d R2|R1 ρ(R1,R2) R2|,whereρ(R 1,R 2)= ψR 2|ψR 1 ρo (R 1,R 2).(5)As is well understood,decoherence arises because the bath particle becomes entangled with the Brownian particle and the two (asymptotic-out)states ψR 2 and ψR 1 resulting from scattering interactions associated with the same bath ket |ψ and different position eigenkets|R 2 and |R 1 can have negligible overlap even for |R 2−R 1|small.The change of the Brownian particle’s reduced density operator by a single collision is ∆ρ(R 1,R 2)≡ρ(R 1,R 2)−ρo (R 1,R 2)= ψR 2|ψR 1 −1 ρo (R 1,R 2).(6)It involves overlap terms of the form ψR 2|ψR 1 = ψ|e −i p ·R 2/ S †o e −i p ·(R 1−R 2)/ S o e i p ·R 1/ |ψ = ψ|S †2S 1|ψ =tr bath S †2S 1|ψ ψ| ,(7)where the operators S j =e −i p ·R j / S o e i p ·R j /(8)for j =1,2are translated scattering operators.We introduce corresponding T j operators according toS j =1+i T j ,(9)and using the unitarity of the S j ,which follows immediately because S o is unitary,we findS †2S 1=1+T †2T 1−12T †2T 2+i 2 T 2+T †2and so ψR 2|ψR 1 =1+ ψ|A|ψ ,(10)where A =T †2T 1−12T †2T 2+i 2T 2+T †2 .Thus the change in the Brownian particle reduced density operator is∆ρ(R1,R2)= ψ|A|ψ ρo(R1,R2)(11)The general strategy is to evaluate the matrix element ψ|A|ψ by inserting complete sets of momentum eigenstates,ψ|A|ψ = d q1d q2 ψ|q2 q2|A|q1 q1|ψ ,(12) determine q2|A|q1 ,and then perform the momentum eigenstate integrals.Writing S o= 1+i T o as well,and using the relations(8)and(9)wefindq2|A|q1 =e i(q1·R1−q2·R2)/ q2|T†o e i p·(R2−R1)/ T o|q1 (13)−1e i(q1−q2)·R2/ q2|T†o T o|q12i+δ(E2−E1)f(q2,q1),(14)2π mwhere f(q2,q1)is the scattering amplitude,we can identifyq2|T o|q1 =1f(q2,q1),2π q2where E i=q2i/(2m).Now in the traditional calculations[1,2,3]one calculates∂ρ(R1,R2)/∂t by considering the change∆ρ(R1,R2)in a time∆t due to collisions with bath particles that would pass in the neighborhood of the Brownian particle,taking the distribution of their velocities from the assumed thermal equilibrium of the bath.To calculate∆ρ(R1,R2)from one of these bath particles,a box-normalized momentum eigenstate, |q is used in place of a localized ket|ψ .Unlike the|φm |ψ states we introduced above,the|φm |q obviously cannot be considered either as asymptotic-in states or as the actual states at t=0since the |q are delocalized.Nonetheless,the traditional approach seems to simplify the calculation because, as is clear from(12),only diagonal elements q|A|q are required if the limit of an infinite boxis taken.But from the expression(13)for q2|A|q1 it is clear that,when a resolution over a complete set of momentum states|q′ is inserted between T†o and T o and the expression(15) for the matrix elements of T o is used,the diagonal elements q|A|q involve the square of Dirac delta functionsδ(q−q′).To evaluate these the“magnitude”ofδ(0)must be somehow set.This is done by relating it to an original normalization volume of the box.While not implausible,such a protocol is certainly not rigorous and is open to question.To avoid the necessity of this kind of maneuver we will employ bath states|ψ that are normalized and localized,as is required by a strict application of scattering theory.Before addressing the full calculation for a bath in thermal equilibrium we consider scattering involving a single state|ψ .A.Scattering of a single bath ketFrom the equations(12)and(13)for ψ|A|ψ in terms of q2|A|q1 it is clear that we require integrals of the formI1= d q1d q2u(q1,q2) q2|T o+T†o|q1 ,(16)I2(R)= d q1d q2u(q1,q2) q2|T†o e i p·R/ T o|q1 ,which we work out in Appendix A for an arbitrary function u(q1,q2)of the two momentum variables.Wefind that we can write these expressions exactly asI1= d q ˆq⊥d∆u(q−∆2)M1(q,∆)(17) andI2(R)= dˆn d q ˆq⊥d∆u(q−∆2)e i Q·R/ M2(q,ˆn,∆),(18) The integration over q covers all momentum space,while∆is a two dimensional momentum vector ranging over the plane perpendicular to q;ˆn is a unit vector with dˆn the associated solid angle element.Moreover,M1(q,∆)=12,q−∆2,q+∆4π2 2Q2)f(Q,q−∆FIG.2:For this as both the total asymptotic-inwithQ=ˆn 4.(21) With these formulas in hand we can address the expression for ψ|A|ψ once|ψ is specified. To do this,we take the bath particle wave function r′|ψ to be a Gaussian wave packet centered at r o in position and p o in momentum,r′|ψ =e i p o·(r′−r o)/√√2 q−∆πb2 3/2e i∆·r o/ e−∆2/(4b2)e−|q−p o|2/b2.We now assume that this wave packet is located far enough away from the regions of space where an initial density operator(3)is concentrated,and with an average momentum directed towards the Brownian particle such that the combined density operator(4)can be taken both as an initial density operator at t=0,and as the asymptotic-in density operator (see Fig.2).Then using the expressions above wefindψ|A|ψ = 1whereB(ˆn,q,∆)=e i q·(R1−R2)/ e i Q·(R2−R1)/ e−i∆·(r o−2e−i∆·(r o−R1)/ e−∆2/(4b2)M2(q,ˆn,∆)−12 e−i∆·(r o−R1)/ −e−i∆·(r o−R2)/ e−∆2/(4b2)M1(q,∆),and where we have put2.(23) The Gaussian functions will keep∆within about b of zero and q within about b of p o.We now assume that the central momentum p o is much greater in magnitude than its variance, p o≫b,and hence q≫b for all q that make a significant contribution;we also assume that the scattering amplitude varies little over the momentum range b.Then we can putM1(q,∆)≈14π2 2|f(qˆn,q)|2.Once these approximation are made the integral over∆of the three terms in B(ˆn,q,∆) can be done immediately.The integral over∆of thefirst term is not so simple because∆still appears in Q.In the exponential we have phase factors that vary asQ·(R2−R1)q2+∆2=qˆn·(R2−R1)8 q+...Thefirst correction term is of orderb2|R2−R1|(q/b)(24) Since q≫b this term will still be much smaller than unity even if the distance between the two positions of the Brownian particle is several widths of the wave packet.We assume that|R2−R1|is indeed such this quantity is much less than unity.Then we can replace the phase by its leading order expansionQ·(R2−R1) ,in the exponentials of thefirst two integrals,and the integration over∆can be done as well.These are two dimensional integrals over a plane perpendicular to q,and so they are of the form ˆq⊥d∆e−i∆·(r o−R),where we have used the fact that ab= and introducedexp − R2−(ˆq·R)2 /a2Γq(R)=A r o(q),(26)(πb2)3/2whereA r o(q)=Γq(r o−(Γq(r o−R1)+Γq(r o−R2)) dˆn|f(qˆn,q)|222πi+R.Moreover,since the integral in(26)restricts q to within a distance of about b of p o,in(27)we can replace q by p o in the scattering amplitudes and in the phase,using the assumption already made thatthey vary little over a range of b;we can also replace theΓq functions by correspondingΓpo functions.The integral in(26)can then be done,and using(11)wefind∆ρ(R1,R2)=−ρo(R1,R2)Γp(r o−oe−ℓ2/a2R)=FIG.3:A configuration where a≫R=|R1−R2|;ℓ=e−βp2/(2m)(29)ΩprovidedΩis much larger than the cube of the thermal de Broglie wave lengthλ= m.(30) The usual convex decomposition of(29)in terms of the delocalized energy eigenstates is the obvious one and,aside from the freedom in choosing orthogonal states from among adegenerate set,it is the only one in terms of orthogonal states.But a host of others can also found.A particularly convenient set of convex decompositions for our problem at hand can be obtained by usinge−βp2/(2m)= ¯β 2πm/ˆβ 3/2e−¯β(p−p)2/(2m)(31) which holds as long asˆβand¯βare both positive and11¯β+ ArrayΩ d pˆµ(p)|ψrp ψrp|,(32) whereˆµ(p)= ˆββ 3/4e−¯β(p−p)2/(4m)|r (34)=¯λ3/2e−¯β(p−p)2/(4m)|r ,are characterized by the length scale¯λ= m,(compare with (30)).One then immediately finds r ′|ψrp =2√¯λ3/2e i p ·(r ′−r )/ e −2π|r ′−r |2/¯λ2(35)so the wave packet |ψrp is centered at r and has an average momentum p .Indeed,it is of the Gaussian form used in the preceding section with minimal uncertainties,b ≡2m =k B ¯T2m =k B ˆT2m +(δp x )22.We see that in the class (32,33)of convex decompositions of ρbath a part of the thermal kinetic energy is associated with the size of the wave packets themselves,while the rest resides inthe motion of the centres of the wave packets.If we take ˆT→0then the wave packets are essentially all at rest characterized by a size ¯λ→λ,which is the thermal de Broglie wavelength.On the other hand,for ˆT≫¯T the wave packets are much larger than the thermal de Broglie wavelength,and essentially all the thermal kinetic energy is associated with the expectation value of the momenta of the wave packets;we have p 2 2mp 2ˆµ(p )d p =33k B ˆT1.Assumptions and choices1.We neglect initial correlations,taking the initial full density operator at t=0to be adirect product of a Brownian particle density operator and a density operator for the bath particles in thermal equilibrium,ρtotal(t=0)=ρo⊗ρbath.(36) 2.We assume that the density of bath particles is much less thanλ−3;then the issue ofparticle degeneracy does not matter and we may consider the density operator of the total bath to be just the product of density operators for individual particles.Thus we can calculate effects‘particle by particle’.We choose a volumeΩmuch larger than any other volume of interest.3.We use a convex decomposition ofρbath for a single bath particle of the type describedabove,with¯T≪T such thatˆT≈Tand thereforeb2≪ p2 .(37) This renders b sufficiently small so that the variation in scattering amplitudes over the momentum spread of a wave packet is negligible for essentially all of the wave packets in the convex decomposition.4.The value of¯T should also be small enough that the neglect of the variation of thescattering amplitudes in the integral(22)is justified,and that we can use the approx-imation of neglecting terms on the order of(24)above.For the latter we needb2q|R2−R1|≪1Now for typical wave packets the average momentum p,and hence q,will be of the order of mv wp=|R2−R1|= 8π λ≪1,vwpand sinceˆT≈T this reduces to¯T≪|R2−R1|5.We choose a coarse-graining time∆t sufficiently large thatv wp∆t≫a,(38)v wp∆t≫|R2−R1|.That is,a typical packet travels a distance much greater than its width and much greater than the distance between the two decohering sites during the coarse graining ing the expressions for v wp and a above,andˆT≈T,thefirst condition readsT∆t≫R(recall(23);see Fig.4).Of course,some of these will completely miss the Brownian particle,but none have had a collision with it in the past.For a given p we refer to this region of space as R(p).Returning to the wave packets,note that those with central positions r close to the R1 or R2of interest will initially be overlapping with regions of space for whichρo(R1,R2)is non-vanishing;here any talk of a collision is inappropriate,since at initiation,at t=0,the Brownian and bath particle would immediately be strongly interacting.This is an artifactΩ ψrp|A|ψrp ,where we assume that the inclusion of the problematic class of wave packets identified above will not lead to serious ing the result(26)from our scattering calculation above, we have∆ρ(R1,R2)=nρo(R1,R2) d pˆµ(p) R(p)d r d q e−|q−p|2/b2But this is not necessary.We can simply note that,by virtue of(37),ˆµ(p)will vary little over the range b that e−|q−p|2/b2peaks and falls.Hence we can replaceˆµ(p)byˆµ(q)and R(p)by R(q),and immediately do the integral over p to yield∆ρ(R1,R2)=nρo(R1,R2) d qˆµ(q) R(q)d r A r(q).Since the only r dependence is in theΓq,see(25),one can now do the r integral for each fixed q,putting d r=d r⊥dr ,where r refers to the distance in the direction−q.Since the integration over r⊥is unrestricted in the region R(q)we haveR(q)d r⊥Γq(r⊥−R⊥i)=1=R1,R2,orfor R∆t,mand so wefind∆ρ(R1,R2)ˆµ(q) dˆn e i(q−qˆn)·(R1−R2)/ −1 |f(qˆn,q)|2.mFinally,we recall thatˆT≈T and therefore putˆµ(q)≈µ(q),whereµ(q)= β,(41)4πand hence on a coarse grained time scale wefind(1,2)withε=1.III.THE TRADITIONAL APPROACH:A REMEDYWe showed in the preceding section how the problem of evaluating a squared Dirac function can be circumvented by expressing the thermal state of the bath particles in an over-complete,non-orthogonal basis of Gaussian wave packets(see Eq.