Convergence of resonances on thin branched quantum wave guides

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 $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on $X_\eps$ approximate those of the Laplacian with ``free'' boundary conditions on $X_0$, the skeleton graph of $X_\eps$.

48 pages, 1 figure

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• Spectral convergence of non-compact quasi-one-dimensional spaces
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• Soliton transport in tubular networks: transmission at vertices in the shrinking limit
• Planar waveguide with "twisted" boundary conditions: small width
• Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains
• Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
• Vertex coupling in quantum graphs: approximations by scaled Schroedinger operators
• Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions
• Nonlinear standing waves on planar branched systems: Shrinking into metric graph