On the Non-Orthonormality of Lippmann-Schwinger-Low States

a r X i v :q u a n t -p h /0002042v 2 18 D e c 2001
ON THE NON-ORTHONORMALITY OF LIPPMANN-SCHWINGER-LOW STATES
V.J.Menon 1and B.K.Patra 2
1Department of Physics,Banaras Hindu University,Varanasi 221005,India
2
Variable Energy Cyclotron Centre,1/AF Bidhan Nagar,Calcutta 700064,India
Abstract
Introduction
We denote the free and full Hamiltonian operators by H o and H≡H o+V respectively with V being a short-range interaction.Their continuum eigenkets obey the Schrodinger (superscript S)equations
(E k−H o)|k =0(1)
(E k−H)|ψS k =0(2)
where the masses are assumed to be renormalized so that energies do not shift.For later convenience we also introduce the free resolvent G o k,the complex projectorηo k onto free
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states of energy E k,πtimes a Dirac delta D o k,related functionsµnk and d nk along with a useful identity via
G o k=1
E k−E n+iǫ
(3)
D o k=πδ(
E k−H o)=ǫG o†k G o k;d nk=
ǫ
E k−E n+iǫ=
E k−E n+2iǫ
“In sharp contrast to the underlying Eq.(1)the LSL states satisfy
(E k−H+iǫ)|ψL k =iǫ|k ,(8) or equivalently
(E k−H)|ψL k =−ηo k V|ψL k ”(9)
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Proof
“The amplitudes T L nk≡ n|V|ψL k fulfill a nonlinear relation
(T L nk−T L∗kn)/(E k−E n+iǫ)=−(1+µnk)C L nk(10)
C L nk= ψL n|V G o†n G o k V|ψk ”(11) Proof
LEMMA C(NONORTHONORMALITY):
Upon using the Low form for ψL n|and the LS form for|ψL k (cfs.Eqs.6,7)onefinds
ψL n|ψL k = n|ψL k + n|V(E n−H−iǫ)−1|ψL k (16)
In the usual Goldberger-Watson treatment(labeled by the superscript G)one erro-neously assumes that H|ψL k =E k|ψL k and reduces Eq.(16)into[6]
I G nk= n|k + n|V 1E n−E k−iǫ |ψL k = n|k (17) In our opinion the use of Eqs.(8),(9)as eigenket statement is quite risky and it is much safer to employ the LS representations(6)for both ψL n|and|ψL k .Then
I L nk= n|k + n|G o k V|ψL k + ψL n|V G o†n|k
+ ψL n|V G o†n G o k V|ψL k (18) which is readily shown to coincide with the Lemma(15)in view of the properties (Eq.(10))and(Eq.(14)).The fact that I L nk reduces to n|k if E n=E k but fails to
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do so if E n=E k is very disturbing because it implies that the set of LSL states|ψL n which are degenerate at a given collision energy E k are mutually nonorthogonal.
LEMMA D(ILLUSTRATION):
(19)
∆∗n∆k
where the form factor g k and Fredholm determinant∆k are defined by
g k= k|g ;∆k=1−λ g|G o k|g ”(20)
Proof
∆∗n∆k ∆k−∆∗n
The mainfindings of the present paper are contained in Lemmas A,B,C,and D.The
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nonorthogonality of the LSL states(for E n=E k,n=k)implies that,even in the absence of bound states,the Moller operator connecting|k to|ψL k may be nonunitary and k|ψL k ψL k|may loose its interpretation as the unit matrix.Several standard results of scattering perturbation theory[1-7]based on the LSL states may require re-examination.Before ending,it may be added that the present work is not concerned with another peculiarity of the LS equation-the Faddeev ambiguity[8]-arising from the noncompactness of the kernel.We also believe that the time-dependence of the LSL states will be much richer than the standard Schrodinger kets|ψS k(t) but this aspect will be dealt-with in a future communication.
ACKNOWLEDGEMENTS:VJM thanks the UGC,Goverment of India,New Delhi forfinancial support.
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References
[1]B.A.Lippmann and J.Schwinger,Phys.Rev.79,469(1950).
F.E.Low,Phys.Rev.97,1392(1955).
[2]Charles J.Joachain,Quantum Collision Theory,North-Holland Publishing Com-
pany(Amsterdam),Lippmann-Schwinger Equation and the Born series,pp.161-162,305-307,409-410.
[3]M.L.Goldberger and K.M.Watson,Collision Theory,John Wiley&Sons,Inc.
(1964),L-S-L Equation,pp197-199,Rearrangement Reactions,page157.
[4]Silvan S.Schweber,An Introduction to Relativistic Quantum Field Theory,Harper
&Row,Publishers,New York,N.Y.,Lippmann Schwinger equation pp.316,327, S-matrix expansion(page330):S or U(∞,−∞)=P exp[−i ∞−∞dtV I(t)]=1+ ∞n=1(−i
company(1966),Lippmann Schwinger Equation,pp.180-181,Low Equation page 190,Faddeev Equation,page556.
9。