(32)).However,it iscertainly reasonable to explore the possibility of using the standard diagonal representation of the thermal bath density operator,which facilitates the formal calculation considerably. After all,all the representations ofρbath are equally valid and should yield the same master equation provided the calculation is done in a correct way.It is therefore worthwhile to search for a way to deal properly with such an ill-defined object as the“square”of a delta distribution function.In this section we show how a proper evaluation of the diagonal momentum basis matrix elements can be implemented.This leads to an alternate derivation of the master equation (1,2),and allows us to highlight the origin of the problem plaguing earlier workers and to discuss further implications.However,rather than attempting a mathematically rigorous formulation,we base our presentation on a simple physical argument.Our point is that such an argument can lead to a prescription for correctly evaluating improper products of Dirac delta functions,although this differs from previous naive treatments.A.A single collisionLet us consider again the action of a single scattering event on the Brownian particle in position representationρo(R1,R2)and in the limit of a large mass.It follows from the discussion in section II that after the collision it differs merely by a factor from the initial Brownian state,ρ(R1,R2)=η(R1,R2)ρo(R1,R2)(42) which is given byη(R1,R2)=tr bath{e−i p R2/ S†o e i p(R2−R1)/ S o e i p R1/ ρbath},(43) (see Eqs.(5)and(7)).In section II only pure statesρbath=|ψ ψ|of the bath particle were considered,but the reasoning is immediately generalized to mixed states.The factorη(R1,R2)may be called the decoherence function,since it describes the ef-fective loss of coherence in the Brownian state which arises from disregarding the scattered bath particle.The normalization ofρbath impliesη(R1,R2)=1(44)lim|R1−R2|→0which means that the collision does not change the position distribution of the Brownian point particle,ρ(R,R)=ρo(R,R).On the other hand,possible quantum correlationsbetween increasingly far separated points will vanish,since a collision may be viewed as a position measurement of the Brownian particle by the bath which destroys superpositions of distant locations:lim|R1−R2|→∞η(R1,R2)=0(45) This complete loss of coherence implies that the collision took place with a probability of one.It could be realized,in particular,by taking the incoming bath particle state to be a momentum eigenket in a box centered on one of the scattering sites.In thermal equilibrium the density operator(29)of the bath particle can be written asρbath=λ3Ω p∈PΩµ(p) |p p|,(46) with the normalized momentum distribution function(40)atβ=1/(k B T).The |p are momentum eigenkets normalized with respect to the bath volumeΩ,|p =(2π )3/2(2π )3/2,(49) which satisfyp|p′ =δ(p−p′)and span the full space, d p|p p|=I.(50)Since the bath state(46)is diagonal in the momentum representation,an explicit expres-sion for the decoherence function(43)is readily obtained:η(R1,R2)→ d pµ(p) p|e−i p·R2/ S†o e i p·(R2−R1)/ S o e i p·R1/ |p= d pµ(p) 1− p|T†o T o |p +e i p·(R1−R2)/ p|T†o e i p·(R2−R1)/ T o |p= d pµ(p) 1−(2π )3Ω p∈PΩ→ d p.In the second line we introduced the operator T o=i(1−S o)and used the unitarity of S o,i(T o−T†o)=−T†o T o,as in section II and in[2,3].The last line follows after inserting a complete set of states (50)and noting the relation(47).The expression in square brackets in(51)should be well-defined andfinite.However, it involves two arbitrarily large quantities,the“quantization volume”Ω,which stems from the normalization of the bath particle,and the squared amplitude of the T o-operator with respect to(improper)momentum kets.The simple matrix element is given by the expression (15)p′|T o|p =δ(p−p′)|R1−R2|→∞.Therefore the limit(45)allows to specify the unknown function g(p)in(53).One obtainsg(p)=Ωσ(p)p2withσ(p)= dˆn f(pˆn,p) 2the total cross section for scattering at momentum p.Formally,this means that one should treat the expression involving the squaredδ-function and scattering amplitude asδ(p−p′)f(p′ˆn,p) 2→Ω∆t=−n d pµ(p)p∂tρ(R1,R2)=−F(R1−R2)ρ(R1,R2).(55) with F given by(2),again withε=1.C.InterpretationIt is clear that the derivation of the decoherence function(43)does not hold rigorously even for volume-normalized(47)momentum states,since their amplitude is uniform in spaceand they cannot be considered as asymptotic-in or asymptotic-out states.Nonetheless, the fact that one obtains the“correct”master equation by using the diagonal momentum representation(46)indicates that it can be reasonable,at least in a formal sense,to extend the applicability of(43)to volume-normalized momentum eigenstates.Then the appearance of the total cross section in the appropriate replacement rule(54) has a clear physical interpretation.The squared matrix element of the T o operator with respect to two orthogonal proper states may be viewed as the probability for a transition between the states due to a collision.The appropriate normalization of the probability necessary in the limit of improper states is then effected by the appearance of the total cross sectionσ(p)in(54),which is absent in the usual naive treatments of the squared delta function.This point of view is confirmed by the fact that the rule(54),which was derived from a simple physical argument(45),implies a conservation condition.Integrating(54)we have (2π )3=1(56)p2σ(p)and hence,using(50)and switching to volume-normalized states,p|T o T†o |p →1.(57) Inserting the identity(48)yieldsp′∈PΩ p′|T o |p 2→1.(58)This is reminiscent of the situation of a multi-junction in mesoscopic physics[8],or of the scattering offa quantum graph[9],where one defines a transition matrix T mn=|t mn|2 which connects afinite number of incoming and outgoing channels.There the t mn are the transmission amplitudes between the incoming and outgoing states and the currentconservation impliesm T mn=1with T mn=|t mn|2,in analogy to(58).The fact that the conservation relation(58)has no meaningful equivalence in the contin-uum limitΩ→∞is closely connected to the difficulty of evaluating the squared scattering amplitude in the momentum representation.It suggests that the diagonal representation ofρbath can be used in a rigorous formulation of the master equation only if the transition of going from a discrete to a continuous set of bath states is delayed until after the square of the scattering matrix element is evaluated.A calculation along this line,albeit in a perturbative framework,is presented in the following section.IV.WEAK-COUPLING CALCULATIONWe now consider an approach that is totally different from the derivation in Section II. Instead of performing a scattering calculation,we obtain a master equation for the reduced density operator from a weak coupling approximation that is very similar to the analyses of quantum optics.Again the assumption of a low density of bath particles will allow us to calculate the effect of the bath particles one particle at a time,so we begin with our Brownian particle and a single bath particle restricted to a box of normalization volumeΩ. While we will take the limitΩ→∞in the course of the calculation,we can do it in such a way that products of Dirac delta functions never appear.In the absence of any interaction between the particles the Hamiltonian readsH o=P22m,where m and M are the bath and Brownian particle masses and p and P are their momentum operators.The normalized eigenstates of H o are direct products |P |p ,where |p is given by(47)with(49),and similarlyR |P =e i P·R/ Ω.(59) The values of p and P are restricted to a discrete set,p,P∈PΩ,so that the wave functions respect periodic boundary conditions.Our full Hamiltonian is thenH=H o+V(r−R),where r and R are respectively the bath and Brownian particle position operators,and V describes the interaction.In the interaction picture the full density operator evolves according toρtotal I (t)=U(t)ρtotalI(0)U†(t),(60)。
Present and Future Bounds on Non-Standard Neutrino Interactions
arXiv:hep-ph/0302093v1 12 Feb 2003
(a) IPPP, University of Durham, Durham DH1 3LE,UK (b) School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 (c) Depto. de F´ ısica Te´ orica and IFIC, Universidad de Valencia-CSIC, Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia, Spain
Abstract We consider Non-Standard neutrino Interactions (NSI), described by four-fermion operators ¯γf ), where f is an electron or first generation quark. We assume these of the form (¯ να γνβ )(f operators are generated at dimension ≥ 8, so the related vertices involving charged leptons, obtained by an SU (2) transformation νδ → eδ , do not appear at tree level. These related vertices necessarily arise at one loop, via W exchange. We catalogue current constraints from sin2 θW measurements in neutrino scattering, from atmospheric neutrino observations, from LEP, and from bounds on the related charged lepton operators. We estimate future bounds from comparing KamLAND and solar neutrino data, and from measuring sin2 θW at the near detector of a neutrino factory. Operators constructed with νµ and νe should not confuse the determination of oscillation parameters at a ν factory, because the processes we consider are more sensitive than oscillations at the far detector. For operators involving ντ , we estimate similar sensitivities at the near and far detector.
