Curvature-induced bound states in Robin waveguides and their asymptotical properties

TL;DR

This paper investigates how the geometry of infinite planar domains influences the bound states of Robin Laplacians, deriving asymptotic formulas and conditions for the existence of discrete spectra based on boundary curvature.

Contribution

It provides a two-term asymptotic formula linking boundary curvature to bound states and characterizes spectral properties for various domain geometries.

Findings

Asymptotic formula relates boundary curvature to bound states.

Discrete spectrum exists for certain domain deformations and convex shapes.

No discrete spectrum for concave domain geometries.

Abstract

We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a two-term asymptotic formula in which the next-to-leading term is determined by the extremum of the boundary curvature. We also discuss the non-asymptotic case of attractive boundary interaction and show that the discrete spectrum is nonempty if the domain is a local deformation of a halfplane or a wedge of angle less than $\pi$, and it is void if the domain is concave.