Periodic Manifolds with Spectral Gaps

2002/07/14 by Olaf Post, Post, Olaf · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.math-ph/0207017

21 pages, 3 eps-figures, LaTeX

arxiv created 2002/07/14 · openalex publication_date 2002/07/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number N we construct periodic (i.e. covering) manifolds such that the essential spectrum of the corresponding Laplacian has at least N open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.

Citations

• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator

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• Gradient bounds for viscosity solutions to certain elliptic equations

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