Parametrically Excited Hamiltonian Partial Differential Equations
Generalised chiral QED2 Anomaly and Exotic Statistics
a r X i v :h e p -t h /9705178v 1 23 M a y 1997Generalized Chiral QED 2:Anomaly and Exotic StatisticsFuad M.SaradzhevInstitute of Physics,Academy of Sciences of Azerbaijan,Huseyn Javid pr.33,370143Baku,AZERBAIJANABSTRACTWe study the influence of the anomaly on the physical quantum picture of the generalized chiral Schwinger model defined on S 1.We show that the anomaly i)results in the background linearly rising electric field and ii)makes the spectrum of the physical Hamiltonian nonrelativistic without a massive boson.The physical matter fields acquire exotic statistics.We construct explicitly the algebra of the Poincare generators and show that it differs from the Poincare one.We exhibit the role of the vacuum Berry phase in the failure of the Poincare algebra to close.We prove that,in spite of the background electric field,such phenomenon as the total screening of external charges characteristic for the standard Schwinger model takes place in the generalized chiral Schwinger model,too.PACS numbers:03.70+k ,11.10.Mn.1IntroductionThe two-dimensional QED with massless fermions,i.e.the Schwinger model(SM),demonstrates such phenomena as the dynamical mass generation and the total screening of the charge[1].Although the Lagrangian of the SM contains only masslessfields,a massive bosonfield emerges out of the interplay of the dynamics that govern the originalfields.This mass generation is due to the complete compensation of any external charge inserted into the vacuum.In the chiral Schwinger model(CSM)[2,3]the right and left chiral components of the fermionic field have different charges.The left-right asymmetric matter content leads to an anomaly.At the quantum level,the local gauge symmetry is not realized by a unitary action of the gauge symmetry group on Hilbert space.The Hilbert space furnishes a projective representation of the symmetry group[4,5,6].In this paper,we aim to study the influence of the anomaly on the physical quantum picture of the CSM.Do the dynamical mass generation and the total screening of charges take place also in the CSM?Are there any new physical effects caused just by the left-right asymmetry?These are the questions which we want to answer.To get the physical quantum picture of the CSM we needfirst to construct a self-consistent quantum theory of the model and then solve all the quantum constraints.In the quantization procedure,the anomaly manifests itself through a special Schwinger term in the commutator algebra of the Gauss law generators.This term changes the nature of the Gauss law constraint:instead of beingfirst-class constraint,it turns into second-class one.As a consequence,the physical quantum states cannot be defined as annihilated by the Gauss law generator.There are different approaches to overcome this problem and to consistently quantize the CSM. The fact that the second class constraint appears only after quantization means that the number of degrees of freedom of the quantum theory is larger than that of the classical theory.To keep the Gauss law constraintfirst-class,Faddeev and Shatashvili proposed adding an auxiliaryfield in such a way that the dynamical content of the model does not change[7].At the same time,after quantization it is the auxiliaryfield that furnishes the additional”irrelevant”quantum degrees of freedom.The auxiliaryfield is described by the Wess-Zumino term.When this term is added to the Lagrangian of the original model,a new,anomaly-free model is obtained.Subsequent canonical quantization of the new model is achieved by the Dirac procedure.For the CSM,the correspondig WZ-term is not defined uniquely.It contains the so called Jackiw-Rajaraman parameter a>1.This parameter reflects an ambiguity in the bosonization procedure and in the construction of the WZ-term.The spectrum of the new,anomaly-free model turns out to be relativistic and contains a relativistic boson.However,the mass of the boson also depends on the Jackiw-Rajaraman parameter[2,3].This mass corresponds therefore to the”irrelevant”quantum degrees of freedom.The quantum theory with such a parameter in the spectrum is not physical,i.e. thatfinal version of the quantum theory which we would like to get.The latter should not contain any nonphysical parameters,otherwise one can not say anything about a physical quantum picture.In another approach also formulated by Faddeev[8],the auxiliaryfield is not added,so the quantum Gauss law constraint remains second-class.The standard Gauss law is assumed to be regained as a statement valid in matrix elements between some states of the total Hilbert space,and it is the states that are called physical.The theory is regularized in such a way that the quantum Hamiltonian commutes with the nonmodified,i.e.second-class quantum Gauss law constraint.The spectrum turns out to be non-relativistic[9,10].Here,we follow the approach given in our previous work[11].The pecularity of the CSM is that its anomalous behaviour is trivial in the sense that the second class constraint which appears afterquantization can be turned intofirst class by a simple redefinition of the canonical variables.This allows us to formulate a modified Gauss law to constrain physical states.The physical states are gauge-invariant up to a phase,the phase being1-cocycle of the gauge symmetry group algebra.In [12,13,14],the modification of the Gauss law constraint is obtained by making use of the adiabatic approach.Contrary to[11]where the CSM is defined on R1,we suppose here that space is a circle of lengthL,−L2,so space-time manifold is a cylinder S1×R1.The gaugefield then acquires aglobal physical degree of freedom represented by the non-integrable phase of the Wilson integral on S1.We show that this brings in the physical quantum picture new features of principle.Another way of making two-dimensional gaugefield dynamics nontrivial is byfixing the spatial asymptotics of the gaugefield[15,16].If we assume that the gaugefield defined on R1diminishes rather rapidly at spatial infinities,then it again acquires a global physical degree of freedom.We will see that the physical quantum picture for the model defined on S1is equivalent