On nonlinear boundary value problem corresponding to $N$-dimensional inverse spectral problem

TL;DR

This paper links an inverse spectral problem for N-dimensional Schrödinger equations to a nonlinear boundary value problem, providing exact solutions, stability analysis, and new existence and uniqueness results.

Contribution

It introduces a novel relationship between inverse spectral problems and nonlinear boundary value problems, leading to exact solutions and theoretical insights.

Findings

Established a relationship between inverse spectral problems and nonlinear boundary value problems.

Provided an exact solution for the inverse spectral problem.

Proved new results on the existence and uniqueness of solutions.

Abstract

We establish a relationship between an inverse optimization spectral problem for N-dimensional Schr\"odinger equation $ -\Delta \psi+q\psi=\lambda \psi $ and a solution of the nonlinear boundary value problem $-\Delta u+q_0 u=\lambda u- u^{\gamma-1},~~u>0,~~ u|_{\partial \Omega}=0$. Using this relationship, we find an exact solution for the inverse optimization spectral problem, investigate its stability and obtain new results on the existence and uniqueness of the solution for the nonlinear boundary value problem.