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

02/16/2022

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by   Rob Rahm, et al.

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We consider the BVP -y" + qy = λ 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 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.