On the spectrum of a Robin Laplacian in a planar waveguide

TL;DR

This paper investigates the spectral properties of the Robin Laplacian in a planar waveguide, focusing on the essential spectrum and conditions for discrete eigenvalues under specific boundary conditions.

Contribution

It provides new insights into the essential spectrum and establishes criteria for the existence of discrete spectrum in Robin Laplacians on waveguides.

Findings

Characterization of the essential spectrum for Robin Laplacians.

Sufficient conditions for discrete spectrum existence.

Analysis of boundary coupling function limits.

Abstract

We consider the Laplace operator in a planar waveguide, i.e., an infinite two-dimensional straight strip of constant width, with particular types of Robin boundary conditions. We study the essential spectrum of the corresponding Laplacian when the boundary coupling function has a limit at infinity. Furthermore, we derive sufficient conditions for the existence of discrete spectrum.