to that obtained in[15,16].We consider the general version of the CSM with a U(1)gaugefield coupled with different charges to both chiral components of a fermionicfield.We show that the charges are not arbitrary,but satisfy a quantization condition.The SM where these charges are equal is a special case of the generalized CSM.This will allow us at each step of our consideration to see the distinction between the two models.We work in the temporal gauge A0=0in the framework of the canonical quantization scheme and the Dirac’s quantization method for the constrained systems[17].We use the system of units where c=1.In Section2,we quantize our model in two steps.First,the matterfields are quantized, while A1is handled as a classical backgroundfield.The gaugefield A1is quantized afterwords,using the functional Schrodinger representation.We derive the anomalous commutators with nonvanishing Schwinger terms which indicate that our model is anomalous.In Section3,we show that the Schwinger term in the commutator of the Gauss law generators is removed by a redefinition of these generators and formulate the modified quantum Gauss law constraint.We prove that this constraint can be also obtained by using the adiabatic approximation and the notion of quantum holonomy.In Section4,we construct the physical quantum Hamiltonian consistent with the modified quan-tum Gauss law constraint,i.e.invariant under the modified gauge transformations both topologically trivial and non-trivial.We introduce the modified topologically non-trivial gauge transformation op-erator and defineθ–states which are its eigenstates.We consider in detail the case of the SM and demonstrate its equivalence to the freefield theory of a massive scalarfield.For the generalized CSM,we define the exotic statistics matterfield and reformulate the quantum theory in terms of thisfield.In Section5,we construct two other Poincare generators,i.e.the momentum and the boost.We act in the same way as before with the Hamiltonian,namely we define the physical generators as those which are invariant under both topologically trivial and non-trivial gauge transformations.We show that the algebra of the constructed generators is not a Poincare one and that the failure of the Poincare algebra to close is connected to the nonvanishing vacuum Berry curvature.In Section6,we study the charge screening.We introduce external charges and calculate(i)the energy of the ground state of the physical Hamiltonian with the external charges and(ii)the current density induced by these charges.Section7contains our conclusions and discussion.2Quantization Procedure2.1Classical TheoryThe Lagrangian density of the generalized CSM isL=−10,1,γ0=σ1,γ1=−iσ2,γ0γ1=γ5=σ3,σi(i=2(1±γ5)ψ.In the temporal gauge A0=0,the Hamiltonian density isH=H EM+H F,(2) where H EM=12)=A1(L2)=ψ±(L¯he±λ}ψ±,generated byG=∂1E+e+j++e−j−,λbeing a gauge function,as well as under global gauge transformations of the right-handed and left-handed Diracfields which are generated byQ±=e± L/2−L/2dxj±(x).Due to the gauge invariance,the Hamiltonian density is not uniquely determined.On the con-strained submanifold G≈0of the full phase space,the Hamiltonian density˜H=H+v H·G,(4) where v H is an arbitrary Lagrange multiplier which can be any function of thefield variables and their momenta,reduces to the Hamiltonian density H.In this sense,our theory cannot distinguish between H and˜H,and so both Hamiltonian densities are physically equivalent to each other.For arbitrary e+,e−the gauge transformations do not respect the boundary conditions 3.The gauge transformations compatible with the boundary conditions must be either of the formλ(L2)+¯h2πe+=N,N∈Z,(6)or of the formλ(L2)+¯h2πe−=N∈Z.(7)Eqs.6or7imply the charge quantization condition for our system.Without loss of generality, we choose the condition 6.For N=1,e−=e+and we have the standard Schwinger model.For N=0,we get the model in which only the right-handed component of the Diracfield is coupled to the gaugefield.From Eq.5we see that the gauge transformations under consideration are divided into topo-logical classes characterized by the integer n.Ifλ(L2),then the gauge transformation istopologically trivial and belongs to the n=0class.If n=0it is nontrivial and has winding number n.Given Eq.5,the nonintegrable phaseΓ(A)=exp{i¯he+L b(t)}.In contrast toΓ(A),the line integralb(t)=1e+Ln.By a non-trivial gauge transformation of the formg n=exp{i2πe+L].The configurations b=0and b=¯h2πe+L.2.2Quantization and AnomalyThe eigenfunctions and the eigenvalues of thefirst quantized fermionic Hamiltonians ared± x|n;± =±εn,± x|n;± ,wherex|n;± =1L exp{i¯hεn,±·x},εn,±=2π2π).We see that the spectrum of the eigenvalues depends on b.For e+b Le+L,the energies ofεn,+decrease by¯h2πL N.Some of energy levels change sign.However,the spectrum atthe configurations b=0and b=¯h2π2π¯h (and e−b L2π¯h]and{e±b L2π¯h],a†n|vac;A;+ =0for n≤[e+b Landb n |vac;A ;− =0for n ≤[e −b L2π¯h ].(11)Excited states are constructed by operating creation operators on the Fock vacuum.In the ζ–function regularization scheme,we define the action of the functional derivative on first quantized fermionic kets and bras byδδA 1(x )|n ;± ·|λεm,±|−s/2,n ;±|←δδA 1(x )|m ;± m ;±|·|λεm,±|−s/2.From 8we get the action ofδδA 1(x )a n =−lim s →0m ∈Zn ;+|δδA 1(x )a †n=lims →0m ∈Zm ;+|δδA 1(x )on b n ,b †n can be written analogously.Next we define the quantum fermionic currents and fermionic parts of the second-quantized Hamiltonian asˆj s ±(x )=12L /2−L /2dx (ψ†s ±d ±ψs ±−ψs ±d ⋆±ψ†s±).Substituting 8into these expressions,we obtainˆj s ±(x )=n ∈Z1Lnx }ρs ±(n ),whereρs +(n )≡k ∈Z12[b †k ,b k +n ]−·|λεk,−|−s/2|λεk +n,−|−s/2are momentum space charge density (or current)operators,andˆH s ±(x )=n ∈Z1Lnx }H s±(n ),H s ±(n )≡H s 0,±(n )∓e ±bρs±(n ),(12)whereH s0,+(n)≡¯hπ2[a†k,a k+n]−·|λεk,+|−s/2|λεk+n,+|−s/2,H s0,−(n)≡¯hπ2[b k+n,b†k]−·|λεk,−|−s/2|λεk+n,−|−s/2. The charges corresponding to the currentsˆj s±(x)areˆQ s±=e± L/2−L/2dxˆj s±(x)=e±ρs±(0).With Eqs.10and11,we have for the vacuum expectation values:vac;A;±|ˆj±(x)|vac;A;± =−12(ξ++ξ−),whereη±≡±lim s→01λ k∈Z|λεk,±|−s+1.Taking the sums,we obtainη±=±22π¯h}−1L(({e±b L2)2−12η±,ˆQ ±=e±:ρ±(0):−L2ξ±,where double dots indicate normal ordering with respect to|vac,A ,ˆH 