Sampling The Lowest Eigenfunction to Recover the Potential in a One-Dimensional Schr\"odinger Equation

TL;DR

This paper introduces a simple method to recover the potential in a one-dimensional Schrödinger equation by sampling the lowest eigenfunction, effectively addressing issues of measurement imprecision in higher modes.

Contribution

The paper presents a novel approach to inverse spectral problems that uses only the lowest eigenfunction to recover the potential, simplifying previous methods.

Findings

Potential can be recovered from a single eigenfunction sample.

Lower modes provide more reliable information due to reduced measurement errors.

Method works with just one boundary condition.

Abstract

We consider the BVP $-y" + qy = \lambda y$ with $y(0)=y(1)=0$. The inverse spectral problems asks one to recover $q$ from spectral information. In this paper, we present a very simple method to recover a potential by sampling one eigenfunction. The spectral asymptotics imply that for larger modes, more and more information is lost due to imprecise measurements (i.e. relative errors \textit{increases}) and so it is advantageous to use data from lower modes. Our method also allows us to recover "any" potential from \textit{one} boundary condition.