Complex potentials and the inverse problem

Abstract

The occurrence of complex potentials with real eigenvalues has implications concerning the inverse problem, i.e. the determination of a potential from its spectrum. First, any complex potential with real eigenvalues has at least one equivalent local potential. Secondly, a real spectrum does not necessarily corresponds to a local real potential. A basic ambiguity arises from the possibility the spectrum to be generated by a complex potential. The purpose of this work is to discuss several aspects of this problem.

An Approach to Solving Direct and Inverse Scattering Problems for Non-Selfadjoint Schrödinger Operators on a Half-Line

“…The overall approach developed in the present work to solve this problem is based on the representation (8). Indeed, the calculation of {ρ k } α k=1 is easily realizable with the aid of the argument principle theorem applied to find zeros of (8) in the unit disc. To the best of our knowledge, there has been no practical way of calculating the normalization polynomials.…”

“…The calculation of the scattering function s(p) requires an analytic extension of the Jost function e(ρ) obtained from (8), onto the strip −ε 0 < Im(ρ) < 0. We explore different possibilities for such an extension, including the Padé approximants (see [28,29]) and the power series analytic continuation [30] (p. 150), [31].…”

“…The approach based on the representations (7) and (8) leads to efficient numerical methods for solving both direct and inverse scattering problems.…”

“…Following [25] (see also [26] (p. 63)), the substitution of ( 23) into (14) and termwise integration lead to the series representation (8) for the Jost solution.…”

“…In the present work, an approach to solving direct and inverse scattering problems for (1) under Condition (2) is developed. Complex-valued potentials arise when studying parity time (PT)-symmetric potentials [1] (Chapter 1), [2], quasi-exactly solvable (QES) potentials [3,4], hydrodynamics, and magnetohydrodynamics [5]; see also [6][7][8].…”