Essential self-adjointness of Schroedinger type operators on manifolds
a rX iv:mat h /21231v3[mat h.SP]29J u l22ESSENTIAL SELF-ADJOINTNESS OF SCHR ¨ODINGER TYPE OPERATORSON MANIFOLDS MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBIN Dedicated to M.I.Vishik on occasion of his 80th birthday Abstract.We obtain several essential self-adjointness conditions for the Schr¨o dinger type operator H V =D ∗D +V ,where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ,and V is a Hermitian bundle endomorphism.These conditions are expressed in terms of completeness of certain metrics on M naturally associated with H V .These results generalize the theorems of E.C.Titchmarsh,D.B.Sears,F.S.Rofe-Beketov,I.M.Oleinik,M.A.Shubin and M.Lesch.We do not assume a priori that M is endowed with a complete Riemannian metric.This allows us to treat e.g.operators acting on bounded domains in R n with the Lebesgue measure.We also allow singular potentials V .In particular,we obtain a new self-adjointness condition for a Schr¨o dinger operator on R n whose potential has the Coulomb-type singularity and is allowed to fall offto −∞at infinity.For a specific case when the principal symbol of D ∗D is scalar,we establish more precise results for operators with singular potentials.The proofs are based on an extension of the Kato inequality which modifies and improves a result of H.Hess,R.Schrader and D.A.Uhlenbrock.Contents 1.Introduction 22.Main results 43.Some examples 94.Proof of the first part of Theorem 2.3125.Kato inequality for Bochner Laplacian 136.The case when the support of V is small187.Proof of the second part of Theorem 2.3228.Quadratic forms and the essential self-adjointness of H V +249.Proof of Theorem 2.73110.Proof of Theorem 2.1333Appendix A.Friedrichs mollifiers33Appendix B.Positivity37Appendix C.Stummel and Kato classes40Appendix D.More history422MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINReferences481.IntroductionSpectral theory of differential operators takes its deep roots in non-relativistic quantum me-chanics,even though thefirst results about eigenvalues and eigenfunctions are much older.Let us recall that according to von Neumann a state of a quantum system is a vectorψin a complex Hilbert space H(ψis defined up to a non-vanishing complex factor),whereas a physical quantity is a self-adjoint operator A in H.A special physical quantity H,called energy or hamiltonian, is responsible for the evolution of any given state which is given by the Schr¨o dinger equation1=Hψ,(1.1)∂twhere we choose units so that the Planck constant is1.Given an initial conditionψ(0)=ψ0,(1.2) we can writeψ(t)=exp(itH)ψ0,where the exponent can be defined by the spectral theorem,and the equation(1.1)is satisfied in the strong sense if we assume thatψ0∈Dom(H).Concrete hamiltonians H come from a vaguely defined procedure which is applied to a classical system and called quantization.This procedure leads to hamiltonians H which are usually second order differential operators,called Schr¨o dinger operators.However,it is a rare situation when one can actually immediately see that such an operator H is self-adjoint in a natural L2space. One can usually only see that H is symmetric(or,in a slightly different terminology,hermitian). The difference between symmetric and self-adjoint operators is often ignored by physicists,even very good ones,but this difference is very important.Indeed,for a symmetric operator H a solution of the initial value problem(1.1)-(1.2)may not exist,even ifψ0∈Dom(H).We can try to extend H to make it a maximal operator(i.e.make it defined on allψ∈L2such that Hψ∈L2where Hψis understood in the sense of distributions).But then it may happen that the solution of(1.1)-(1.2)is not unique.Existence and uniqueness of the solution of(1.1)-(1.2)is guaranteed(forψ0∈Dom(H))only if H is self-adjoint.It is usual,however,that one cannot immediately see what the natural domain of H is.Then one starts from smooth functions with compact support and tries to see whether H can be uniquely extended to a self-adjoint operator.If this is the case,then H is called essentially self-adjoint.Under weak restrictions on regularity of coefficients this is equivalent to saying that the closure of the operator H from the smooth functions with compact support coincides with the maximal operator.Then one has the only natural self-adjoint operator associated with the given differential operator,and the evolution of the quantum system is naturally well defined. By this reason essential self-adjointness is often referred to as quantum completeness.ESSENTIAL SELF-ADJOINTNESS OF SCHR¨ODINGER TYPE OPERATORS ON MANIFOLDS3 It is clear from the arguments above that establishing essential self-adjointness of a Schr¨o dinger type operator must be a starting point of any further investigation of the corresponding quantum system,where this operator serves as a hamiltonian.Therefore it is a fundamental problem of mathematical physics to provide precise and effective conditions for essential self-adjointness. The problem of essential self-adjointness for Schr¨o dinger type operators was mainly studied in L2(R n).There are thousands of papers devoted to such a study,thefirst one(though in a different terminology)is due to H.Weyl(1909).But it makes perfect sense to consider the situation in a curved space or,more generally,on manifolds,especially on Riemannian manifolds. Here one usually meets an interesting interaction of analysis and geometry,since geometry of the manifold can play an important role.Thefirst paper in this direction is due to M.Gaffney (1954),where essential self-adjointness of the Laplacian on any complete Riemannian manifold was proven.In this paper we attempt an approach which unifies and extends almost all earlier results on essential self-adjointness of Schr¨o dinger type operators in vector bundles on manifolds.The main results are of two types.An important feature of thefirst type is that we do not assume a Riemannian metric to be given a priori,but construct it from the operator.Another important feature comes from a close watch of trajectories of the corresponding classical hamiltonian and goes back to I.Oleinik(1993,1994).Our work was mainly inspired by his results and also by recent work by M.Lesch(2000)who extended Oleinik’s results and methods to a much more general context of Schr¨o dinger type operators in sections of vector bundles.Adding non-isotropy in momentum variables we eventually came tofirst essential self-adjointness results which are non-isotropic both in space and momentum variables for the Lesch type operators with singular potentials in sections of vector bundles.The second type of results is obtained by an extension of the Kato inequality to vector bundles which wasfirst proclaimed by H.Hess,R.Schrader and D.A.Uhlenbrock(1980).However they only stated their inequality for smooth sections,and this is not sufficient for applications to essential self-adjointness if a non-smooth potential is added.We rectify and improve their result,then apply it to transfer essential self-adjointness results which were previously known for scalar operators only,to operators which have scalar principal symbol.One of the main tools which allows application of the Kato inequality is the positivity of the Schwartz kernel of the resolvent of the Laplacian.In fact,a slightly stronger result is needed. Such a result is obtained in Appendix B,where we also discuss other related questions and formulate a conjecture,which,if proven,would simplify our proof.This conjecture also has an independent interest.All main results of this paper are actually new not only on manifolds but in R n as well.We tried to make this paper as self-contained as possible.To this end we sometimes chose to repeat arguments of older papers,even in case where only minor modifications were needed to obtain what was needed for our goals.We did it sometimes also for some obviously known results if no references were easily available.However in most cases we improved the exposition given in older papers and rendered them in a more modern language.The most standard material is moved to appendices.4MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBIN2.Main results2.1.The setting.Let M be a C∞-manifold without boundary,dim M=n.We will always assume that M is connected.We will also assume that we are given a positive smooth measure dµ,i.e.in any local coordinates x1,x2,...,x n there exists a strictly positive C∞-densityρ(x) such that dµ=ρ(x)dx1dx2...dx