0,+=¯h2π2π¯h]ka†k a k|λεk,+|−s− k≤[e+b LLlims→0{k>[e−b L2π¯h]kb†k b k|λεk,−|−s}and:ρ+(0):=lims→0{ k>[e+b L2π¯h]a k a†k|λεk,+|−s},:ρ−(0):=lims→0{ k≤[e−b L2π¯h]b k b†k|λεk,−|−s}.The operators:ˆj±(x):and:ˆH±:are well defined when acting onfinitely excited states which have only afinite number of excitations relative to the Fock vacuum.To construct the quantum electromagnetic Hamiltonian,we quantize the gaugefield using the functional Schrodinger representation.In this representation,when the vacuum and excited fermionic Fock states are functionals of A1,the gaugefield operators are represented asˆA1(x)→A1(x),ˆE(x)→−i¯hδL pxαp.Since A1(x)is a real function,αp satisfiesαp=α⋆−p.The Fourier expansion for the canonical momentum conjugate to A1(x)is thenˆE(x)=1L¯h p∈Z p=0e−i2πdαp, whereˆπb≡−i¯h dL exp{i2πL ¯h2ddb−1dα−p+qd2Lˆπ2b−1dαqd2(ξ++ξ−).If we multiply two operators that arefinite linear combinations of the fermionic creation and annihilation operators,theζ–function regulated operator product agrees with the naive product. However,if the operators involve infinite summations their naive product is not generally well defined. We then define the operator product by mutiplying the regulated operators with s large and positive and analytically continue the result to s=0.In this way we obtain the following relations[ρ±(m),ρ±(n)]−=±mδm,−n,(15) [H0,±(n),H0,±(m)]−=±¯h2π[ˆH0,±,ρ±(m)]−=∓¯h2πdbρ±(m)=0,d2π¯hδp,±m,d2π¯hδp,±m,(p>0).(16) The quantum Gauss operator isˆG=ˆG0+2πLpx−ˆG−(p)e−i2πLe+ρN(0),ˆG ±(p)≡¯h pd2πρN(±p)andρN=ρ++Nρ−is momentum space total charge density operator.Using15and16,we easily get thatρ+(±p)(andρ−(±p))are gauge invariant.For example, forρ+(±p)we have:[ˆG+(p),ρ+(±q)]−=0,[ˆG−(p),ρ+(±q)]−=0,(p>0,q>0).The operatorsˆG±(p)don’t commute with themselves,[ˆG+(p),ˆG−(q)]−=(1−N2)e2+L24π2d3Quantum Constraints3.1Quantum SymmetryIn non-anomalous gauge theories,Gauss law is considered to be valid for physical states only.This identifies physical states as those which are gauge-invariant.The problem with the anomalous be-haviour of the generalized CSM,in terms of states in Hilbert space,is apparent:owing to theSchwinger terms we cannot require that states be annihilated by the Gauss law generators ˆG±(p ).Let us represent the action of the topologically trivial gauge transformations by the operatorsU 0(τ)=exp {i¯hp>0(ˆG+τ++ˆG −τ−)}(17)with τ0,τ±(p )smooth,thenU −10(τ)α±p U 0(τ)=α±−ipτ∓(p ),U −1(τ)d dα±p∓i 2π)2τ±(p ),(p >0).The composition law for the operators U 0isU 0(τ(1))U 0(τ(2))=exp {2πiω2(τ(1),τ(2))}U 0(τ(1)+τ(2)),whereω2(τ(1),τ(2))≡−i2π¯h )2p>0p (τ(1)−τ(2)+−τ(1)+τ(2)−)is a 2-cocycle of the gauge group algebra.Thus for N =±1we are dealing with a projectiverepresentation.The 2-cocycle ω2(τ(1),τ(2))is trivial,since it can be removed by a simple redefinition of U 0(τ).Indeed,the modified operators˜U0(τ)=exp {i 2πα1(γ;τ)}·U 0(τ),(18)whereα1(γ,τ)≡−12π¯h )2p>0(α−p τ−−αp τ+)is a 1-cocycle,satisfy the ordinary composition law˜U0(τ(1))˜U 0(τ(2))=˜U 0(τ(1)+τ(2)),i.e.the action of the topologically trivial gauge transformations represented by 18is unitary.The modified Gauss law generators corresponding to 18areˆ˜G±(p )=ˆG ±(p )±18π2α±p .(19)The generators ˆ˜G±(p )commute:[ˆ˜G+(p ),ˆ˜G −(q )]−=0.This means that Gauss law can be maintained at the quantum level for N=±1,too.We define physical states as those which are annihilated byˆ˜G±(p)[11]:ˆ˜G(p)|phys;A =0.(20)±The zero componentˆG0is a sum of quantum generators of the global gauge transformations of the right-handed and left-handed fermionicfields,so the other quantum constraints are:ρ±(0):|phys;A =0.(21) It follows from20that the physical states|phys;A respond to a gauge transformation from the zero topological class with a phase:U0(τ)|phys;A =exp{−i2πα1(γ;τ)}|phys;A .(22) Only for models without anomaly,i.e.for N=±1,this equation translates into the statement that physical states are gauge invariant.Equation22expresses in an exact form the nature of anomaly in the CSM.At the quantum level the gauge invariance is not broken,but realized projectively.The1-cocycleα1occuring in the projective representation contributes to the commutator of the Gauss law generators by a Schwinger term and produces therefore the anomaly.3.2Adiabatic ApproachLet us show now that we can come to the quantum constraints20and21in a different way,using the adiabatic approximation[23,24].In the adiabatic approach,the dynamical variables are divided into two sets,one which we call fast variables and the other which we call slow variables.In our case, we treat the fermions as fast variables and the gaugefields as slow variables.Let A1be a manifold of all static gaugefield configurations A1(x).On A1a time-dependent gaugefield A1(x,t)corresponds to a path and a periodic gaugefield to a closed loop.We consider the fermionic part of the second-quantized Hamiltonian:ˆH F:which depends on t through the background gaugefield A1and so changes very slowly with time.We consider next the periodic gaugefield A1(x,t)(0≤t<T).After a time T the periodicfield A1(x,t)returns to its original value:A1(x,0)=A1(x,T),so that:ˆH F:(0)=:ˆH F:(T).At each instant t we define eigenstates for:ˆH F:(t)by:ˆH F:(t)|F,A(t) =εF(t)|F,A(t) .The state|F=0,A(t) ≡|vac,A(t) is a ground state of:ˆH F:(t),:ˆH F:(t)|vac,A(t) =0.The Fock states|F,A(t) depend on t only through their implicit dependence on A1.They are assumed to be periodic in time,|F,A(T) =|F,A(0) ,orthonormalized,F′,A(t)|F,A(t) =δF,F′,and nondegenerate.The time evolution of the wave function of our system(fermions in a background gaugefield)is clearly governed by the Schrodinger equation:∂ψ(t)i¯h¯h T0dt·εF(t),whileT0dt L/2−L/2dx˙A1(x,t) F,A(t)|iδγBerryF≡δA1(x,t)|F,A(t) ,(24) then= T0dt L/2−L/2dx˙A1(x,t)A F(x,t).