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

2025/08/18 by Bakharev, Fedor, Matveenko, Sergey
#60J65 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35J05 #Secondary: 35K05 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.12696

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.

Citations

• Fractional Laplacian in V-shaped waveguide
• Spectra of Dirichlet Laplacian in 3-dimensional polyhedral layers
• Existence of the discrete spectrum in the Fichera layers and crosses of arbitrary dimension
• Plane waveguides with corners in the small angle limit
• Quantum waveguides with corners

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