n.Let E be any hermitian vector bundle over M.We denote by L2(E)the Hilbert space of square integrable sections of E with respect to the scalar product(u,v)= M u(x),v(x) E x dµ(x).denotes thefiberwise inner product.Here ·,· ExLet F be another hermitian vector bundle on M.Consider afirst order differential operator D:C∞c(E)→C∞c(F)(here C∞c denotes the space of smooth compactly supported sections). We assume that the principal symbol of D is injective.In other words,D is elliptic(possibly overdetermined).Let D∗be the formal adjoint of D,so D∗:C∞c(F)→C∞c(E)is a differential operator such that(Du,v)=(u,D∗v),for all u∈C∞c(E),v∈C∞c(F).The main purpose of this paper is to give a sufficient condition for the essential self-adjointness of the operatorH V=D∗D+V,(2.1) where V∈L2loc(End E)is a linear self-adjoint bundle endomorphism1,i.e.,for all x∈M,the operator V(x):E x→E x is self-adjoint.Let us recall definitions of the minimal and maximal operators associated with the differential expression H V.The minimal operator H V,min is the closure of H V from the initial domain C∞c(E),whereas the maximal operator can be defined as H V,max=(H V,min)∗.In particular, the domain Dom(H V,max)of H V,max coincides with the set of sections u∈L2(E),such that H V u∈L2(E)(here the expression H V u is understood in the sense of distributions).Then showing that H V is essentially self-adjoint amounts to proving that H V,max is a symmetric operator.We make the following assumption on V.Assumption A.V=V++V−,wherei.V+(x)≥0,V−(x)≤0as linear operators E x→E x for every x∈M.ii.for every compact K⊂M there exist positive constants a K<1and C K such thatK|V−|2|u|2dµ 1/2≤a K ∆M u +C K u ,for all u∈C∞c(M),(2.2)ESSENTIAL SELF-ADJOINTNESS OF SCHR¨ODINGER TYPE OPERATORS ON MANIFOLDS5 where|V−(x)|denotes the norm of the linear map V−(x):E x→E x,∆M:=d∗d is the scalar Laplacian of an arbitrary Riemannian metric on M(cf.Definition5.4),and · is the norm in L2(E).Remark2.2.The condition(2.2)is automatically satisfied if V−∈L p loc(End E)with p≥n2,for n≥4;and p=2,for n≤3.Then Dom(H V,max)⊂W2,2loc (E).ii.Suppose V satisfies Assumption A and the operator D∗D has a scalar principal symbol.Then Dom(H V,max)⊂W1,2loc(E).Note that due to Remark2.2the requirement on V in Theorem2.3(i)implies that the As-sumption A is satisfied for V.The proof of Theorem2.3(i)is quite simple and is given in Sect.4.The proof of Theorem2.3(ii) is given in Sect.7and is based on the generalization of the Kato inequality which we obtain in Sect.5.Motivated by Theorem2.3we suggest the following conjecture(which is not trivial even in case M=R n):Conjecture2.4.If V satisfies Assumption A,then Dom(H V,max)⊂W1,2loc(E).Note that Theorem2.3(i)implies,in particular,that the conjecture holds if n≤3.Also,by Theorem2.3(ii),the conjecture is true if H V acts on scalar functions(e.g.,if H V is the magnetic Schr¨o dinger operator).Since we were not able to prove the conjecture in its full generality,we will work in this paper under the additionalAssumption B.Dom(H V,max)⊂W1,2loc (E)and Dom(H V+,max)⊂W1,2loc(E).Before formulating the main result,we shall introduce some additional notation. 2.5.The metric associated to D.Let D:T∗M⊗E→F be a morphism of vector bundles defined byD(φu)= D(dφ)u+φDu,(2.3) where u∈C∞(E),φ∈C∞(M),andˆD(dφ)(u)is identified withˆD(dφ⊗u),soˆD=−iσ(D), whereσ(D)is the principal symbol of D.Note that D∗(ξ)=−( D(ξ))∗,whereξ∈T∗x M.6MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINIfξ∈T∗x M,then D(ξ)defines a linear operator E x→F x.Forξ,η∈T∗x M,define ξ,η =1√√√ESSENTIAL SELF-ADJOINTNESS OF SCHR¨ODINGER TYPE OPERATORS ON MANIFOLDS7 Corollary2.9.Suppose that V≥−q where q:M→R satisfies assumptions(i)and(iii)of the above theorem.Then H V is essentially self-adjoint.Corollary2.10.If the metric g T M is complete,then the operator D∗D is essentially self-adjoint.Remark2.11.Note that both the space L2(E)and the operator H V depend on the measure dµon M.However,this measure did not appear in the formulation of Corollary2.9.Hence,if the conditions of this corollary are satisfied,the operator D∗D+V will be essentially self-adjoint for any choice of the measure.In particular,M may have afinite volume with respect to this measure.Remark2.12.Theorem2.7covers some interesting operators,appearing in differential geometry. In particular,if D=d:C∞(M)→Ω1(M)is the de Rham differential,then D∗D=∆M is the scalar Laplacian on M,cf.Definition5.4.More generally,we can consider the de Rham differential d:Ωj(M)→Ωj+1(M)on forms of arbitrary degree.Let d∗be the formal adjoint of d and set D=d+d∗.Then D∗D is the Laplace-Beltrami operator on differential forms,cf.[6,§3.6].Also if D=∇:C∞(E)→Ω1(M,E)is a covariant derivative corresponding to a metric connection on Hermitian vector bundle E(Sect.5.2),then D∗D is the Bochner Laplacian on E,cf.Definition5.4.Note,however,that the operators above are generally different from the classical ones since they are defined by using an arbitrary positive smooth measure dµwhich does not necessarily coincide with the standard Riemannian measure.The restrictionδ<1is essential in the condition(ii)of Theorem2.7.In Sect.3.4,we present an example of an operator,which is not essentially self-adjoint,but satisfies the conditions of Theorem2.7with anyδ>1.Unfortunately,we do not know whether Theorem2.7remains true if we allowδ=1in condition(ii).However,in Sect.10we prove the followingTheorem2.13.Suppose that V satisfies Assumptions A and B.Assume that the metric g T M is complete and that the operator H V=D∗D+V is semi-bounded below on C∞c(E).Then H V is essentially self-adjoint.For the case when H V is a magnetic Schr¨o dinger operator acting on scalar functions,this result was formulated in[79].Unfortunately,in case of singular potentials not all details of the proof were presented.These details,however,can be found in the present paper.Theorem2.7immediately implies the followingCorollary2.14.Assume that V∈L2loc(End E)satisfies Assumption A.Suppose that V= V1+V2,where V1,V2∈L2loc(End E)are such that the operatorδD∗D+V1is semi-bounded below on C∞c(E)for someδ<1,and V2≥−q,where q satisfies conditions(i)and(iii)of Theorem2.7.Then the operator H V=D∗D+V is essentially self-adjoint.8MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINRemark2.15.If we assume that M=R n with the standard metric and measure,H V=−∆+V where∆is the standard Laplacian and V is a scalar(real-valued)function,the following more explicit condition on V1implies the semi-boundedness condition given in the formulation of Corollary2.14:V1∈L p(R n),where p≥nc√ESSENTIAL SELF-ADJOINTNESS OF SCHR ¨ODINGER TYPE OPERATORS ON MANIFOLDS 93.Some examples3.1.Example:an operator with non-isotropic symbol.We now consider an example of an operator with a non-isotropic symbol (a function on T ∗M depending not only on the norm of acovector but also on its direction).We shall use Corollary 2.10to prove that this operator is essentially self-adjoint.This result cannot be obtained by “isotropic estimates”of I.Oleinik[58],M.Shubin [77],and M.Lesch [53].Let M =R 2with the standard metric and measure,and V =0.Consider the operator D =a (x,y )∂∂y,where a,b are smooth real-valued nowhere vanishing functions in R 2.It is elliptic (overdeter-mined).We are interested in the operator H :=D ∗D =−∂∂x −∂∂y.The matrix of the inner product on T ∗M defined by D via (2.4)is diag(a 2/2,b 2/2).The matrix of the corresponding Riemannian metric g T M on M is diag(2a −2,2b −2),i.e.the metric itself is ds 2=2a −2dx 2+2b −2dy 2.By Corollary 2.10,to prove that the operator H is essentially self-adjoint,it is enough to show that the metric g T M is complete,i.e.,that ∞ds =∞,where ds is the arc-length element associated to g T M .(3.1)Let γ(t )=(x (t ),y (t )),where t ∈[0,∞)is a curve in R 2which goes to infinity,cf.Sect.2.5.Then the completeness condition (3.1)can be written as ∞0|x ′(t )|b dt =∞.The purpose of this example is to show that the later condition can be satisfied even when the integral in the left hand side of condition (ii)of Theorem 2.17converges.Roughly speaking,we would like to construct an example,where at least one of the integrals of 1/a and 1/b is large,while the integral of 1/√a dt ≥ x (t n )dx a dt ≥ ∞0dxa=∞.10MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINDenote by r k the solutions of cos(2πe x)=1,i.e.r k=ln k,k=1,2,....Then there is an interval I k around r k such that1−cos(2πe x)≤x−2,i.e.such that|sin(πe x)|≤(√a(x,y)≥1√2√4√4√(1−cos(2πe x))x2+1≥14√a2+b2}.Let us consider the curveγ(t)=(t,t).Then at a pointγ(t)we have a2+b2≥(3−2√∂x(1−y2)∂∂x (1−x2)2∂∂y(1−y2)2∂|x i|+N i<j1ESSENTIAL SELF-ADJOINTNESS OF SCHR¨ODINGER TYPE OPERATORS ON MANIFOLDS11 globally Lipschitz andV(x)≥−q(x), ∞ds/√|x i|+Ni<j1r(t)=∞.