γBerryFWe see that upon parallel transport around a closed loop on A1the Fock state|F,A(t) acquiresan additional phase which is integrated exponential of A F(x,t).Whereas the dynamical phaseγdynF provides information about the duration of the evolution,the Berry’s phase reflects the nontrivial holonomy of the Fock states on A1.However,a direct computation of the diagonal matrix elements ofδδδA1(x,t)A F(y,t)−2π2¯h2 n>01L n(x−y))=(1−N2)e2+2ǫ(x−y)−1The corresponding U(1)connection is easily deduced asA F=0(x,t)=−12 T0dt L/2−L/2dx L/2−L/2dy˙A1(x,t)F F=0(x,y,t)A1(y,t).In terms of the Fourier components,the connection A F=0is rewritten as vac,A(t)|ddα±p(t)|vac,A(t) ≡A±(p,t)=±(1−N2)e2+L2pα∓p,so the nonvanishing curvature isF+−(p)≡d dαpA−=(1−N2)e2+L2p.A parallel transportation of the vacuum|vac,A(t) around a closed loop in(αp,α−p)–space(p>0) yields back the same vacuum state multiplied by the phaseγBerry F=0=(1−N2)e2+L2piαp˙α−p.This phase is associated with the projective representation of the gauge group.For N=±1,when the representation is unitary,the curvature F+−and the Berry phase vanish.As mentioned in the beginning of this Section,the projective representation is trivial and the2-cocycle in the composition law of the gauge transformation operators can be removed by a redefinition of these operators.Analogously,if we redefine the momentum operators asddα±p≡d8π2¯h21 dα±p|vac,A(t) =0,˜F+−=˜ddαp˜A−=0.However,the nonvanishing curvature F+−(p)shows itself in the algebra of the modified momentum operators which are noncommuting:[˜ddα−q]−=F+−(p)δp,q.Following27,we modify the Gauss law generators asˆG ±(p)−→ˆ˜G±(p)=¯h p˜d2πρN(±p)that coincides with19.The modified Gauss law generators have vanishing vacuum expectation values,vac,A(t)|ˆ˜G±(p,t)|vac,A(t) =0.This justifies the definition20.For the zero componentˆG0,the vacuum expectation valuevac,A(t)|ˆG0|vac,A(t) =−12(e+η++e−η−)=1The quantum theory consistently describing the dynamics of the CSM should be definitely compatible with20.The corresponding quantum Hamiltonian is then defined by the conditions[ˆ˜H,ˆ˜G±(p)]−=0(p>0)(29)which specify thatˆ˜H must be invariant under the modified topologically trivial gauge transformations generated byˆ˜G±(p).We have in29a system of non-homogeneous equations in the Lagrange multipliersˆv H,±which become operators at the quantum level.The solution of these equations isˆv H,±(p)=¯hp2{pd4π¯h)2α∓p}.Substituting this expression forˆv H,±(p)into the quantum counterpart of28,on the physical states |phys;A we obtain1L2¯h2 p>0(d dα−p−1dαp,˜ddα±by˜d2L ˆπ2b−1dαp,˜d2Lˆπ2b+V(ρN;ρN),whereV(ρN;ρN)≡e2+Lp2ρN(−p)ρN(p)is the energy of the Coulomb current-current interaction.In order to make the dependence on N for the Hamiltonian more obvious,let us representρN asρN=12(1−N)σ,whereρ≡ρ1=ρ++ρ−,σ≡ρ−1=ρ+−ρ−,and[ρ(p),ρ(q)]−=[σ(p),σ(q)]−=0,[σ(p),ρ(q)]−=2pδp,−q.Then the Coulomb interaction energy takes the formV(ρN;ρN)=14(1−N)2V(σ;σ)+12Lˆπ2b+V(ρ;ρ).For N=−1,the momentum space electric charge density operator isσ(p)andˆ˜H EM =12π¯h:[e+b L2π¯h]+n,ˆψ+→exp{i2πn2π¯h ]→[e−b LLx}ˆψ−.The action of the topologically nontrivial gauge transformations on the states can be represented by the operatorsU n=exp{−i2π¯h ]−2πd[e+b L nρN(n)and U0is given by17.To identify the gauge transformation as belonging to the n th topological class we use the index n in31.The case n=0corresponds to the topologically trivial gauge transformations.The topologically nontrivial gauge transformation operators satisfy the same composition law as the topologically trivial ones.The modified operators are˜U n =exp{−i¯hˆTb})n|phys;A .Among all states|phys;A one may identify the eigenstates of the operators of the physical variables.The action of the topologically nontrivial gauge transformations on such states may, generally speaking,change only the phase of these states by a C–number,since with any gauge transformations both topologically trivial and nontrivial,the operators of the physical variables and the observables cannot be ing|phys;θ to designate these physical states,we haveexp{∓i¯h ˆTb})n|phys;A(so calledθ–states[26,27]),where|phys;A is an arbitrary physical state from20.In one dimension the parameterθis related to a constant background electricfield.To show this, let us introduce states which are invariant even against the topologically nontrivial gauge transfor-mations.Recalling that[e+b L2π¯h]θ}|phys;θ .(32)The new states|phys continue to be annihilated byˆ˜G±(p),and are also invariant under the topo-logically nontrivial gauge transformations.The electromagnetic part of the Hamiltonian transforms asˆHEM→exp{i[e+b L2π¯h]θ}=12L¯h2 p>0[˜d dα−p]+,i.e.in the new Hamiltonian the momentumˆπb is supplemented by the electricfield strength Eθ≡e+The condition34can be then rewritten as a system of linear equations in(β0,β±).We can easilyfind a solution of these equations,which gives us(β0,β±)as functions of[e+b L2π¯h}.However,these constants are irrelevant for our consideration and we neglect them.Finding(β0,β±)from34and substituting them into the expression33,on the physical states we obtainˆ˜H|phys;A =ˆHphys|phys;AwhereˆH phys =ˆH physF+ˆH physEM,ˆH physF=ˆH0,++ˆH0,−−1L¯h(1+N2)([e+b LL ¯h[e+b L2L ˆπ2b+V(ρN;ρN)+e2+L2π¯h] p∈Z p=0(−1)p 24(1−N2)2([e+b LL¯h p>0|λεp,±|−sρs±(−p)ρs±(p).Eqs.35and36give us a physical Hamiltonian invariant under both topologically trivial andnontrivial gauge transformations,ˆH physF andˆH physEMbeing invariant separately.The last two terms in35make invariant the free fermionic part of the Hamiltonian,while the ones in36the electromagnetic part.For N=±1,the last two terms in36vanish.These terms are therefore caused by the anomaly and represent new types of interaction which are absent in the nonanomalous models.The new interactions admit the following interpretation.Let us combine the last term in36with the kinetic part of the electromagnetic Hamiltonian,then124(1−N2)2([e+b L2L2 L/2−L/2dx(ˆπb−L E(x))2,i.e.the momentumˆπb is supplemented by the linearly rising electricfield strengthE(x)≡−e+2π¯h].As in four-dimensional models of a relativistic particle moving in an externalfield,we may define a generalized momentum operator in the formˆ˜πb(x)≡ˆπb−L E(x).The commutation relations for ˆ˜πb are[ˆ˜πb (x ),ˆ˜πb (y )]−=i (1−N 2)e 2+LL(1−N 2)[e +b L2L 2ˆ˜π2b→14π2(1−N 2)[e +b Lp 2ρN (p )=−e 2+L2p 2ρbgrd ·ρN (p ).It is just the background linearly rising electric field that couples b to the fermionic physical degreesof freedom in the Coulomb interaction.As a consequence,the eigenstates of the physical Hamiltonian are not a direct product of the purely fermionic Fock states and wave functionals of b .This is a common feature of gauge theories with anomaly.That the Hilbert space in such theories is not a tensor product of the Hilbert space for a gauge field and the fixed Hilbert space for fermions was shown in [6],[7].The background charge interpretation is related to the definition of the Fock vacuum.The definition given in Eqs.10-11depends on [e +b L2π¯h]is fixed.The values of the gauge field in regions of different [e +b L2π¯h]changes,then there is a nontrivial spectral flow,i.e.some of energy levels of the first quantized fermionic Hamiltonians cross zero and change sign.This means that the definition of the Fock vacuum changes.The charge operators ˆQ ±also change.Let :ˆQ (0)±:be charge operators defined in the region where[e +b L 2π¯h]the charge operators become :ˆQ (0)±:∓e ±[e ±b L。
