On an optimal potential of Schr\"odinger operator with prescribed $m$ eigenvalue

TL;DR

This paper introduces a new inverse spectral problem for Schrödinger operators, establishing the existence of solutions that match prescribed eigenvalues by solving nonlinear differential equations, linking inverse problems with nonlinear dynamics.

Contribution

It formulates a novel inverse spectral problem with prescribed eigenvalues and proves that solutions can be obtained through nonlinear differential equations, connecting inverse spectral theory with nonlinear analysis.

Findings

Existence of solutions to the inverse spectral problem.

Solutions can be explicitly constructed via nonlinear differential equations.

New relationship between inverse problems and nonlinear differential equations.

Abstract

The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential $V_0$ find the closest function $\hat{V}$ such that $m$ eigenvalues of one-dimensional space Schrodinger operator with potential $\hat{V}$ would coincide with the given values $ E_1 $, $ \ldots $, $ E_m \in \mathbb {R} $. In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solution can be directly found by solving a system of nonlinear differential equations.