Ground states on a fractured strip and one dimensional reduction

TL;DR

This paper studies ground states of a nonlinear Schrödinger equation on a strip with boundary conditions, showing their existence and convergence to line ground states as the strip narrows.

Contribution

It establishes the existence of ground states on a strip with specific boundary conditions and demonstrates their convergence to line ground states in the narrow strip limit.

Findings

Existence of ground states on the strip with Neumann and delta boundary conditions

Convergence of energy minimizers to line ground states as strip width approaches zero

Ground states characterized as minimizers of action or energy under constraints

Abstract

We consider the nonlinear Schr\''odinger equation on a strip with Neumann boundary conditions and a delta condition on the $x$-axis. First, we show the existence of ground states as minimizers of the action or of the energy under suitable constraints. Second, we prove that the energy minimizers converge to the ground state on the line with a delta condition as the amplitude of the strip shrinks to zero.