《2024年无穷维Hamilton算子的拟谱》范文
《无穷维Hamilton算子的拟谱》篇一一、引言在数学物理领域，无穷维Hamilton算子是一个重要的研究对象。
它涉及到量子力学、统计力学、场论等多个领域，是描述物理系统动态行为的关键工具。
近年来，随着科学技术的飞速发展，对无穷维Hamilton算子的研究也日益深入。
本文旨在探讨无穷维Hamilton算子的拟谱问题，分析其研究现状及未来发展方向。
二、无穷维Hamilton算子的基本概念无穷维Hamilton算子是一种描述物理系统动态行为的数学工具，其基本思想是将系统的能量函数（即Hamilton函数）与时间演化算子相结合，从而得到系统的动态演化规律。
在无穷维空间中，Hamilton算子具有丰富的谱结构和动力学性质，对于理解物理系统的行为具有重要意义。
三、无穷维Hamilton算子的拟谱研究拟谱是研究Hamilton算子谱结构的一种重要方法。
通过拟谱方法，可以了解Hamilton算子的本征值、本征函数以及谱的分布情况，从而揭示系统的动态行为和稳定性。
目前，对于无穷维Hamilton算子的拟谱研究已经取得了一定的成果。
首先，针对不同类型的无穷维Hamilton系统，研究者们提出了各种拟谱方法。
例如，对于具有周期性边界条件的系统，可以采用Floquet理论；对于具有混沌特性的系统，可以利用Lyapunov指数等方法进行分析。
这些方法的应用使得我们能够更深入地了解无穷维Hamilton算子的谱结构。
其次，在拟谱研究过程中，还涉及到了许多数学技巧和工具。
例如，利用函数分析、微分方程、线性代数等数学知识，可以更好地描述和解决无穷维Hamilton算子的谱问题。
此外，计算机技术的发展也为拟谱研究提供了强大的支持，使得我们可以进行更加精确和高效的数值计算。
四、无穷维Hamilton算子拟谱的研究现状目前，无穷维Hamilton算子的拟谱研究已经取得了重要的进展。
研究者们针对不同类型的系统和问题，提出了各种拟谱方法和技巧。
2Stochastic averaging of quasi-generalized Hamiltonian systems
ABSTRACT
A stochastic averaging method for generalized Hamiltonian systems (GHS) subject to light dampings and weak stochastic excitations is proposed. First, the GHS are briefly reviewed and classified into five classes, i.e., non-integrable GHS, completely integrable and non-resonant GHS, completely integrable and resonant GHS, partially integrable and non-resonant GHS and partially integrable and resonant GHS. Then, the averaged Itoˆ and FPK equations and the drift and diffusion coefficients for the five classes of quasi-GHS are derived. Finally, the stochastic averaging for a nine-dimensional quasi-partially integrable GHS is given to illustrate the application of the proposed procedure, and the results are confirmed by using those from Monte Carlo simulation.
《2024年无穷维Hamilton算子的特征值问题》范文
《无穷维Hamilton算子的特征值问题》篇一摘要：本文探讨了无穷维Hamilton算子的特征值问题，首先对相关概念进行了阐述，接着对问题的基本性质进行了分析，然后利用数学分析方法和技巧对问题进行了解析和求解，最后对研究结果进行了总结和展望。
一、引言在数学物理和量子力学中，Hamilton算子是一个重要的概念，它描述了系统的能量和动力学特性。
随着研究的深入，人们开始关注无穷维Hamilton算子的特征值问题，这涉及到更广泛的物理系统和更复杂的数学结构。
本文旨在探讨无穷维Hamilton算子的特征值问题，为相关研究提供理论依据。
二、Hamilton算子及其基本性质Hamilton算子是一个自伴的线性算子，其定义在Hilbert空间上。
在无穷维的情况下，Hamilton算子具有更复杂的性质和更广泛的应用。
特征值问题通常指的是寻找满足特定条件的算子特征向量的问题。
对于Hamilton算子而言，其特征值和特征向量描述了系统的能量状态和波函数。
三、无穷维Hamilton算子的特征值问题无穷维Hamilton算子的特征值问题是一个复杂的数学问题，涉及到无穷维Hilbert空间中的自伴算子。
在这个问题中，我们需要找到满足一定条件的特征向量和特征值，这些特征向量和特征值描述了系统的能级和对应的波函数。
这个问题具有挑战性，因为需要处理无穷维的Hilbert空间和自伴算子。
四、问题的分析和求解为了解决无穷维Hamilton算子的特征值问题，我们采用了数学分析的方法和技巧。
首先，我们分析了Hamilton算子的基本性质和结构，包括其自伴性、正定性等。
然后，我们利用变分法、微分方程等数学工具对问题进行求解。
具体而言，我们首先通过构造适当的试探函数空间，然后利用自伴性和正定性等性质将原问题转化为一个有限维的优化问题。
接着，我们利用微分方程等工具对优化问题进行求解，得到了一组特征向量和特征值的近似解。
最后，我们通过数值分析和实验验证了我们的解的正确性和有效性。
《2024年无穷维Hamilton算子的拟谱》范文
《无穷维Hamilton算子的拟谱》篇一摘要：本文旨在探讨无穷维Hamilton算子的拟谱问题。
首先，我们将介绍Hamilton算子的基本概念及其在物理和数学领域的重要性。
随后，我们将阐述拟谱方法的基本原理和在处理无穷维系统中的优势。
最后，我们将详细描述我们的研究方法和结果，以及这些结果对无穷维系统理论和相关领域研究的潜在贡献。
一、引言Hamilton算子是一种广泛应用于量子力学、光学、电磁学等领域的数学工具。
在处理具有无穷维度的系统时，Hamilton算子的谱问题变得尤为重要。
然而，由于无穷维系统的复杂性，直接求解其谱往往面临巨大挑战。
因此，寻求有效的拟谱方法成为研究的关键。
二、Hamilton算子的基本概念Hamilton算子是一种描述系统动力学的算子，具有特定的形式和性质。
在量子力学中，它描述了粒子的能量和动量关系。
在光学和电磁学中，它用于描述光场或电磁场的演化。
由于系统的复杂性，Hamilton算子往往具有无穷维度，使得其谱的求解变得困难。
三、拟谱方法的基本原理及优势拟谱方法是一种用于处理无穷维系统的数学方法。
它通过将系统在一定的近似空间中进行展开，将原本复杂的无穷维问题转化为有限维问题进行处理。
这种方法在处理具有复杂相互作用的系统时具有显著优势，能够有效地降低问题的复杂度。
四、无穷维Hamilton算子的拟谱研究针对无穷维Hamilton算子的拟谱问题，我们采用了一种基于拟谱方法的解决方案。
首先，我们选择了一个合适的近似空间，将Hamilton算子在这个空间中进行展开。
然后，我们利用数值方法求解展开后的有限维问题，得到Hamilton算子的近似谱。
最后，我们通过分析近似谱的性质，了解原系统的动力学特性。
五、研究方法与结果我们采用了一种基于多项式展开的拟谱方法。
首先，我们选择了一组合适的多项式基函数作为近似空间的基底。
然后，我们将Hamilton算子在这组基底上进行展开，得到一个有限维的矩阵表示。
一类相对非线性薛定谔方程解的存在性
2019,39A (1):95-104数学物理学报http://actam s.w 一类相对非线性薛定谔方程解的存在性+W 邱雯W 张贻民 **3Abdelgadir Ahmed Adam(工武汉理工大学数学科学研究中心武汉430070; 2武汉理工大学理学院数学系 武汉430070;3Department of Mathematics, Taibah University Madinah Munawwarah, Saudi Arabia)摘要：利用临界点理论考虑了一类相对非线性薛定谔方程，主要通过变量代换将相对非线性薛 定谔方程转化成半线性椭圆型方程.首先考虑位势函数为零时，将经典的场方程结果推广到了 相对非线性薛定谔方程；而后利用临界点理论得到了有界位势情形方程非平凡解的存在性，在此情形，改进了文献[12-13]中的超线性条件.关键词：山路引理；相对非线性薛定谔方程；集中紧引理.M R (2010)主题分类：35J20; 35J62 中图分类号：O175.25 文献标识码：A文章编号：1003-3998(2019)01-95-101引言考虑如下相对非线性薛定谔方程izt = —A z + W (x )z — h (\z \2)z — k A \/l + z 2x G R N ,(1.1)其中z : R x R N — C , 是给定的常数，W : R N — R 是给定的位势，h 是实函数.方程(1.1)来源于物质中的高功率超短激光方程（参见文献[1-2, 4])•文献[1-2]研究了方程（1.1) 极小解的整体存在性以及一维柯西问题解的整体存在性和渐近行为.C h e n 和Sudan 证 明了参数小于零时，方程（1.1)具有负能量的解，且当时间足够大时，解不能消散.关于 方程（1.1)更多的背景可以参见文献[1-2, 4]及其参考文献•如果考虑方程（1.1)的驻波解，即考虑z (x ,t ) = e x p (-iEt )u (x ),其中E G R , u > 0是实函数•易知函数z 满足方程（1.1)当且仅当函数u 满足—A u + V {x )u — ZcA \/1 -\- u 2—, U= h (u ), x G (1.2)2\/1 + u 2收稿日期：2017-12-27;修订日期：2018-04-23E-mail: loiolo@; zhangym802@; ahmedguangzhou17@*基金项目：国家自然科学基金（11471330, 11501555, 11771127)和中央高校基本科研业务费专项基金（2017IVA-075, 2017IVA076, 2018-zy-138, 2018IB014)Supported by the NSFC (11471330, 11501555, 11771127), the Fundamental Research Funds for the Central Universities (2017IVA075, 2017IVA076, 2018-zy-138, 2018IB014)**通讯作者96数学物理学报Vol.39A其中V=W-五是新的位势函数，^表示新的非线性函数.当k=0,方程就是经典的半线 性椭圆型方程，不同情形下方程解的存在性和多解性结果非常多丨10-11，15-16.若k = 0,不 失一般性，假设k = 1.对于方程（1.2)，S h e n和W a n g [12]首先引入新的变换得到了方程在 超线性次临界情形下驻波解的存在性.C h e n g和Y a n g[5-6],C h e n g和Y a o[7]将文献[12]的 结果推广到了更一般的非线性情形.接着，S h e n和W a n g[14]利用山路引理，得到了临界情 形下方程非平凡解的存在性，并进而在文献[13]中考虑了参数k < 0时方程非平凡解的存 在性.本文中，首先考虑方程（1.2)当V(x) =0时解的存在性，希望将经典的B e r e s t y c k i-L io n s 场方程[3]的结果推广到方程（1.2).即考虑如下问题：—A u AVi+w2广92Vl+U1=h(u),x G R N,(1.3)其中h(u)G C(R，R)满足：(h0)当N 2 3 时，—00<lim<s—0_lim 幽=s—0+ s=—m< 0;当N =1,2 时，lim ^=-m G(-〇〇,0)，其中m是正常数.s—0s(h i)当W 23时，-〇〇<l i m哮S 〇,其中Z=#写；当W =2时，对任意的a > 0,s—>•+〇〇存在Ca > 0,使得对所有的s 20,有|h(s)| s C aeas2.(h2)当N2 2 时，存在Z >0,使得H(C) =/0C h(s)dx >0;当N =1时，存在Z o>0, 使得对所有Z G(0，C o),有H(C) <0,且H(Co) = 0, h(Co) > 0.可得到相似B erestycki-Lions场方程的结果为定理1.1假设方程（1.3)中函数h满足条件（h0)-(h2),则方程（1.3)存在一个非平凡解 w(x) G妒取、且满足如下性质：(i) Vx G R N, w(x) >0;(ii) ^是施瓦兹球对称的，即^卜）=[^),厂=卜|,并且^卜）随着厂递减；(iii) w G C2(R n);(iv) w(x)和它的二阶导数在无穷远处对于某些C满足指数衰减|D aw(x)| <Ce-5|x|, x G R n,其中 d >0,|a|< 2.