Monotonicity of discrete spectra of Dirichlet Laplacian in 3-dimensional layers

TL;DR

This paper studies how the eigenvalues of the Dirichlet Laplacian in 3D polyhedral layers change with geometry, showing monotonicity under certain conditions and revealing complex behaviors with asymmetry.

Contribution

It extends known monotonicity results to 3D polyhedral layers and provides explicit examples of non-monotonic spectral behavior due to asymmetry.

Findings

Eigenvalues below the essential spectrum depend monotonically on geometric parameters.

Explicit example of non-monotone spectral behavior caused by asymmetric perturbations.

Analysis of eigenvalue limits as geometric parameters approach critical configurations.

Abstract

We investigate monotonicity properties of eigenvalues of the Dirichlet Laplacian in polyhedral layers of fixed width. We establish that eigenvalues below the essential spectrum threshold monotonically depend on geometric parameters defining the polyhedral layer, generalizing previous results known for planar V-shaped waveguides and conical layers. Moreover, we demonstrate non-monotone spectral behavior arising from asymmetric geometric perturbations, providing an explicit example where unfolding the polyhedral layer unexpectedly leads to the emergence of discrete eigenvalues. The limiting behavior of eigenvalues as the geometric parameters approach critical configurations is also rigorously analyzed.