Inverse resonance scattering on the real line

Abstract

We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems (including characterization): in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). We prove that a potential is uniquely determined by zeros of reflection coefficients and there exist distinct potentials with the same resonances. We describe the set of "isoresonance potentials". Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of zeros of the reflection coefficient for an other compactly supported potential, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonances counting function is determined, 4) these results are applied to canonical systems. Contents 25 7. Canonical systems 29 References 36

Jost functions and Jost solutions for Jacobi matrices, II. Decay and analyticity

“…There is a large literature on this question [6,7,17,18,21,29,30]. In particular in [17,18], Korotyaev makes some progress in classifying all Jost functions in this case.…”

Resonances for steplike potentials: Forward and inverse results

“…The relationships (5) and (8) Thus applying Theorem 2.2, we obtain, using the notation of that theorem,…”

“…The papers [1] and [9] have studied a similar question for compactly supported potentials on a half-line. Here we give an example of our inverse results for steplike potentials.…”

Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

“…The results in Section 2.1 are taken from [186]. For further aspects of the theory related to the Regge problem we refer the reader to [122], [115] [255], [256], [150], [142], [243], [148] and [251]. In particular, statement 4 of Theorem 2.1.2 was obtained in [251,Theorem 6].…”

“…Since asymptotics of eigenvalues in the classical Regge problem are known only in special cases, e. g., if the potential q is continuous and q(a) = 0, the corresponding inverse problem is solved also in these particular cases [148].…”