注1.1文献[13]中的注1.3在关于N 2 3时，给出了场方程的相似结论，但是没有证 明.本文给出了定理1.1 一个简短的证明，并且考虑了 N= 1,2的情形.从S h e n和W a n g[12-14] —系列的工作可以看出，他们在用尽量简洁的条件去得到方程(1.2)次临界和临界情形非平凡解的存在性，方法是经典的山路引理，因此不可避免的假设 了一个相似的（A R)条件：存在一个M使得对任意的s >〇，成立〇 < M好(s)S SM S).本文希望可以对他们提出的条件进行一定的改进，因此考虑如下相对非线性薛定谔方程—A m+V[x)u —A V1+m2—=f(x,m),x G R^, (1.4)2^1十u2其中/(x，u)G C(R x R N,R),位势V :R N — R是连续的，函数V与f满足如下条件 (V0)存在V〇 >0,使得对任意的x G R N都有V(x) 2V〇 >0;No .1邱雯等：一类相对非线性薛定谔方程解的存在性97(V1)lim V (x )= V (⑴)，且对任意的 x G R w 都有 V (x ) < V (⑴);|x |—(fO) lim ’(¥) = 0，即 /(x ，s) = o(|s|)，s — 0+;s —0 s(f 1)存在常数2 < q i < 2*及C > 0,使得|f (x, s)| < C (1 + |s|q 1-1)；(f 2)存在常数</2 >及函数h G ^(R #)，使得^=/(x , s)s - F (x , s) > s® - h i{x ),V6其中（x , s ) G R N x [0, +t o );(f 3)当 s — o o 时，有^^ — o o .定理1.2假设函数V 满足（V 0)-(V 1)，函数f 满足（f 0)-(f 3).则问题（1.4)在空间 Hi(RN )上存在一个非平凡解.-些基本性质方程（1.3)和（1.4)对应的能量泛函分别为J 〇(u )J r n1十2(1+ u 2)|V u |2dx ^H (u)dx,J r nI 〇(u )7r n2(1十 u 2)|V w |2d x + — / V (x )u 2 dx — F (x , u )dx :2 JRN J r n其中 H (u ) = J 〇u h(s)dx,F (x ,u ) =J 〇u /(x ,s )d x .由于泛函 J 〇和 I 〇 在通常的 Sobolev 空间H 1^，中不好定义，令g (u )，1十2(1十 u 2)7相似文献[12]引入变董代换v = G (u )= /〇ug (t )dt ，得到新的能董泛函为J (v ) = J 〇(G ~1(v )) = ~ [ |V v |2d x — f H (G ~1(v ))dx :2 J r nJ r n(2.1)2u 2u I (v )=I 〇(G -1(v )) = l [2 J R N 则方程（1.3)和（1.4)变为|V v |2dx 十12V (x )|G -1(v )|2dx F (x, G -1(v))dx.(2.2)—A v 十 V (x )—A v =V ) _，-1W 厂h (G ^giG -1-W )(v ))，(2.3)f (x ,G -g (G -—1㈦）1㈦）’v G H 1(R n ).(2.4)引理2.1函数g 与变董代换G -1⑷具有如下性质: (g 〇)对任意 s g R ，有 y f |S| < |G -^丨 < |s|;(gl ) (G H 5))7 = ^(G -H s ))=/2+2(G -1⑷)2 ^ i .V 2+3(G-i (S))2 S 丄，98数学物理学报Vol .39A(g2) lin i s—0(g3) limg一1 ⑷=sG 一1⑷1;(g 4)对任意 s G R ,有 fG —i (s) S g(G_s 1(s)) S G —i (s);(g 5)对任意 s 2 0,有 s S G —Y s X G —1^)) S (6 - 2a /^)s .注2.1 —方面，由引理2.1，函数h , V (x ), /(x ，s )的定义及满足的条件，可知泛函J (v )和I (v )在空间Hi(R N )中有定义，且J (v ), I (v ) G C 1.另一方面，相似文献[13]中第二节易 知求方程（13)的解等价于求方程（2.3)的解，求方程（14)的解等价于求方程（2.4)的解.弓丨理2.2如果函数h (s )满足条件（h 〇)-(h 2)，则函数啊-養y勝卜s与函数h 具有相似的条件.证（h 〇)的相似条件.由函数h 和^的定义可知函数k (v ) G C (R +，R ).进一步，由引 理2.1中性质（g 2)，可知lim G -» = 0,因此lim g (G -») = g (0) = 1.结合引理2.1中性v ——0 v ——0质（g 2)和条件（h 〇)可得limk (v )limh (G ~1(v)) G -1(v)limh (G -1(v))->0 vv —0 G -1(v ) v g (G -1(v ))G -1(v )—0 G -1(v )(h 1)的相似条件.当N 2 3时，由引理2.1中性质（g 3)，可知lim G -1(v )=⑴，因此lim g-1(G -1(v))结合引理2.1中性质（g3)和条件（h i )可得limk(v)/N 十2)^N — 2)lim2h (G -1(v))G -1(v)1^，、（N 十2)^N —2)G -1(v )(N 十2)/(N -2) v (N 十2)/(N —2)g (G —1(v))3lim h (G -1(v))0.G -i(v )—⑵ G -1(v )(N+2)AN —2)当N = 2时，由引理2.1中性质（g 0)和（g1)，结合条件（h1)，对a > 0, Ca > 0,可得1k(v)g (G -1(v))h (G —1(v)) < h (G —1(v)) < C aea(^ (v)) < C ae°(h 2)的相似条件.由函数g 和G 的定义可知，函数G 和G -1是单调递增的.因此，当N 2 2时，有h (G —1(s))dG —1(s)h (t)d t.同理可得 W = 1 的情形，且 fc(G(C。
Hamilton力学的辛算法 ppt课件
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Euler中点法
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《2024年无穷维Hamilton算子的拟谱》范文
《无穷维Hamilton算子的拟谱》篇一一、引言在物理学和数学中，Hamilton算子是一个重要的概念，尤其在量子力学和经典力学中扮演着核心角色。
随着研究的深入，无穷维Hamilton算子成为了研究的热点。
然而，由于无穷维空间的复杂性，其谱问题的研究变得十分困难。
为了解决这一问题，拟谱方法被引入到无穷维Hamilton算子的研究中。
本文旨在探讨无穷维Hamilton算子的拟谱问题，分析其性质和特点，为相关领域的研究提供理论支持。
二、无穷维Hamilton算子的基本概念无穷维Hamilton算子是一种描述量子系统动力学的算子，其具有无穷多个本征值和本征函数。
在经典力学中，Hamilton算子被用来描述系统的能量，其表达式包含系统的动能和势能。
在量子力学中，Hamilton算子则是描述波函数随时间演化的算符。
由于实际物理系统的复杂性，我们通常需要考虑无穷维空间中的Hamilton算子。
三、拟谱方法的基本原理拟谱方法是一种用于处理无穷维问题的数值方法。
其基本思想是将无穷维空间进行离散化处理，将无穷维问题转化为有限维问题。
通过选取适当的基函数，将原问题表示为一系列线性方程的组合，从而实现对原问题的近似求解。
拟谱方法在处理无穷维Hamilton算子问题时，可以有效地降低问题的复杂度，提高求解的精度。
四、无穷维Hamilton算子的拟谱分析针对无穷维Hamilton算子的拟谱问题，我们采用拟谱方法进行分析。
首先，我们将Hamilton算子在一定的基函数下进行展开，得到一系列的系数。
然后，利用这些系数构建一个有限维的矩阵问题。
通过求解这个矩阵问题，我们可以得到原问题的近似解。
在实际操作中，我们需要根据具体的问题选择合适的基函数和离散化方法，以获得更好的求解效果。
五、结果与讨论通过拟谱方法，我们得到了无穷维Hamilton算子的近似解。
结果表明，拟谱方法可以有效地降低问题的复杂度，提高求解的精度。
同时，我们还发现拟谱方法的求解效果与基函数的选择和离散化方法的选取密切相关。
亥姆霍兹函数
亥姆霍兹函数
又称卡尔曼-希尔伯特函数，是一个带螺旋形衰减系数的定制函数，可以衡量一个系
统在改变激发条件时，随着时间推移，过渡到其新状态的情况。
该函数由卡尔曼-希尔伯
特所发明，是对应理想系统的动态特性的一种可以证明的估计。
它有可能帮助设计师预测
情况下，一个被发展出来的被激发的系统的输出，并在该输出不适合的情况下，进行系统
的调整以适应输出需要的改变。
在某一时刻，一个系统状态可以表达如下：
U(t) = K * U * Exp(-p * t)
其中，K为一个正值，用于表示函数偏移；U为函数初始值；p是衰减系数，该值影响指数衰减的速度。
用上述函数描述被激发系统的动态特性就可以得到卡尔曼-希尔伯特函数：
U(t) = K * U * Exp(-2 * p * t) * Cos(p * t + \phis)
在上述式子中加入了一个相位项（\phis），该多项用于描述级联（连续）相位调整（幅度），是被激发系统本身输出和激发件输出之间的延迟。
这就意味着随着\phis增加，系统的时间衰减会较慢，而且延迟也会随之而增加。
另外，K，U和p三个参数也可以在调节系统特性时使用。
K可以用于增加激发强度，
U可以用于改变初始值，而p则可以用于衰减趋势的变化程度的控制。
因此，卡尔曼-希尔伯特函数可以用于表征被激发系统的动态变化，并且它也常常被
用于评估系统的表现情况，从而为系统的调节与设计提供有用的参考。
同时，卡尔曼-希
尔伯特函数也广泛应用于工程控制理论方面的研究，特别是模糊控制理论。
由于其独特简
单的特性，该函数仍是工程中极具重要性的函数。
《无穷维Hamilton算子的特征值问题》范文
《无穷维Hamilton算子的特征值问题》篇一摘要：本文旨在探讨无穷维Hamilton算子的特征值问题。
首先，我们将简要介绍Hamilton算子的基本概念和性质。
接着，我们将深入分析无穷维Hamilton算子的特征值问题的数学框架和基本假设。
最后，我们将通过数学推导和实例分析，探讨该问题的解法及实际应用。
一、引言Hamilton算子是一种在量子力学和经典力学中广泛应用的算子。
近年来，随着数学物理的深入研究，无穷维Hamilton算子的特征值问题逐渐成为研究的热点。
本文将重点关注这一问题的数学框架、基本假设以及解法。
二、Hamilton算子的基本概念及性质Hamilton算子是一种描述物理系统动力学特性的算子，它在量子力学和经典力学中发挥着重要作用。
其基本形式包括动量算子和势能算子等。
Hamilton算子具有自伴性、正定性等重要性质，这些性质使得我们能够更好地理解和分析其特征值问题。
三、无穷维Hamilton算子的特征值问题的数学框架无穷维Hamilton算子的特征值问题是在无穷维空间中研究Hamilton算子的特征值和特征函数的问题。
在这个问题中，我们需要构建适当的数学框架，包括定义函数空间、内积、范数等。
此外，我们还需要假设系统满足一定的物理条件和数学条件，如系统的边界条件、对称性等。
四、无穷维Hamilton算子的特征值问题的基本假设与解法针对无穷维Hamilton算子的特征值问题，我们首先需要做出一些基本假设。
例如，我们假设系统是线性的、自伴的，且具有某些特定的对称性。
基于这些假设，我们可以采用一系列数学方法来解决这个问题，如变分法、谱方法、数值分析等。
在具体解法上，我们可以先通过变分法将特征值问题转化为相应的变分问题。
然后，利用谱方法求解该变分问题，得到一系列的特征值和特征函数。
此外，我们还可以采用数值分析方法，如有限元法、差分法等，对问题进行离散化处理，从而得到数值解。
五、实例分析为了更好地理解无穷维Hamilton算子的特征值问题，我们可以通过具体实例进行分析。
《无穷维Hamilton算子的特征值问题》范文
《无穷维Hamilton算子的特征值问题》篇一摘要：本文旨在探讨无穷维Hamilton算子的特征值问题。
首先，我们将介绍Hamilton算子的基本概念及其在物理学中的应用。
接着，我们将详细阐述无穷维Hamilton算子的数学模型，并分析其特征值问题的求解方法。
最后，我们将通过实例分析来验证所提方法的可行性和有效性。
一、引言Hamilton算子在量子力学、光学和许多其他物理领域中有着广泛的应用。
随着研究的深入，人们开始关注无穷维Hamilton算子的特征值问题。
由于其在数学和物理领域的重大意义，这一问题的研究成为了学术界的热点。
二、Hamilton算子及其基本概念Hamilton算子是一种在量子力学中用于描述粒子运动状态的算子。
它描述了粒子在给定势能下的能量状态，并决定了波函数的演化。
在有限维的情况下，Hamilton算子的特征值问题可以通过数值方法进行求解。
然而，在无穷维空间中，由于维度的增加，问题的复杂性大大增加，需要采用新的方法和理论来处理。
三、无穷维Hamilton算子的数学模型无穷维Hamilton算子可以描述为在无穷维空间中，粒子在给定势能下的能量状态。
由于维度的增加，该问题的求解变得极为复杂。
为了解决这一问题，我们需要建立相应的数学模型。
该模型通常包括一个描述粒子运动状态的偏微分方程和一个描述势能的函数。
通过求解这个偏微分方程，我们可以得到Hamilton算子的特征值和特征函数。
四、特征值问题的求解方法针对无穷维Hamilton算子的特征值问题，我们需要采用一些特殊的求解方法。
一种常用的方法是变分法。
通过将原问题转化为一个变分问题，我们可以利用已有的数值方法来求解。
另外，谱方法也是一种有效的求解方法。
通过将原问题转化为一个谱问题，我们可以利用谱理论来求解特征值和特征函数。
此外，还有一些其他的方法，如差分法、有限元法等，也可以用于求解该问题。
五、实例分析为了验证所提方法的可行性和有效性，我们采用具体实例进行分析。
2 Fermi Liquids and Luttinger Liquids
10
Heinz J. Schulz et al.
