Inverse spectrum problem for quasi-stationary states

TL;DR

This paper introduces a semi-classical method to reconstruct potentials with quasi-stationary states from spectral data, extending inverse techniques for wave equations and applicable in physics fields like gravitational wave research.

Contribution

It generalizes inverse methods for potential wells and barriers to reconstruct complex potentials from spectral data with small imaginary parts.

Findings

Successfully reconstructs potentials from spectra with small imaginary parts.

Analytic spectra are effectively analyzed using the proposed method.

Application demonstrated in gravitational wave physics.

Abstract

In this work we present a semi-classical approach to solve the inverse spectrum problem for one-dimensional wave equations for a specific class of potentials that admits quasi-stationary states. We show how inverse methods for potential wells and potential barriers can be generalized to reconstruct significant parts for the combined potentials. For the reconstruction one assumes the knowledge of the complex valued spectrum and uses the exponential smallness of its imaginary part. Analytic spectra are studied and a recent application of the method in the literature for gravitational wave physics is discussed. The method allows for a simple reconstruction of quasi-stationary state potentials from a given spectrum. Thus it might be interesting for different branches of physics and related fields.