(3.5)As an example,we can set V(x)=−|x|2log(1+|x|).Note that we can also consider potentials which satisfy(3.4)but not(3.5).Constructions of rich families of such potentials can be found in[68]and[58].Note that self-adjointness of(3.3) in this case can not be established by the method of[39].3.4.Example:a not essentially self-adjoint operator.In this subsection we present an example of an operator,which is not essentially self-adjoint,but satisfies the conditions of Theorem2.7with everyδ>1.Consider an operator D:C∞c(R)→C∞c(R),where D=ddx+|x|α,and henceD∗D=−d2dx2−α|x|α−1sgn x.This operator is not essentially self-adjoint sinceα−1>2(cf.Example1.1in section3.1of [5]).However,for anyδ>1,δD∗D+V=−δd212MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINSince2α>α−1andδ−1>0,there exists Cδ>0such that−δα|x|α−1sgn x+(δ−1)|x|2α>−Cδfor all x∈R.Hence,δD∗D+V>−CδWe conclude thatδD∗D+V is semi-bounded below for everyδ>1.Therefore,Theorem2.7 does not hold withδ>1.4.Proof of the first part of Theorem2.3In this section assumptions on the potential are as in hypotheses of Theorem2.3(i). Lemma4.1.For any u∈Dom(H V,max),we have V u∈L2loc(E).Proof.Let us assume that u∈Dom(H V,max),i.e.u∈L2(E)andD∗Du+V u=f∈L2(E).(4.1)Define t1:=2.Since V∈L ploc (End E),the H¨o lder inequality implies V u∈L s1loc(E),where12+1t1+1t2=1n=1p−2s2=1p=1n+1s1−22p .In this way we can continue to obtain a series of inclusions u∈L t k loc(E),V u∈L s k loc(E),k= 1,2,...,with1s k−2s k+1=1n 1−nESSENTIAL SELF-ADJOINTNESS OF SCHR¨ODINGER TYPE OPERATORS ON MANIFOLDS13 [87,Ch.6.5])we obtain u∈W1,˜q loc(E).In particular,u∈L˜q loc(E)for such˜q.Since˜q>1,we can argue as in thefirst part of the proof to come to the same conclusion that V u∈L2loc(E).2 4.2.Proof of Theorem2.3(i).If u∈D(H V,max),then D∗Du=H V u−V u∈L2loc(E).By elliptic regularity,u∈W2,2(E).loc5.Kato inequality for Bochner LaplacianIn this section we prove a generalization of the Kato inequality to the Bochner Laplacian on a manifold M endowed with a Riemannian metric g=g T M and a measure dµ=ρdx.Here,dx denotes the Riemannian volume form on M andρ:M→(0,∞)is a smooth function.As afirst step,we establish a“smooth version”of the Kato inequality,cf.Proposition5.9. In the case when dµis the Riemannian volume form on M,a similar inequality was proven in H.Hess,R.Schrader and D.A.Uhlenbrock[35].Then,in Sect.5.15we prove“L1loc”version of the Kato inequality.5.1.A pairing on the space of bundle-valued forms.Let E be a Hermitian vector bundle over M and letE is called complex conjugation and defines an anti-linear isomorphism E→E→C,a⊗b := a,b .(5.1) LetΛi=T∗M∧···∧T∗M be the i-th exterior power of the cotangent bundle to M.The space of smooth sections of the tensor productΛi⊗E is called the space of E-valued differential i-forms on M and is denoted byΩi(M,E).The map(5.1)extends naturally to the maps :Λk⊗E⊗E)→Ωk(M,C).Ifα∈Ωi(M,E),β∈Ωj(M,E),then E),α∧E),and α∧β .Let∗:Λi→Λn−i be the Hodge-star operator,cf.[89].This operator extends naturally to the spacesΛi⊗E andΩi(M,E).The formulaα,β Λi⊗E:=∗−1 α∧∗2Note that to apply the elliptic regularity to the equation D∗Du=f∈W−1,t(E)we must have t>1.Ifloct=1,then the last equation does not imply u∈W1,t(E),cf.[36,Th.7.9.8].This is also the reason we need tolocstart the proof of the lemma with showing that V u∈L t loc(E)with t>1.14MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINSimilarly,forα,β∈Ωi(M,E),one defines α,βΛi⊗E∈C ∞(M)and|α|∈C(M).5.2.Covariant derivative and its dual.Let∇be a connection on E.It defines a linear map∇:Ω∗(M,E)→Ω∗+1(M,E)such that∇(ω∧β)=dω∧β+(−1)iω∧∇β,ω∈Ωi(M),β∈Ωj(M,E).In this paper we will always assume that∇is a Hermitian connection,i.e.,that the following equality is satisfiedd α∧β +(−1)i α∧β ρ,α,β∈Ωi comp(M,E).Let∇∗:Ω∗(M,E)→Ω∗−1(M,E)denote the formal adjoint of∇,i.e.,we have(∇α,β)= (α,∇∗β)for allα,β∈Ω∗comp(M,E).Lemma5.3.Supposeβ∈Ωj(M,E).Then∇∗β=(−1)j∗−1∇∗β+(−1)j∗−1dρβ ρ= M d αρ∧∗β −(−1)j−1 M αρ∧ρ∧∗∇∗β ρ=(−1)j α,∗−1∇∗β+∗−1dρβ =0by the Stokes theorem.To simplify the notation let us denote Aρβ=∗−1dρβ = αAρESSENTIAL SELF-ADJOINTNESS OF SCHR¨ODINGER TYPE OPERATORS ON MANIFOLDS15 Definition5.4.The Bochner Laplacian is the operator∇∗∇:C∞(E)→C∞(E).Similarly, we define the Laplacian∆M=d∗d:C∞(M)→C∞(M).Note,that if dµis the Riemannian volume form on M,then∆M=−∆g,where∆g is the metric Laplacian,∆g u=div(grad u).Remark5.5.In Sect.2.5we introduced a metric g T M associated with afirst order elliptic differ-ential operator D.It is easy to check that for D=∇:C∞(E)→Ω1(M,E)this metric coincides with the metric g which we now assume to be given a priori.5.6.Kato inequality.Recall that a distributionνon M is called positive(notationν≥0), if for every non-negative functionφ∈C∞c(M),we have(ν,φ)≥0.It follows thatνis in fact a positive Radon measure(see e.g.[31],Theorem1in Sect.2,Ch.II).We writeν1≥ν2if ν1−ν2≥0.The main result of this section is the followingTheorem5.7.Assume that u∈L1loc(E)and∇∗∇u∈L1loc(E).Then∆M|u|≤Re ∇∗∇u,sign u ,(5.6) wheresign u(x)= u(x)∇u =2Re (∇u)¯u =2Re u16MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINFurthermore,using(5.5),we getd∗(|u|ǫd|u|ǫ)=−∗−1d∗(|u|ǫd|u|ǫ)−|u|ǫAρd|u|ǫ=−∗−1d(|u|ǫ∗d|u|ǫ)−|u|ǫAρd|u|ǫ=−∗−1 d|u|ǫ∧∗d|u|ǫ −|u|ǫ∗−1d∗d|u|ǫ−|u|ǫAρd|u|ǫ=− d|u|ǫ 2−|u|ǫ(∗−1d∗+Aρ)d|u|ǫ=− d|u|ǫ 2+|u|ǫ∆M|u|ǫ.(5.11) Similarly,d∗ u∇u −Aρ u∇u − uAρ∇u −∗−1 u∇u =− ∇u 2+ u∇∗∇u =− ∇u 2+Re (∇∗∇u)¯u , which,in view of(5.10),implies(5.8).Corollary5.10.Assume that u∈C∞(E).Then the following distributional inequality holds:∆M|u|≤Re ∇∗∇u,sign u .Remark5.11.In the case when dµis the Riemannian volume form on M(so thatρ≡1) analogues of Proposition5.9and Corollary5.10were obtained by H.Hess,R.Schrader and D.A.Uhlenbrock[35].They used a slightly more complicated definition of sign u,which is,in fact,unnecessary,because the difference with our definition occurs only on a set of measure0.5.12.Friedrichs mollifiers.We will now deduce Theorem5.7from Proposition5.9using the Friedrichs mollifiers technique.For reader’s convenience we will review basic definitions and results of K.Friedrichs[29].Suppose that j∈C∞c(R n),j(z)≥0for all z∈R n,j(z)=0for|z|≥1,and R n j(z)dz=1. Forρ>0and x∈R n,define jρ(x)=ρ−n j(ρ−1x).Then jρ∈C∞c(R n),jρ≥0,jρ(x)=0if |x|≥ρ,and R n jρ(x)dx=1.LetΩ⊂R n be an open set.Suppose that f∈L1loc(Ω).Define(Jρf)(x)=fρ(x)= f(x−y)jρ(y)dy= jρ(x−y)f(y)dy,(5.13) where the integration is taken over R n.Then Jρf(x)is defined for x∈Ωρ,whereΩρ= x:x∈Ω,dist(x,∂Ω)>ρ .We recall the following standard Lemma(cf.[36,Theorem1.3.2]):Lemma5.13.Let1≤p<∞.With the notations from(5.13),i.If f∈L ploc (Ω),then Jρf∈C∞(Ωρ)ii.If f∈L p loc(Ω),then Jρf→f asρ→0,in the norm of L p(K),where K⊂Ωis any compact set.iii.If f∈C(Ω),then Jρf→f asρ→0,uniformly on any compact set K⊂Ω.ESSENTIAL SELF-ADJOINTNESS OF SCHR ¨ODINGER TYPE OPERATORS ON MANIFOLDS 17To apply the Friedrichs mollifiers technique to the operator ∇∗∇,let us choose a coordinate neighborhood U of M with coordinates x 1,x 2,...x n .Choosing a local frame of E over U we will identify the space of sections of E over U with the space of vector valued functions on U .Under this identification the Bochner Laplacian ∇∗∇becomes a second order elliptic differential operator which can be written in the form i,k a ik (x )∂i ∂k +ib i (x )∂i +c (x ),where a ik ,b i and c are m ×m matrices of smooth functions (m =dim E ).Alternatively,we can write it in the form i,k ∂i a ik (x )∂k + ib i (x )∂i +c (x ),(5.14)with possibly different a ik ,b i and c .The later form is sometimes more convenient if the coeffi-cients are not smooth.We can also consider u ρ=J ρu ,where J ρis an integral operator whose integral kernel is j ρ(x −y )Id,where Id is m ×m identity matrix.We now state a key proposition whose proof is given in Appendix A.Proposition 5.14.Assume that u ∈L 1loc (E )and ∇∗∇u ∈L 1loc (E ).Then ∇∗∇u ρ→∇∗∇u in L 1loc (E )over U ,as ρ→0+.5.15.Proof of Theorem 5.7.Notefirstthatthestatement is actually ly,using partition of unity we see that it suffices to prove the inequality over every coordinate neigh-borhood from a covering of M .Let us fix such a neighborhood U and apply Proposition 5.14.Clearly u ρ∈C ∞(E |U ρ).By Proposition 5.9,for any ǫ>0,∆M |u ρ|ǫ≤Re ∇∗∇u ρ,u ρ|u ρ|ǫ −Re ∇∗∇u,u|u ρ|ǫ +Re ∇∗∇u,u ρ|u |ǫ .