Some spectral properties of Rooms and Passages domains and their skeletons

TL;DR

This paper studies the spectral properties of Laplacians on Rooms and Passages domains, revealing asymptotic behaviors and relationships with Sturm-Liouville operators, advancing understanding of spectral geometry in complex domains.

Contribution

It provides new asymptotic estimates for Neumann Laplacians and explores their connection to Sturm-Liouville operators on domain skeletons, using Dirichlet-Neumann bracketing techniques.

Findings

Second term of Neumann Laplacian's eigenvalue counting function is of order √λ.

Upper estimate for Dirichlet Laplacian's eigenvalue counting function is of order √λ.

Established relationship between Neumann Laplacians and Sturm-Liouville operators on skeletons.

Abstract

In this paper we investigate spectral properties of Lapla- cians on Rooms and Passages domains. In the first part, we use Dirichlet- Neumann bracketing techniques to show that for the Neumann Lapla- cian in certain Rooms and Passages domains the second term of the asymptotic expansion of the counting function is of order $\sqrt{\lambda}$. For the Dirichlet Laplacian our methods only give an upper estimate of the form $\sqrt{\lambda}$. In the second part of the paper, we consider the relation- ship between Neumann Laplacians on Rooms and Passages domains and Sturm-Liouville operators on the skeleton.