On the spectrum of sets made of cores and tubes

TL;DR

This paper investigates the spectral properties of unbounded sets composed of a bounded core with attached tubes, using variational methods to analyze eigenfunctions and disprove a conjecture related to spectral bounds.

Contribution

It introduces a variational approach to study spectral properties of core-and-tube sets and disproves a conjecture about the Makai-Hayman inequality using singular perturbation techniques.

Findings

Eigenfunctions exhibit exponential decay at infinity.

Disproves the conjecture that a specific planar set provides the sharp Makai-Hayman constant.

Analyzes the compactness properties of level sets of the Dirichlet integral.

Abstract

We analyze the spectral properties of a particular class of unbounded open sets. These are made of a central bounded ``core'', with finitely many unbounded tubes attached to it. We adopt an elementary and purely variational point of view, studying the compactness (or the defect of compactness) of level sets of the relevant constrained Dirichlet integral. As a byproduct of our argument, we also get exponential decay at infinity of variational eigenfunctions. Our analysis includes as a particular case a planar set (sometimes called ``bookcover''), already encountered in the literature on curved quantum waveguides. J. Hersch suggested that this set could provide the sharp constant in the {\it Makai-Hayman inequality} for the bottom of the spectrum of the Dirichlet-Laplacian of planar simply connected sets. We disprove this fact, by means of a singular perturbation technique.