(5.18)18MAXIM BRAVERMAN,OGNJEN MILATOVIC,MIKHAIL SHUBINLetρ→0+.By Proposition5.14,∇∗∇uρ→∇∗∇u in L1loc(E|U).We also know that |uρ|−1ǫuρ <1.Therefore,thefirst term of the sum in(5.18)converges to0in L1loc(U)(hence in distributional sense).We know that∇∗∇u∈L1loc(E).We have already shown that asρ→0+,|uρ|ǫ→|u|ǫin L1loc(U),and hence,after passing to a subsequence,almost everywhere.Therefore,along the same subsequence,|uρ|−1ǫuρ→|u|−1ǫu almost everywhere.So the dominated convergence theorem implies that the second term of the sum in(5.18)goes to0in L1loc(U)(hence in distributional sense)along a sequence of positiveρ’s.Therefore,for all u∈L1loc(E)such that∇∗∇u∈L1loc(E),we obtain∆M|u|ǫ≤Re ∇∗∇u,uW denote the closure of W in M.By(2.2),there exist a W<1,and CW ∆M v +CESSENTIAL SELF-ADJOINTNESS OF SCHR ¨ODINGER TYPE OPERATORS ON MANIFOLDS 19In other words the operator |V −|is dominated by ∆M .Hence by Theorem X.18from [64]the quadratic form of |V −|will be dominated by the quadratic form of ∆M .More precisely,for any a ∈(aU of U is contained in W .Let φ,˜φ:M →[0,1]be smooth functions such that φ2+˜φ2≡1,the restriction of φto U is identically equal to 1,and supp φ⊂W .It follows that supp ˜φ⊂M \U .By IMS localization formula (cf.[23,§3.1],[75,Lemma 3.1]),we have ∇∗∇=φ∇∗∇φ+˜φ∇∗∇˜φ+12[∇∗∇,˜φ],˜φ ,(6.2)where [·,·]denotes the commutator bracket.Set J =12 [∇∗∇,˜φ],˜φ =−12|d ˜φ|2.Then J is a smooth function on M ,whose support is contained in W \U .With this notation (6.2)takes the form∇∗∇=φ∇∗∇φ+˜φ∇∗∇˜φ+J.(6.3)Let C >0be as in Corollary 6.3.Let b ≫C be a large number,which will be chosen later.Suppose that u ∈L 2(E )satisfies the equality(H V +b )u =0.(6.4)。
单位内部认证船舶英语考试(试卷编号211)
单位内部认证船舶英语考试(试卷编号211)1.[单选题]The AUTOCHIEF-IV main engine remote control system includes ______.A)AC-5B)AC-5C)digitalD)hydraulic答案:C解析:2.[单选题]The star formation is most commonly used and requires _____ on the alternator.A)oneB)twoC)threeD)four答案:D解析:3.[单选题]The difference between the fire detectors of the traditional and bus control type fire alarm systems is ______.A)theyB)theC)theD)the答案:D解析:4.[单选题]VLCC stands for______.A)veryB)veryC)veryD)very答案:C解析:5.[单选题]Once the power is recovered after blackout, the sequential start of automatic power plant would enable the motors in operation before the breakdown to start ______ automatically.C)respectivelyD)immediately答案:A解析:6.[单选题]Voltage will always lead current in a/an _____.A)capacitiveB)inductiveC)magneticD)resistive答案:B解析:【注】在电感性电路中，电压总是超前电流。
inductive circuit：电感性电路；capacitive circuit：电容电路7.[单选题]Which one is the function of steering gear?A)ToB)ToC)ToD)To答案:C解析:8.[单选题]When the voltage remains constant and the resistance is increased in a series circuit, the flow of current _____.A)increasesB)increasesC)remainsD)decreases答案:D解析:9.[单选题]The emergency generator or emergency battery is connected to _____ on most large ships.A)distributionB)sectionC)emergencyD)main答案:C解析:答案:B解析:11.[单选题]Switchboards may be of the dead-front type in which all live parts are installed behind _____ and only the operation handles and instruments are on the front.A)theB)theC)theD)the答案:C解析:12.[单选题]The Maritime Labour Convention, 2006, was issued by the _____.A)UNB)IMOC)ILOD)ITU答案:C解析:13.[单选题]The difference between magnetic heading and compass heading is called______.A)variationB)deviationC)compassD)drift答案:B解析:14.[单选题]Internet Explorer, Firefox, Google Chrome, Safari, and Opera are the major ______.A)webB)uniformC)fileD)Java答案:A解析:D)It's答案:D解析:16.[单选题]The number of cycles per second occurring in AC voltage is known as the_____.A)phaseB)frequencyC)waveD)half答案:B解析:17.[单选题]Copper is often used as an electrical conductor because it _____.A)hasB)hasC)isD)holds答案:C解析:【注】electrical conductor：导电体；opposition：阻挠，反对18.[单选题]A ground can be defined as an electrical connection between the wiring of a motor and its _____.A)shuntB)circuitC)metalD)inter-pole答案:C解析:19.[单选题]In more recent years, ______ has been used by civilians in many new ways to determine positions, such as in automobile and boat navigation, hiking, emergency rescue, and precision agriculture and mining.A)GPSB)GMDSSC)AISD)Navtex20.[单选题]The podded propulsor is widely adopted in the electric propulsion system. In this system, ______.A)theB)theC)theD)the答案:A解析:21.[单选题]_____ is used to produce electric power.A)AnB)AC)AD)A答案:A解析:22.[单选题]Prior to closing the breaker when paralleling two AC generators, the recommended practice is to have the frequency of the incoming machine _____.A)slightlyB)theC)slightlyD)have答案:C解析:23.[单选题]All echo-sounders can measure the ______.A)actualB)actualC)averageD)average答案:B解析:24.[单选题]The field coils _____ and the armature is _____. This is in fact the arrangement adopted for large, heavy duty alternators.A)stationaryB)stationaryC)rotate25.[单选题]What feature(s) may be found on certain satellite EPIRB units?A)StrobeB)EmergencyC)Float-freeD)All答案:D解析:【注】卫星EPIRB有闸门照明，406MHz紧急发射和自浮释放支架。
Counterpropagating Two-Soliton Solutions in the FPU Lattice
1
Introduction
It has long been of interest in applied math and physics to study how energy and mass are scattered or transfered during the collision of two or more coherent objects. We study this question for the Fermi-PastaUlam (FPU) problem: q ¨j = V ′ (qj +1 − qj ) − V ′ (qj − qj −1 ) j∈Z (1) which models an infinite chain of anharmonic oscillators with nearest-neighbor interaction potential V . On making the change of variables rj = qj +1 − qj and pj = q ˙j , the state variable u = (r, p) satisfies a system of first order Hamiltonian ODEs, ut = JH ′ (u) (2) where the Hamiltonian H is given by H (r, p) =
Counterpropagating two-soliton solutions in the FPU lattice
A. Hoffman and C.E.Wayne Boston University Department of Mathematics and Statistics and Center for BioDynamics 111 Cummington St. Boston, MA 02135 June 10, 2008
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Operatorial approach to the non-Archimedean stability of a PexiderK-quadratic functional equationA.B.C HAHBI a,A.C HARIFI a,*,B.B OUIKHALENE b,S.K ABBAJ aa Department of Mathematics,Faculty of Sciences,University of Ibn Tofail,Kenitra,Moroccob Laboratory LIRST,Polydisciplinary Faculty,Department ofMathematics,University Sultan Moulay Slimane,Beni-Mellal,MoroccoReceived6May2013;revised24November2013;accepted5January2014Available online17January2014Abstract.We use the operatorial approach to obtain,in non-Archimedean spaces,the Hyers–Ulam stability of the Pexider K-quadratic functional equationXfðxþkÁyÞ¼j gðxÞþj hðyÞ;x;y2E;k2Kwhere f;g;h:E!F are applications and K is afinite subgroup of the group of auto-morphisms of E and j is its order.Mathematics Subject Classification:Primary39B82;Secondary39B52Keywords:Group automorphisms;Jensen functional equation;Quadratic functionalequation;K-quadratic functional equation;Pexider functional equation;Non-Archimedean Hyers–Ulam stability1.I NTRODUCTIONThe concept of the stability for functional equations was introduced for thefirst time by Ulam in1940[17].Ulam started the stability by the following question Given a group G,a metric groupðG0;dÞ,a number d>0and a mapping f:G!G0 which satisfies the inequality dðfðxyÞ;fðxÞfðyÞÞ<d for all x;y2G,does there exist an*Corresponding author.Tel.:+2122012613279017.E-mail addresses:ab-1980@live.fr(A.B.Chahbi),charifi2000@yahoo.fr(A.Charifi),bbouikhalene@yahoo.fr(B.Bouikhalene),samkabbaj@yahoo.fr(S.Kabbaj).1319-5166ª2014King Saud University.Production and hosting by Elsevier B.V.All rights reserved. /10.1016/j.ajmsc.2014.01.001homomorphism h :G !G 0and a constant c >0,depending only on G and G 0such that d ðf ðx Þ;h ðx ÞÞ6cd for all x in G ?In 1941,Hyers [17]gave the partial solution to Ulam’s question in Banach spaces.The result of Hyers was extended,for additive mappings by Aoki [1]and later,for lin-ear mappings by Rassias [29].For more information on the history of the concept see[4,5,7,11,15,18–20,23,25,32–34,36–38]and especially the recent developments of the stability in [6,7].The first stability theorem for the K-quadratic functional equation was proved for K ¼f id g by Hyers–Ulam (1941)[17]and Rassias (1978)[31]and for K ¼fÀid ;id g by Skof (1983)[35]in Banach spaces.Cholewa (1984)[12]extended Skof’s result to an abelian group.Czerwik (1992)[13],in the spirit of Hyers–Ulam–Rassias generalized Skof’s theorem.Recently,the stability problem of the K-quadratic functional equation has been investigated by a number of mathematicians,the interested reader should refer to Ait Sibaha et al.[3],Bouikhalene et al.[8],Charifiet al.