Infinitely many solutions to linearly coupled Schr\"{o}dinger equations with non-symmetric potential

TL;DR

This paper proves the existence of infinitely many positive synchronized solutions for a linearly coupled Schrödinger system in low-dimensional space, even without symmetry in the potentials, extending previous nonlinear coupling results.

Contribution

It introduces a novel application of Lyapunov-Schmidt reduction and localized energy methods to linearly coupled Schrödinger equations without symmetry assumptions.

Findings

Existence of infinitely many solutions established.

Solutions are positive and synchronized.

Extends previous nonlinear coupling results to linear coupling.

Abstract

We study a linearly coupled Schr\"{o}dinger system in $\R^N(N\leq3).$ Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties and the parameters in the system satisfy some suitable restrictions. Using the Liapunov-Schmidt reduction methods two times and combing localized energy method, we prove that the problem has infinitely many positive synchronized solutions, which extends the result Theorem 1.2 about nonlinearly coupled Schr\"{o}dinger equations in \cite{aw} to our linearly coupled problem.