The inverse problem for a spectral asymmetry function of the Schr\"odinger operator on a finite interval

TL;DR

This paper investigates the inverse problem of reconstructing potentials in the Schrödinger equation from a spectral asymmetry function, establishing uniqueness results for classes of potentials sharing the same asymmetry function.

Contribution

It characterizes classes of square-integrable potentials with identical spectral asymmetry functions and shows the existence of a unique potential for each Dirichlet spectral sequence given a fixed asymmetry function.

Findings

Identifies classes of potentials with the same asymmetry function.

Proves the existence of a unique potential for each spectral sequence.

Characterizes the asymmetry function as an entire function of order 1/2 and type 1.

Abstract

For the Schr\"odinger equation $-d^2 u/dx^2 + q(x)u = \lambda u$ on a finite $x$-interval, there is defined an "asymmetry function" $a(\lambda;q)$, which is entire of order $1/2$ and type $1$ in $\lambda$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function. For any given $a(\lambda)$, there is one potential for each Dirichlet spectral sequence.