2002/06/18 by D. I. Borisov, D. Borisov, Pavel Exner +3 · 80 citations
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Boundary value problem #Continuous spectrum #Coupling (piping) #Dirichlet boundary condition #Dirichlet distribution #Eigenfunction #Eigenvalues and eigenvectors #Laplace operator #Mathematical analysis #Mathematics #Neumann boundary condition #Operator (biology) #Physics #Pure mathematics #Quantum mechanics #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #Spectrum (functional analysis) #cond-mat #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1063/1.1519941
published in Journal of Mathematical Physics 43(12), 6265-6278 (American Institute of Physics)
arxiv created 2002/06/18 · openalex publication_date 2002/12/01 · arxiv updated 2014/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the Laplacian in a strip R×(0,d) with the boundary condition which is Dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to Neumann. This operator is known to have a non-empty and simple discrete spectrum for any a>0. There is a sequence 0<a1<a2<⋯ of critical values at which new eigenvalues emerge from the continuum when the Neumann window expands. We find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to (a−an)2 with an explicit coefficient expressed in terms of the corresponding threshold-energy resonance eigenfunction.