Subsequently, I will first briefly discuss the case of a noninteracting manyfermion system (the Fermi gas), and then turn to Landau’s theory of the interacting case (the liquid), first from a phenomenological point of view, and then microscopically. A much more detailed and complete exposition of these subjects can be found in the literature [5–9]. 2.2.1 The Fermi Gas
2
Fermi Liquids and Luttinger Liquids
Heinz J. Schulz, Gianaurelio Cuniberti, and Pierbiagio Pieri
2.1
Introduction
In these lecture notes, corresponding roughly to lectures given at the summer school in Chia Laguna, Italy, in September 1997, an attempt is made to present the physics of three-dimensional interacting fermion systems (very roughly) and that of their one-dimensional counterparts, the so-called Luttinger liquids (in some more detail). These subjects play a crucial role in a number of currently highly active areas of research: high temperature and organic superconductors, quantum phase transitions, correlated fermion systems, quantum wires, the quantum Hall effect, low-dimensional magnetism, and probably some others. Some understanding of this physics thus certainly should be useful in a variety of areas, and it is hoped that these notes will be helpful in this. As the subject of these lectures was quite similar to those delivered at Les Houches, some overlap in the notes [1] was unavoidable. However, a number of improvements have been made, for example a discussion of the “Klein factors” occurring in the bosonization of one-dimensional fermions, and new material added, mainly concerning spin chains and coupled Luttinger liquids. Some attempt has been made to keep references up to date, but this certainly has not always been successful, so we apologize in advance for any omissions (but then, these are lecture notes, not a review article).
庞特里亚金极大值原理是偏微分方程
庞特里亚金极大值原理是偏微分方程The Pontryagin maximum principle is a fundamental concept in the field of optimal control theory. It provides a powerful tool for determining the optimal control strategies for dynamical systems subject to constraints. Originally developed by Russian mathematician Lev Pontryagin in the 1950s, this principle has had a significant impact on various areas of science and engineering.庞特里亚金极大值原理是最优控制理论中的一个基本概念。
它为确定受约束动态系统的最佳控制策略提供了一个强大的工具。
这一原理最初由俄罗斯数学家列夫·庞特里亚金在20世纪50年代提出，对科学和工程的各个领域都产生了重要的影响。
The central idea behind the Pontryagin maximum principle is to find the optimal control that maximizes a certain objective function, subject to the dynamics of the system and any constraints that may be present. By formulating the optimal control problem in terms of a Hamiltonian function, one can derive a set of differential equations known as the Pontryagin equations, which must be satisfied by the optimal control.庞特里亚金极大值原理的核心思想是寻找最优控制，从而最大化一个特定的目标函数，同时要考虑系统的动态性质和可能存在的约束。
《2024年无穷维Hamilton算子的拟谱》范文
《无穷维Hamilton算子的拟谱》篇一一、引言在数学物理的诸多领域中，Hamilton算子因其描述了系统的能量和动量而备受关注。
尤其在量子力学和经典力学中，Hamilton算子扮演着至关重要的角色。
随着研究的深入，无穷维Hamilton算子逐渐成为研究的热点，其拟谱问题更是引起了广泛的关注。
本文旨在探讨无穷维Hamilton算子的拟谱问题，并为其提供一个高质量的范文。
二、背景与预备知识无穷维Hamilton算子通常出现在量子场论、量子力学、统计物理等领域。
它描述了具有无穷多自由度的系统的动力学特性。
在处理这类问题时，我们需要借助泛函分析、算子理论等数学工具。
此外，为了更好地理解无穷维Hamilton算子的性质，我们需要了解一些预备知识，如希尔伯特空间、自伴算子等。
三、无穷维Hamilton算子的定义与基本性质无穷维Hamilton算子可以定义为在适当的功能空间上的自伴算子。
它具有一些基本性质，如对称性、保谱性等。
这些性质使得我们能够更好地理解其动力学特性和物理意义。
此外，我们还需要探讨无穷维Hamilton算子的谱的性质，如离散谱、连续谱等。
四、拟谱的概念与性质拟谱是描述算子谱的一种方法，它能够帮助我们更好地理解算子的性质和动力学特性。
在无穷维Hamilton算子的情况下，我们可以通过拟谱来研究其谱的性质，包括离散谱的分布、连续谱的取值范围等。
此外，我们还需要探讨拟谱与系统动力学行为之间的关系。
五、无穷维Hamilton算子的拟谱方法针对无穷维Hamilton算子的拟谱问题，我们可以采用一些具体的方法进行研究。
例如，我们可以利用傅里叶变换将无穷维系统转化为有限维系统，从而简化问题的求解过程。
此外，我们还可以采用其他数值方法或近似方法进行研究，如变分法、迭代法等。
六、应用实例与实验结果分析为了验证我们的方法的有效性，我们可以选择一些具体的物理系统进行实验研究。
例如，我们可以考虑量子场论中的某些模型，如无质量场、谐振子场等。
python 希尔伯特变换 求瞬时振幅 相位
希尔伯特变换是一种数学工具，用于从实数信号中提取出其瞬时频率、相位和振幅信息。
在Python中，可以使用一些库来实现希尔伯特变换。
以下是一个使用scipy.signal.hilbert()函数进行希尔伯特变换的简单示例，然后计算瞬时幅度和相位：pythonimport numpy as npfrom scipy.signal import hilbert# 创建一个简单的正弦波信号fs = 1000 # 采样率T = 1 / fs # 采样周期t = np.arange(0, 1, T) # 时间向量# 正弦波参数f = 50 # 频率A = 1 # 振幅# 创建正弦波信号x = A * np.sin(2 * np.pi * f * t)# 计算希尔伯特变换analytic_signal = hilbert(x)instantaneous_amplitude = np.abs(analytic_signal)instantaneous_phase = np.unwrap(np.angle(analytic_signal))# 绘制结果import matplotlib.pyplot as pltplt.figure(figsize=(10, 6))plt.plot(t, x, label='Original Signal')plt.plot(t, instantaneous_amplitude, label='Instantaneous Amplitude')plt.plot(t, instantaneous_phase, label='Instantaneous Phase [rad]') plt.xlabel('Time [s]')plt.legend()plt.grid(True)plt.show()这个例子首先创建了一个简单的正弦波信号，然后使用scipy.signal.hilbert()函数对它进行了希尔伯特变换。
哈密顿函数雅可比贝尔曼
哈密顿函数雅可比贝尔曼方程
哈密顿函数雅可比贝尔曼方程，也称为哈密顿-雅可比方程，是一种常微分方程，由18世纪英国物理学家、数学家、化学家约翰·哈密顿（John Hamilton）于1834年提出。
它描述了一个系统的动力学行为，即特定状态下系统的本征变化。
它是一种多元非线性常微分方程，定义在多元空间上的实值函数的泰勒展开，形式如下：
H(q,p)=∑i=1n[pi*dqi/dt-Li(q)]=0
其中，qi是系统的位置变量，pi是关联的动量，Li 是每个位置变量的动能函数。
因此，它可以描述系统的本征变化，这是系统动力学研究的基础。
非保守系统的拟hamilton原理及其应用
非保守系统的拟Hamilton原理及其应用1. 介绍在经典力学中，Hamilton原理是描述保守力系统运动的重要原理之一。
然而，在某些情况下，系统中的力并非全都是保守力，这时需要使用非保守系统的拟Hamilton原理来描述系统的运动。
本文将介绍非保守系统的拟Hamilton原理及其应用。
2. 非保守系统的拟Hamilton原理拟Hamilton原理是根据非保守系统的平衡原理推导出来的一种描述系统运动的方法。
在非保守系统中，除了保守力外，还存在非保守力，如摩擦力、阻力等。
拟Hamilton原理考虑了非保守力的影响，以使得系统的运动在非保守力的作用下也能满足系统的平衡条件。
拟Hamilton原理的数学表达如下：$$\\delta W + \\delta K + \\delta U = 0$$其中， - $\\delta W$代表外力做功的变化量 - $\\delta K$代表系统动能的变化量 - $\\delta U$代表系统势能的变化量该原理表明，在非保守力作用下，系统的总能量仍然守恒。
3. 非保守系统的拟Hamilton原理的应用非保守系统的拟Hamilton原理在各个领域均有广泛的应用。
以下列举了一些常见的应用场景：3.1 摩擦力系统在摩擦力系统中，摩擦力是一个典型的非保守力。
使用拟Hamilton原理可以描述摩擦力系统的运动。
例如，摩擦力对物体的减速、停止和滑动等现象可以通过拟Hamilton原理得到合理的解释。
3.2 阻力系统阻力也是一个常见的非保守力。
拟Hamilton原理可以用于描述阻力系统的运动。
例如，在空气中行进的物体受到阻力的影响，可以使用拟Hamilton原理分析物体的运动。
3.3 电路系统在电路系统中，电阻产生的热能可以看作是非保守力的作用。
拟Hamilton原理可以应用于电路系统的动态分析。
例如，研究电阻产生的热量对电路元件的影响、电池耗电量等问题都可以使用拟Hamilton原理进行描述。
二维氢原子中的基态奇异特性数值精确对角化法
二维氢原子中的基态奇异特性数值精确对角化法刘褚航;强百强;季育琛;李炜【期刊名称】《物理学报》【年(卷),期】2017(66)23【摘要】利用数值有限差分法处理二维氢原子的基态波函数时,计算结果发现其存在着数值奇异特性.本文通过构造一套具有正交完备性的离散贝塞尔基函数,并结合基于Lanczos技术的数值精确对角化方法研究二维氢原子中的基态波函数的数值奇异特性,得到的波函数数值解及其相应的本征能量均与解析结果相一致.这套新的完备的离散贝塞尔基函数,可以在研究一些波函数具有数值奇异特性的体系中发挥至关重要的作用.%With the development of computing technology, numerical exact diagonalization method plays a vital role in mod-ern computational condensed matter physics, especially in the research area of strongly correlated electron systems: it becomes a benchmark for other numerical computational techniques, such as quantum Monte Carlo, numerical renormal-ization group, density matrix renormalization group, and dynamic mean field theory. In this paper, we first numerically exactly diagonalize the three-dimensional hydrogen atom with the combination of finite-difference method, and find that the numerical wave function of ground state is in good agreement with the analytical calculations. We then turn to discuss the space dimension confinement hydrogen system, two-dimensional hydrogen atom, and notice that the numer-ical wave function is no longer in agreement with the analytical calculation, wherethe ground state wave function has a numerical singularity as radius approaches to zero. Compared with the case of the three-dimensional hydrogen atom, this issue mainly comes from the nature of space dimension confinement. To resolve such