Recovering a part of potential by partial information on spectra of boundary problems

Abstract

Under additional conditions uniqueness of the solution is proved for the following problem. Given 1) the spectrum of the Dirichlet problem for the Sturm-Liouville equation on $[0,a]$ with real potential $q(x)\in L_2(0,a)$, 2) a certain part of the spectrum of the Dirichlet problem for the same equation on $[\frac{a}{3},a]$ and 3) the potential on $[0,\frac{a}{3}]$. The aim is to find the potential on $[\frac{a}{3},a]$.