A semi-classical inverse problem II: reconstruction of the potential

TL;DR

This paper demonstrates that, under generic conditions, the potential of a 1D Schrödinger operator can be uniquely reconstructed from its semi-classical spectrum, providing an explicit method for the reconstruction.

Contribution

It extends previous work by showing the potential is determined by the spectrum under generic assumptions and offers an explicit reconstruction method.

Findings

Potential is uniquely determined by semi-classical spectrum in 1D.

Explicit reconstruction method provided.

Results depend on genericity assumptions.

Abstract

This paper is the continuation of our work with Victor Guillemin; Victor and I proved that the Taylor expansion of the potential at a generic non degenerate critical point is determined by the semi-classical spectrum of the associated Schr\"odinger operator near the corresponding critical value. Here, I show that, under some genericity assumptions, the potential of the 1D Schroedinger operator is determined by its semi-classical spectrum. Moreover, there is an explicit reconstruction. This paper is strongly related to a paper of David Gurarie (J. Math. Phys. 36:1934--1944 (1995)).