an issue of numerical singularity in two-dimensional hydrogen atom, we need to construct a new discrete and normalized Bessel function as a basis to study the ground state behavior of dimension confinement system based on the framework of Lanczos-type numerical exact diagonalization. The constructed normalized Bessel basis is orthogonal and discrete, and thus becomes suitable for practical calculation. Besides, these prominent properties of such a Bessel basis greatly reduce the complexity and difficulty in practical calculation, and thus makes computing work efficient. In addition, Lanczos-type numerical exact diagonalization method can extremely speed up the process of solving the eigenvalue equation. As a result, such a high efficient calculation of our method demonstrates the consistence between numerical and analytical ground state energy value, and the corresponding wave function with enough truncated basis number. Since this kind of numerical singularity occurs in many space dimension confinement systems, our finding for constructing a new discrete Bessel basis function may be helpful in studying the quantum systems with numerical singularity behaviors in wavefunctions in future. On the other hand, it should be pointed out that the Bessel basis is incorporated into the linear augment plane wave method in the density functional theory to study the electronic band structure of the condensed material and obtain high accurate results,especially in the theoretical prediction of topological insulators and in experimental realization as well.【总页数】8页(P9-16)【作者】刘褚航;强百强;季育琛;李炜【作者单位】上海科技大学物质科学与技术学院, 上海 201210;上海科技大学物质科学与技术学院, 上海 201210;上海科技大学物质科学与技术学院, 上海 201210;中国科学院上海微系统与信息技术研究所, 信息功能材料国家重点实验室, 上海200050【正文语种】中文【相关文献】1.高频强激光场中氢原子基态电离特性研究 [J], 高婉琴;王加祥;张雅芸2.氢原子基态极化率的精确计算 [J], 陈昌远3.基于去奇异边界元法的二维数值波浪水池计算参数影响分析 [J], 杨师宇;吴静萍;汪敏;陈昌哲4.二维边界元法中几乎奇异积分的解析法 [J], 牛忠荣;王左辉;胡宗军;周焕林5.氢原子(e,2e)反应中基态三重微分散射截面的计算 [J], 王号;张春光;李晖因版权原因，仅展示原文概要，查看原文内容请购买。
局部Lipschitz条件下的布朗运动和泊松过程混合驱动的正倒向随机微分方程
局部Lipschitz条件下的布朗运动和泊松过程混合驱动的正倒
向随机微分方程
李娟;吴臻
【期刊名称】《应用数学》
【年(卷),期】2002(15)2
【摘要】本文得到在局部Lipschitz条件下的布朗运动和泊松过程混合驱动的倒向随机微分方程解的存在唯一性 ;同时也证明了布朗运动和泊松过程混合驱动的完全藕合的正倒向随机微分方程在局部Lipschitz条件下的解的存在唯一性 .
【总页数】8页(P40-47)
【关键词】局部Lipschiz条件;布朗运动;随机微分方程;随机测度;泊松过程;解;存在性;唯一性
【作者】李娟;吴臻
【作者单位】山东大学数学与系统科学学院
【正文语种】中文
【中图分类】O211.63
【相关文献】
1.非Lipschitz条件下由泊松过程驱动的随机微分方程Euler方法的依概率收敛性[J], 于辉
2.非Lipschitz条件下由连续局部鞅驱动的倒向随机微分方程 [J], 李师煜;李文学;高武军
3.局部Lipschitz条件下的正倒向随机微分方程 [J], 吴臻;谷艳玲
4.非Lipschitz条件下由Lévy过程驱动的倒向随机微分方程解的存在唯一及其稳定性(英文) [J], 任永;胡兰英;夏宁茂
5.局部Lipschitz条件下的正倒向重随机微分方程 [J], 朱庆峰;石玉峰
因版权原因，仅展示原文概要，查看原文内容请购买。
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1
(1.4)
(iii) P (t) ∼ t
2
for |t| >> (ε2 Γ)−1 , for some α > 0.
See the appendix in section 9 as well as [2], [9] for definitions and results on almost periodic functions. (f, g ) denotes the inner product of f and g . If ψ0 is normalized then P (t) has the quantum mechanical interpretation of the probability that the system at time t is in the state ψ0 . 3 We do not discuss the short time behavior in this article; see [12]. This small time behavior is related to the ”watched pot” effect in quantum measurement theory [15].
was considered. The method used is a time dependent / dynamical systems approach introduced in [21], [23] for the problem of perturbations of operators with embedded eigenvalues in their continuous spectra, [24], in the context of resonant radiation damping of nonlinear systems, as well as in [22]; see also [12]. New analytical questions must be addressed in view of (i) the presence of infinitely many frequencies which may resonate with the continuum as well as (ii) the possible accumulation of such resonances in the continuous spectrum. This leads to a careful use of almost periodic properties of the perturbation (Theorems 2.1 and 2.2) and hypothesis (H6) (Theorem 2.3), which is easily seen to hold when the perturbation, W (t), consists of a sum over finite number of frequencies, µj . A special case for which the hypotheses of our theorems are verified is the case of the Schr¨ odinger operator H0 = −∆ + V (x). Here, V (x) is a real-valued function of x ∈ IR3 which decays sufficiently rapidly as |x| → ∞. In this setting Soffer and Weinstein [22] studied
∗
1. Introduction
1.1. Overview Consider a dynamical system of the form: i∂t φ = H0 φ, (1.1)
where H0 denotes a self-adjoint operator on a Hilbert space H. We further assume that H0 has only one eigenstate ψ0 ∈ H with corresponding simple eigenvalue λ0 . Thus, b∗ (t) = e−iλ0 t ψ0 (1.2)
is a time-periodic bound state solution of the dynamical system (1.1). We next introduce the perturbed dynamical system: i∂t φ = ( H0 + εW (t) ) φ. (1.3)
In this paper we prove that if the perturbation , εW (t), is a small, ”generic” and almost periodic in time 1 , then solutions of the perturbed dynamical system (1.3) tend to zero as t → ±∞. It follows that the state, b∗ (t), does not continue or deform to a time periodic or even time almost periodic state. Thus, b∗ (t) is structurally unstable with respect to this class of perturbations. Our methods yield a detailed description of the transient (t large but finite) and long time (t → ±∞) behavior solutions to the initial value problem. Theorems 2.1-2.3 contain precise statements of our main results. The following picture emerges concerning time evolution (1.3) for initial data given by the bound state, ψ0 , of the unperturbed problem. Let P (t) = |( ψ0 , φ(t) )|2 , the modulus square of the projection of the solution at time t onto the state ψ0 2 . Then, (i) P (t) ∼ 1 − CW |t|2 , for |t| small,3 (ii) P (t) ∼ exp(−2ε2 Γt) for t ≤ O((ε2 Γ)−1 ), Γ = O(W 2 ), and
Parametrically Excited Hamiltonian Partial Differential Equations
arXiv:nlin/0012021v1 [nlin.PS] 11 Dec 2000
E. Kirr
∗
பைடு நூலகம்
and M.I. Weinstein
†
February 8, 2008
Abstract Consider a linear autonomous Hamiltonian system with a time periodic bound state solution. In this paper we study the structural instability of this bound state relative to time almost periodic perturbations which are small, localized and Hamiltonian. This class of perturbations includes those whose time dependence is periodic, but encompasses a large class of those with finite (quasiperiodic) or infinitely many non-commensurate frequencies. Problems of the type considered arise in many areas of application including ionization physics and the propagation of light in optical fibers in the presence of defects. The mechanism of instability is radiation damping due to resonant coupling of the bound state to the continuum modes by the time-dependent perturbation. This results in a transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. These results generalize those of A. Soffer and M.I. Weinstein, who treated localized time-periodic perturbations of a particular form. In the present work, new analytical issues need to be addressed in view of (i) the presence of infinitely many frequencies which may resonate with the continuum as well as (ii) the possible accumulation of such resonances in the continuous spectrum. The theory is applied to a general class of Schr¨ odinger operators.