Eigenvalues in Spectral Gaps of a Perturbed Periodic Manifold

2002/07/14 by Olaf Post, Post, Olaf
Computer Science · Mathematics · Physics and Astronomy · #35P20 #58J37 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:35P20 #msc:58J37

paper · pdf · doi:10.48550/arxiv.math-ph/0207018

30 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 consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the number of eigenvalue branches crossing a fixed level is established in terms of a discrete eigenvalue problem. Furthermore, we discuss examples of perturbations leading to infinitely many eigenvalue branches coming from above resp. finitely many branches coming from below.

Related

• Periodic Manifolds with Spectral Gaps
• Generating spectral gaps by geometry
• Gaps between consecutive eigenvalues for compact metric graphs
• The Gaps of Consecutive Eigenvalues of Laplacian on Riemannian Manifolds
• Spectral band bracketing for Laplacians on periodic metric graphs