Sampling The Lowest Eigenfunction to Recover the Potential in a One-Dimensional Schrödinger Equation

2022/02/16 by Rob Rahm, Rahm, Rob
Engineering · Mathematics · #FOS: Mathematics #Microwave Imaging and Scattering Analysis #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Ultrasonics and Acoustic Wave Propagation

paper · pdf · doi:10.48550/arxiv.2202.08191

openalex publication_date 2022/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the BVP -y" + qy = λy 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 increases) and so it is advantageous to use data from lower modes. Our method also allows us to recover "any" potential from one boundary condition.

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