Convergence of resonances on thin branched quantum waveguides

2007/02/21 by Pavel Exner, Olaf Post
Mathematics · Physics and Astronomy · #Numerical methods in inverse problems #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.DG #math.MP #math.SP

paper · pdf · doi:10.1063/1.2749703

48 pages, 1 figure

arxiv created 2007/02/21 · openalex publication_date 2007/09/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family Xε of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on Xε approximate those of the Laplacian with “free” boundary conditions on X0, the skeleton graph of Xε.

Citations

• Kirchhoff's rule for quantum wires
• Quantum Wires with Magnetic Fluxes
• Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
• A class of analytic perturbations for one-body Schrödinger Hamiltonians
• Convergence of spectra of graph-like thin manifolds