Nonlinear Schr\"odinger equations near an infinite well potential

TL;DR

This paper studies the existence and approximation of standing wave solutions of nonlinear Schrödinger equations with potentials close to an infinite well, establishing conditions under which solutions for the infinite well extend to finite well cases.

Contribution

It provides a rigorous analysis of how solutions for the idealized infinite well potential approximate solutions for large but finite potentials in nonlinear Schrödinger equations.

Findings

Established conditions for the continuation of solutions from infinite to large finite potentials.

Proved singular continuation results for nonlinear Schrödinger equations near an infinite well.

Demonstrated the approximation of solutions in the limit as potential well depth tends to infinity.

Abstract

The paper deals with standing wave solutions of the dimensionless nonlinear Schr\"odinger equation \label{eq:abs1} i\Phi_t(x,t) = -\Delta_x\Phi +V_\la(x)\Phi + f(x,\Phi), \quad x\in\R^N,\ t\in\R,\tag{$NLS_\la$} where the potential $V_\la:\R^N\to\R$ is close to an infinite well potential $V_\infty:\R^N\to\R$, i. e. $V_\infty=\infty$ on an exterior domain $\R^N\setminus\Om$, $V_\infty|_\Om\in L^\infty(\Om)$, and $V_\la\to V_\infty$ as $\la\to\infty$ in a sense to be made precise. The nonlinearity may be of Gross-Pitaevskii type. A solution of \eqref{eq:abs1} with $\la=\infty$ vanishes on $\R^N\setminus\Om$ and satisfies Dirichlet boundary conditions, hence it solves \label{eq:abs2} i\Phi_t(x,t) &= -\Delta_x\Phi +V_\la(x)\Phi + f(x,\Phi), &&\quad x\in\Om,\ t\in\R \Phi(x,t) &= 0 &&\quad x\in\pa\Om,\ t\in\R. \tag{$NLS_\infty$}. We investigate when a solution $\Phi_\infty$ of the infinite well potential \eqref{eq:abs2} gives rise to nearby solutions $\Phi_\la$ of the finite well potential \eqref{eq:abs1} with $\la\gg1$ large. Considering \eqref{eq:abs2} as a singular limit of \eqref{eq:abs1} we prove a kind of singular continuation type results.