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Parity-dependent potentials for the one-dimensional Schrödinger equation obtained from inverse spectral theory. (English) Zbl 0541.47008

In the usual theory of inverse scattering associated with one dimensional Schrödinger equation a triangularity is imposed on the Gelfand-Levitan kernel which implies boundary conditions on the associated eigen- functions and leads to a determination of the (local) potential. Let \(H_ 0=-d^ 2/dx^ 2,\quad -\infty<x<+\infty\) and \(A_ 0\) be an operator with eigenvalues \(\pm 1\) corresponding to the directions of momentum. The authors introduce the generalized eigenfunctions \(((2\pi)^{{1\over2}}E^{1/4})^{-1}\exp iaE^{{1\over2}}x,\quad 0<E<\infty,\quad a=\pm 1,\) corresponding to simultaneous eigenfunctions of \(H_ 0\) and \(A_ 0\), with associated Gelfand-Levitan kernel \(K_ 0\). They introduce a perturbed Gelfand-Levitan kernel K with supposed properties which should reflect parity and impose the condition that \(K(x,x^ 1)=K_ 0(x,x^ 1)=0\) for \(| x'|>| x|.\) As a consequence they construct various examples of parity dependent potentials V(x), i.e. \(V(x)\psi(x)=V_ 1(x)\psi(x)+V_ 2(x)P\psi(x),\quad P\psi(x)=\psi(-x).\)
For certain of these examples all the transmission and reflection coefficients are zero so that there are parity-dependent potentials which do not scatter. A further example exhibits a parity-dependent potential whose scattering operator is identical to the reflectionless potentials considered in the paper of Kay and Moses [J. Appl. Phys. 27, 1503 (1956)] and this shows that knowledge of the scattering operator and position of the point eigenvalues is not sufficient to determine whether a potential is non-local or not.
Reviewer: M.Thompson

MSC:

47A40 Scattering theory of linear operators
47A10 Spectrum, resolvent
34L99 Ordinary differential operators
47A65 Structure theory of linear operators
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