Inverse spectral results for Schr\"odinger operators on the unit interval with potentials in L^P spaces

TL;DR

This paper establishes conditions under which two Schr"odinger potentials on the interval [0,1], known on a subinterval and with differences in L^p, are identical based on their eigenvalues, highlighting the role of p.

Contribution

It provides explicit bounds on the number of common eigenvalues needed to guarantee potential equality, depending on p and the subinterval parameter a.

Findings

Explicit eigenvalue count depending on p and a

Potential equality from partial spectral data

Role of L^p spaces in inverse spectral problems

Abstract

We consider the Schr\"odinger operator on $[0,1]$ with potential in $L^1$. We prove that two potentials already known on $[a,1]$ ($a\in(0,{1/2}]$) and having their difference in $L^p$ are equal if the number of their common eigenvalues is sufficiently large. The result here is to write down explicitly this number in terms of $p$ (and $a$) showing the role of $p$.