[9,10]and Lukasik [26],see also [6,20,22–24,31].In 1897,Hensel [16]discovered the p-adic numbers.Let p be a fixed prime number and x a nonzero rational number,there exists a unique integer v p ðx Þ2Z such that x ¼p v p ðx Þa b where a and b are integers co-prime to p.The function defined in Q by j x j p ¼p Àv p ðx Þis called a p-adic,a ultrametric or simply a non-Archimedean absolute va-lue on Q .So,with the p-adic absolute value Q is called a p-adic or a non-Archimedean field.The completion,denoted by Q p of Q with respect to the metric defined by the p-adic absolute value is called the p-adic numbers.Their elements are the formal series p v p ðx ÞP i P 0a i p i ,with a 0–0and j a i j 6p À1are integers.In general,by a non-Archimedean field,we mean a field k equipped with a function j j :k !½0;þ1Þ,called a non-Archimedean absolute value on k and satisfying the fol-lowing conditionsi.j x j ¼0()x ¼0ii.j xy j ¼j x jj y j ;x ;y 2kiii.j x þy j 6max ðj x j ;j y jÞ;x ;y 2k .We assume,throughout this paper that this value absolute is non-trivial i.e.,there exists an element k of k such that,j k j –0;1.By a non-Archimedean vector space,we mean a vector space E over a non-Archi-medean field k equipped with a function k k :E !½0;þ1Þcalled a non-Archimedean norm on E and satisfying the following propertiesi.k x k ¼0()x ¼0,ii.k k x k ¼j k jk x k ;ðk ;x Þ2k ÂE ,iii.k x þy k 6max ðk x k ;k y kÞ;x ;y 2E .The particularity of a non-Archimedean norm is the fact that they do not satisfy the Archimedean axiom and a sequence f x n g is Cauchy if and only if f x n þ1Àx n g converges to zero.In 2005Arriola and Beyer in [2],initiated the stability of Cauchy functional equa-tion over p-adic fields.In 2007,Sal Moslehian and Rassias [29]studied the stability68 A.B.Chahbi et al.Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation69of Cauchy and quadratic functional equations in non-Archimedean spaces.In[30],the author investigated,by using thefixed point method the non-Archimedean stability of a quadratic functional equation.Following this investigation,we deal with the operatorial approach,in a non-Archimedean space,the Hyers–Ulam stability of a Pexiderized version of the K-quadratic functional equation,XfðxþkÁyÞ¼j gðxÞþj hðyÞ;x;y2E;ð1:1Þk2Kwhere f;g;h:E!F are applications from a normed space E into a non-Archimedean space F;K is afinite abelian subgroup of the group of automorphisms of E and j denotes the order of K.The present paper is a continuation,in a non-Archimedean space of the previous work by Charifiet al.[9,10].The paper is organized as follows:in the second section we give some notions, notations and preliminary results.In the third section,we derive the non-Archimedean stability of Eq.(1.1).2.N OTATIONS AND PRELIMINARY RESULTSIn this section,we introduce some notions and notations.We give necessary results for the proof of Theorem 2.6.They are a faithful translation,in terms of a non-Archimedean norm of results which were given in the case of a usual norm by Hyers in[21].A function A:E!F between vector spaces E and F is said to be additive provided if AðxþyÞ¼AðxÞþAðyÞfor all x;y2E;in this case it is easily seen that AðrxÞ¼rAðxÞfor all x2E and all r2Q.Let k2N and A:E k!F be a function,then we say that A is k-additive provided if it is additive in each variable;in addition we say that A is symmetric provided if Aðx rð1Þ;x rð2Þ;...;x rðkÞÞ¼Aðx1;x2;...;x kÞwhenever x1;x2;...;x k2E and r is a permutation ofð1;2;...;kÞ.Let k2N and A:E k!F be symmetric and k-additive and let A kðxÞ¼Aðx;x;...;xÞfor x2E and note that A kðrxÞ¼r k A kðxÞwhenever x2E and r2Q.In this way a function A k:E!F which satisfies for all k2Q and x2E;A kðk xÞ¼k k A k will be called a rational-homogeneous form of de-gree k(assuming A k–0).A function p:E!F is called a generalized polynomial(GP)function of degree m2N if there exist a02E and a rational-homogeneous form A k:E!F(for 16k6m)of degree k,such thatX mA kðxÞpðxÞ¼a0þk¼1for x2E.Let F E denote the vector space(over afield K)consisting of all maps from E into F. For h2E define the linear difference operator D h on F E byD h fðxÞ¼fðxþhÞÀfðxÞð2:1Þfor f2F E and x2E.Notice that these difference operators commute(D h1D h2¼D h2D h1for all h1;h22E)and if h2E and n2N,then D nhthe n th iterate of D h satisfiesD n h fðxÞ¼X nk¼0ðÀ1ÞnÀknkfðxþkhÞfor f2F E and x;h2E.The following theorems were proved by Mazur and Orlicz[27,28],and in greater generality by Djokovic[14].Theorem2.1.Let n2N and f:E!F be a function between a vector space E and F,then the following assertions are equivalent,(1)D nh fðxÞ¼0for all x;h2E.(2)D hn ...D h1fðxÞ¼0for all x;h1;...;h n2E.(3)f is a GP function of degree at most nÀ1.Theorem2.2.Let A k:E!F be a rational-homogeneous form of degree k,then there exists a unique symmetric k-additive transformation A:E k!F such thatA kðxÞ¼Aðx;...;xÞ:The k-additive transformation is often called the polar of transformation A k and it is given by the formulaAðx1;...;x kÞ¼1k!D kx1...x kA kðxÞ:Lemma 2.3.Let E be a vector space,F a non-Archimedean Banach space and p2NÃ;j p j–1.Let d be afixed positive number and f:E!F be a function satisfying one of two conditions1ÞD2h fðxÞ6d;x;h2E;2ÞD h fðxÞÀD h fð0Þk k6d;x;h2E;ð2:2Þthen there exists an additive mapping A:E!F given byAðxÞ¼limn!þ1p n fðpÀn xÞand such thatk AðxÞÀfðxÞþfð0Þk6dProof.The proof is the same on the Assumption1)or2).Assume that1)is true andput g¼fÀfð0Þ,so by(2.2)we have D2h gðxÞ6d for all x and h in E.Replacing x by0 and h by x,we get70 A.B.Chahbi et al.k gð2xÞÀ2gðxÞk6dð2:3Þfor all x in E.Replacing h by x we obtaink gð3xÞÀ2gð2xÞþgðxÞk6d:ð2:4ÞTherefore,taking into account(2.3)and(2.4),we obtaink gð3xÞÀ3gðxÞk6d:ð2:5ÞWe will prove by mathematical induction thatk gðpxÞÀpgðxÞk6d:ð2:6ÞWe suppose that(2.6)true for all k6p.Replacing x byðpÀ1Þx and h by x in(2.2)we getjj gððpþ1ÞxÞÀ2gððpÞxÞþgððpÀ1ÞxÞjj6dð2:7Þand by hypothesis of induction we havejj gððpÀ1ÞxÞÀðpÀ1Þ2gðxÞjj6d:ð2:8ÞBy using the inequalities2.6,2.7and2.8,we get the resultk gðpxÞÀpgðxÞk6d;p2NÃ;x2E:We put qnðxÞ¼p n gðpÀn xÞ,we have when replaced x by pÀnÀ1x in(2.6) k gðpÀn xÞÀpgðpÀðnþ1ÞxÞk6d:ð2:9ÞBy multiplying this inequality by p n we getk q nþ1ðxÞÀq nðxÞk6j p n j d:ð2:10ÞThus,since j p j–0;1;qn ðxÞis a Cauchy sequence,as F is complete hence q nðxÞcon-verge to AðxÞ.Now we haveD2 h AðxÞ¼limn!þ1p n D2pÀn hgðpÀn xÞ6limn!þ1j p n j d¼0:We see that D2h AðxÞ¼0for all x and h in E.Thus,from Theorem1.1A is additive onE.By using(2.6)we have k qnðxÞÀgðxÞk6d,and taking limits as n!1,we obtain k AðxÞÀfðxÞþfð0Þk6dAðxÞ¼limn!þ1p n D pÀn x fð0Þ:This ends the proof of the lemma.hLemma2.4.Let E be a vector space,F a non-Archimedean space and p2NÃ;j p j–1. Let h:E2!F be either identically zero or else a rational-homogeneous form of degree kÀ1ðk>1Þin x for each y;q:E2!F a transformation of degree at most kÀ2in x which vanishes for x¼0and f:E!F be a function satisfying the inequality k fðxþyÞÀfðxÞÀfðyÞþfð0ÞÀqðx;yÞÀhðx;yÞk6d;x;y2E;ð2:11ÞThen hðx;xÞ¼kA kðxÞ,where A k:E!F is either identically zero or else a homogeneous form of degree k,Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation71A kðxÞ¼1k!limn!þ1j p kn j D kpÀn xfð0Þ:Moreover hðx;yÞis given by the formulahðx;yÞ¼1ðkÀ1Þ!limn!þ1j pðkÀ1Þn j D kÀ1pÀn xD yðfÞð0Þ:Proof.By the hypothesis made on h,there exists a map A:E k!F which is additive and symmetric in itsfirst kÀ1arguments,such thathðx;yÞ¼1ðkÀ1Þ!Aðx;...;x;yÞ:ð2:12ÞIn view of(2.1)and(2.11),treating y as a constant and using the increments x1;...;x kÀ1,we haveD x k1...x kÀ1yfðxÞÀDx kÀ11...x kÀ1qðx;yÞÀD kÀ1x1...x kÀ1hðx;yÞ6d:Since qðx;yÞis of degree at most kÀ2in x,by Theorem(2.1)D kÀ1x1...x kÀ1qðx;yÞ¼0:Also from(2.12)and Theorem(2.2)it follows thatD kÀ1x1...x kÀ1hðx;yÞ¼Aðx1;...;x kÀ1;yÞ:Thus we haveD kx1...x kÀ1y fðxÞÀAðx1;...;x kÀ1;yÞ6d:ð2:13ÞUsing the fact that,for each j;16j6kÀ1the k th difference in(2.13)is symmetric in all of its increments,then we obtain thatD kx1...x kÀ1y fðxÞÀAðx1;...;x jÀ1;y;x jþ1;...;x k;x jÞ6d:ð2:14ÞNow,from(2.13)and(2.14)we getk Aðx1;...;x kÀ1;yÞÀAðx1;...;x jÀ1;y;x jþ1;...;x k;x jÞk6dð2:15ÞIn order,to prove that A is additive in its last argument and symmetric in all of its arguments we distinguish two cases.(1)Case k>2.Since k>2,there exists an index i;16i6kÀ1such that i–j.In(2.13),replacing x i by pÀn x i,multiplying this inequality by p n and taking the limitas n!1,we obtain that A is symmetric in all of its arguments.Obviously A must be necessarily additive in its last argument.From inequality(2.13)we haveD kÀ1x1...x kÀ1D y fð0ÞÀAðx1;...;x kÀ1;yÞ6d:Now,take each x j¼pÀn x,multiply the last inequality by p nðkÀ1ÞðkÀ1Þ!,and then let n tend toinfinity.By(2.12)we get72 A.B.Chahbi et al.hðx;yÞ¼1ðkÀ1Þ!limn!