Spectrally determined singularities in a potential with an inverse square initial term

TL;DR

This paper investigates the inverse spectral problem for Bessel-type operators with a potential that includes an inverse square term, demonstrating that the asymptotic expansion coefficients of the potential are uniquely determined by the spectral data.

Contribution

It establishes that the coefficients of the potential's asymptotic expansion are spectrally determined for Bessel-type operators with inverse square singularities.

Findings

Coefficients of the potential's asymptotic expansion are uniquely determined by the spectrum.

Expansion of the resolvent trace relates directly to the potential's asymptotics.

Method uses singular asymptotics lemma to connect spectral data with potential coefficients.

Abstract

We study the inverse spectral problem for Bessel type operators with potential (v(x)): (H_\kappa=-\partial_x^2+\frac{k}{x^2}+v(x)). The potential is assumed smooth in ((0,R)) and with an asymptotic expansion in powers and logarithms as (x\rightarrow 0^+, v(x)=O(x^\alpha), \alpha >-2). Specifically we show that the coefficients of the asymptotic expansion of the potential are spectrally determined. This is achieved by computing the expansion of the trace of the resolvent of this operator which is spectrally determined and elaborating the relation of the expansion of the resolvent with that of the potential, through the singular asymptotics lemma.