Thin waveguides with Robin boundary conditions

TL;DR

This paper analyzes the spectral behavior of the Laplace operator in thin 3D tubes with Robin boundary conditions, revealing different asymptotic behaviors based on symmetry conditions of the cross section.

Contribution

It provides a detailed asymptotic analysis of the spectrum for Robin boundary conditions, highlighting the impact of symmetry on localization and effective potential formation.

Findings

Localization of low-energy levels near minimum points when symmetry fails

Explicit form of effective potential in symmetric case

Different spectral behaviors depending on symmetry criterion

Abstract

We consider the Laplace operator in a thin three dimensional tube with a Robin type condition on its boundary and study, asymptotically, the spectrum of such operator as the diameter of the tube's cross section becomes infinitesimal. In contrast with the Dirichlet condition case, we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section. If that symmetry condition fails, then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube's axis depending on the curvature and the rotation angle. In the symmetric case, the behavior of lower energy modes is shown to be ruled by a one dimensional Sturm-Liouville problem involving an effective potential given in explicit form.