þ1pðkÀ1Þn D kÀ1pÀn xD y fð0Þ:In a similar way,if we define A kðxÞ¼kÀ1hðx;xÞand use the fact that A is additive in each of its arguments,from(2.13)we obtain thatA kðxÞ¼1k!limn!þ1p kn D kpÀn xfð0Þ;ð2:16Þwhich gives the sought result.(2)Case k¼2.Then(2.15)becomesk Aðx1;yÞÀAðy;x1Þk6dfor all x1and y in E,where A is additive in thefirst argument.Replacing x1by pÀn x1, and multiplying by p n,where n and p are any positive integer,we obtain k Aðx1;yÞÀp n Aðy;pÀn xÞk6j p j n dand so by letting n tend to infinityAðx1;yÞ¼limn!þ1p n Aðy;pÀn x1Þ:ð2:17ÞThusAðx1;yþzÞ¼limn!þ1p n Aðyþz;pÀn x1Þ¼limn!þ1p n Aðy;pÀn x1Þþlimn!þ1p n Aðz;pÀn x1Þ¼Aðx1;yÞþAðx1;zÞso that A is additive in its second argument.Now,the symmetry is given by(2.17)and additivity of A,which completes the proof of Lemma2.2.hProposition2.5.Let E be a vector space,F be a non-Archimedean Banach space and p2NÃ;j p j–1.Let d be afixed positive number and f:E!F be a function satisfying the inequalityD mh1...h m fðxÞ6d;x;h1;...;h m2E:ð2:18ÞThen there exists a GP function pmÀ1:E!F which is of degree at most mÀ1,such that,k fðxÞÀp mÀ1ðxÞk6d for all x in E:ð2:19ÞMoreover pmÀ1is given by the formulapmÀ1ðxÞ¼fð0ÞþA1ðxÞþ...þA mÀ1ðxÞð2:20Þwhere each A k is either a homogeneous form of degree k or else identically zero.In addi-tion,the A k are given by the formulasOperatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation73A mÀ1ðxÞ¼1ðmÀ1Þ!limn!þ1pðmÀ1Þn D kpÀn xfð0Þ;ð2:21ÞA kðxÞ¼1k!limn!þ1p kn D kpÀn xfð0ÞÀX mÀ1j¼kþ1D kpÀn xA jð0Þ();ð2:22Þfor16k6mÀ2.Proof.We shall proceed by induction on m.From Lemma(2.1),the proposition holds for m¼2,with A1ðxÞ¼AðxÞ.Assuming that the theorem holds for a given positive integer m,we shall prove it for mþ1.By the hypothesis,we haveD mþ1h1...h mþ1fðxÞ6dfor all x and h j in E;ðj¼1...mþ1Þ.Putgðx;yÞ¼D y fðxÞ¼fðxþyÞÀfðxÞ:ð2:23ÞThen,treating y as afixed parameter we haveD mh1...h m gðx;yÞ¼D m h1...h m D y fðxÞ6dð2:24Þfor eachfixed y and all x and h j in E;ðj¼1...mþ1Þ.By(2.24)and the induction hypothesis,there exists,for eachfixed y2E,a map p:E!F defined by pðx;yÞfor all x in E which is of degree at most mÀ1in x such thatk gðx;yÞÀpðx;yÞk6dð2:25Þfor all x and y in E.More precisely pðx;yÞhas the formpðx;yÞ¼gð0;yÞþqðx;yÞþhðx;yÞð2:26Þwhere hðx;yÞis a homogeneous form of degree mÀ1or else is identically zero,while qðx;yÞis a map of degree at most mÀ2in x,and qð0;yÞ¼0.From,(2.23)and(2.26) and(2.25)we obtaink fðxþyÞÀfðxÞÀfðyÞþfð0ÞÀqðx;yÞÀhðx;yÞk6dð2:27Þfor all x and y in E.Now,in view of(2.27)and Lemma(2.24),the map A m:E!F defined by A mðxÞ¼mÀ1Hðx;xÞ;x2E,is either zero or else a homogeneous form of degree m.In addition,we haveA mðxÞ¼1m!limn!þ1p mn D mpÀn xfð0Þ;ð2:28ÞAccording to Lemma2[21]if we putf1ðxÞ¼fðxÞÀA mðxÞð2:29Þ74 A.B.Chahbi et al.then the map f1satisfies the conditions of Lemma(2.4)for k¼mÀ1;consequently, there exists the map A mÀ1:E!F given byA mÀ1ðxÞ¼1ðmÀ1Þ!limn!þ1pðmÀ1Þn D mÀ1pÀn xfð0ÞÀD mÀ1pÀn xA mð0Þn oð2:30Þwhich is either identically zero or else a homogeneous form of degree mÀ1.Again by Lemma2of[21],if we putf2ðxÞ¼f1ðxÞÀA mÀ1ðxÞð2:31Þthen f2satisfies the conditions of Lemma(2.4)for k¼mÀ2which leads to the exis-tence of the limitA mÀ1ðxÞ¼1m!limn!þ1pðmÀ1Þn D mÀ1pÀn xf2ð0Þ;ð2:32ÞandA mÀ2ðxÞ¼1ðmÀ2Þ!limn!þ1pðmÀ2Þn D mÀ2pÀn xfð0ÞÀD mÀ2pÀn xA mÀ2ð0Þn oð2:33Þcontinuing in this way,we arrive at the mapf mÀ2ðxÞ¼fðxÞÀA3ðxÞÀ...ÀA mðxÞð2:34Þwhere the A kðxÞare given by formula(2.22)in the statement of our theorem and where f mÀ2satisfies the inequalityk f mÀ2ðxþyÞÀf mÀ2ðxÞÀf mÀ2ðyÞþf mÀ2ð0ÞÀhðx;yÞk6dð2:35Þin which hðx;yÞis either identically zero or a homogeneous form of degree one in x. Applying Lemma(2.4)once more and putting A2ðxÞ¼1hðx;xÞ,we haveA2ðxÞ¼12limn!þ1p2n D2pÀn xf mÀ2ð0Þwhich,in view of(2.34),also agrees with formula(2.22)of the theorem.Finally on puttingf mÀ1ðxÞ¼f mÀ2ðxÞÀA2ðxÞ¼fðxÞÀA2ðxÞÀ...ÀA mðxÞð2:36Þand in view of Lemma(2)[14]for the case k¼2,we get the inequality k f mÀ1ðxþyÞÀf mÀ1ðxÞÀf mÀ1ðyÞþf mÀ1ð0Þk6dð2:37Þfor all x and y in E.Since f mÀ1satisfies(2.37),it follows from the Lemma(2.3)that there exists an additive mapA1ðxÞ¼limn!þ1p n D pÀn x f mÀ1ð0Þð2:38Þsatisfying the inequalityk f mÀ1ðxÞÀf mÀ1ð0ÞÀA1ðxÞk6dð2:39Þfor all x in E.Obviously A1ðxÞagrees with formula(2.22)by(2.36)and(2.34).By substituting(2.36)into(2.39)and observing that f mÀ1ð0Þ¼fð0Þ,we obtain Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation75k fðxÞÀfð0ÞÀA1ðxÞÀ...ÀA mðxÞk6dwhich is equivalent to conditions(2.21)and(2.22)of our proposition with m replaced by mþ1.Thus the induction proof has been completed and Proposition(2.5)estab-lished.hTheorem2.6.Let E be a vector space,F be a non-Archimedean Banach space and p2NÃ;j p j–1.Let d be afixed positive number and f:E!F be a function satisfying the inequalityD m h fðxÞ6d;x;h2E:ð2:40ÞThen there exists a GP function pmÀ1:E!F which is of degree at most mÀ1,such that, k fðxÞÀp mÀ1ðxÞk6d for all x in E:ð2:41ÞProof.We have f satisfyD m h fðxÞ¼X mk¼0ðÀ1ÞmÀkmkfðxþkhÞ6d;x;h2E:By putting g k¼ðÀ1ÞmÀkmkf,we haveX m k¼0gkðxþkhÞ6d;x;h2E:ð2:42ÞFor06j6k6m let a jk¼kÀj so that a jk–0if j<k and a kk¼0.For06k6m and x;y;h12E,ðxþmh1ÞþkðyÀh1Þ¼xþkyþa km h1From(2.40)and(2.42)we get thatX m k¼0ðg kðxþkyþa km h1ÞÀg kðxþkyÞÞ6maxX mk¼0gkðxþkyþd km h1Þ;X mk¼0gkðxþkyÞ()6dand since a mm¼0,we obtain thatX mÀ1 k¼0D akm h1g kðxþkyÞ6dð2:43ÞRepeating the argument that led from(2.40)to(2.41)wefind thatX mÀ2 k¼0D ak;mÀ1h2D akm h1g kðxþkyÞ6dð2:44Þ76 A.B.Chahbi et al.for all x ;y ;h 1;h 22E .Repeating this reasoning m À2times,we obtain thatk D a 01h m ...D a 0m h 1g 0ðx Þk 6dð2:45Þfor all x ;y ;h 1;...h m 2E .Since g 0¼ðÀ1Þmf and a 0k –0for 16k 6m ,the inequality (2.45)simply asserts thatk D h m ...D h 1f ðx Þk 6dð2:46Þfor all x ;h 1;...;h m 2E .Thus,by Proposition (2.5),there exists a GP function p m À1:E !F ,of degree at most m À1such thatk f ðx ÞÀp m À1ðx Þk 6d ;ð2:47Þwhich completes the proof of Theorem 2.6.h3.M AINRESULTIn this section we obtain the non-Archimedean Hyers–Ulam stability of the K-qua-dratic functional equation.Lemma 3.1.Let E be a vector space,F a non-Archimedean Banach space,K a finite subgroup of the group of automorphisms of E and j ¼cardK.Let f :E !F satisfyk R k 2K f ðx þk Áy ÞÀR k 2K f ðk Áy ÞÀj f ðx Þk 6d ;x ;y 2E :ð3:1ÞThenD j vf ðu ÞÀg ðv Þ6d j j j;u ;v 2E ;ð3:2Þwith g ðx Þ¼ÀP j À1i ¼0ðÀ1Þj Ài P j ij ¼1f ðR k 2K ij k Áx Þand K ij &K are pairwise different sets such that cardK ij ¼j Ài for j 21;...;ji;i 2f 0;...;j g .Proof.We haveX j i j ¼1X k 2Kf X k 2K ijk k Áx !¼j X j ij ¼1f Xk 2K ijk Áx !;x 2E ;ð3:3Þsince for all b 2K ,b K ij ¼K ik ;i 2f 0;...;j g ;j ;k 21;...;j i:Now,fix u ;v 2E .Letx i ¼u þiv ;y ij ¼Xk 2K ijk Áv ;i 2f 0;...;j g ;j 21;...;ji:For all b 2K ;i 2f 0;...:;j g ;j 21;...:;jiwe have the two following casesCase 1,b À12K ij .Thus i –j ,let k 21;...;j i þ1be such thatK ij ¼K ði þ1Þk S f b À1g .So,we havex i þb y ij ¼u þiv þX l 2K ijb l Áv ¼u þði þ1Þv þXl 2K ði þ1Þkb l Áv¼x i þ1þb y ði þ1ÞkCase 2,b À1R K ij .Since i –0,let k 21;...;ji À1besuchthat K ði À1Þk ¼K ij Sf b À1g .By a similar calculation to the previous we obtainx i þb y ij ¼x i À1þb y ði À1ÞkConsequently,from the above consideration we getXj À1i ¼0ðÀ1Þj ÀiX j i j ¼1X k 2Kf ðx i þk y ij Þ¼0:ð3:4ÞNow,in view of (3.1),(3.3)and (3.4)we havej D j vf ðu ÞÀjg ðv Þ ¼j X j i ¼0ðÀ1Þj Ài j i f ðu þiv Þþj X j À1i ¼0X j ij ¼1ðÀ1Þj Ài f X k 2K ij k Áv !¼j X j i ¼0ðÀ1Þj Ài j i f ðu þiv ÞþX j À10X j ij ¼1X k 2K ðÀ1Þj Àif X k 2K ij k k Áv !¼X j À1i ¼0ðÀ1Þj ÀiX j ij ¼1X k 2K f ðx i þk y ij ÞÀj f ðx i ÞÀX k 2K f ðk y ij Þ"#6d :This ends the proof.hTheorem 3.2.Let E be a vector space,F a non-Archimedean Banach space,K a finite sub-group of the group of automorphisms of E and j ¼cardK.Let f :E !F satisfyk R k 2K f ðx þk Áy ÞÀR k 2K f ðk Áy ÞÀj f ðx Þk 6d ;x ;y 2E :ð3:5ÞThen there exists a unique GP function p :E !F solution of (1,1),of degree at most j ,such thatk f ðx ÞÀf ð0ÞÀp ðx Þk 6d j j j:ð3:6ÞProof.According to (3.5),we haveD j vf ðu ÞÀg ðv Þ6d j j j ;u ;v 2E :ð3:7ÞReplacing u by u þv we getD j vf ðu þv ÞÀg ðv Þ6d j j j :ð3:8ÞBy (3.7)and (3.8)we obtainD j þ1vf ðu Þ6d j j j:ð3:9ÞThen by Theorem (2.6)there exists a GP function q :E !F ,of degree at most j ,such thatk f ðx ÞÀq ðx Þk 6d :ð3:10ÞFor 06k 6j ,there is a rational-homogeneous form of degree k A k :E !F such thatq ðx Þ¼f ð0ÞþXm ¼j k ¼1A k ðx Þ:ð3:11ÞBy (3.5)and (3.10),for all x ;y 2E ,Xk 2Kq ðx þk :y ÞÀj q ðx ÞÀj q ðy Þð3:12Þ6max Xk 2Kðq ðx þk :y ÞÀf ðx þk :y ÞÞ(;X k 2K ðq ðk :y ÞÀf ðk :y ÞÞ ;k j ðq ðx ÞÀf ðx ÞÞk ;k R k 2K f ðx þk Áy ÞÀR k 2K f ðk Áy ÞÀj f ðx Þk6d j j j:ð3:13ÞNow (3.11)says,in light of (3.12)that,for all x ;y 2E ,。