Spectrum of the Dirichlet Laplacian in waveguides with parallel cross-sections

2020/05/10 by Alessandra A. Verri, Verri, Alessandra A.
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2005.04772

openalex publication_date 2020/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ \mathbb R³ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider -Δ_Ω^D the Dirichlet Laplacian operator in Ω. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of -Δ_Ω^D. Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for -Δ_Ω^D; in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.

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