Inverse Spectral Problem for Schr\"odinger Operators

TL;DR

This paper advances inverse spectral theory for Schrödinger operators by reducing symmetry constraints needed to recover potential Taylor coefficients from low-lying eigenvalues, especially in one dimension.

Contribution

It generalizes previous results by Guillemin and Uribe, showing fewer symmetry assumptions are necessary, and provides explicit formulas for wave invariants at the potential's minimum.

Findings

Potential Taylor coefficients can be recovered with fewer symmetry assumptions.

In one dimension, no symmetry assumptions are needed for recovery.

Explicit formulas for wave invariants at the bottom of the well are derived.

Abstract

In this article we improve some of the inverse spectral results proved by Guillemin and Uribe in \cite{GU}. They proved that under some symmetry assumptions on the potential $V(x)$, the Taylor expansion of $V(x)$ near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schr\"odinger operator in $\mathbb R^n$. We prove some similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of $V(x)$. We establish our results by finding some explicit formulas for wave invariants at the bottom of the well.