Nontrivial coupling at quantum graph vertices obtained through squeezing of Dirichlet networks

Nontrivial coupling at quantum graph vertices obtained through squeezing of Dirichlet networks

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P. Exner, Doppler Institute, Prague
Jadwin Hall 343

The problem discussed in this talk is motivated by efforts to understand approximation of quantum graph Hamiltonians by Laplacians on families of "fat graphs."  The emphasis is on new results in the Dirichlet case, however, first we review the background and explain the importance of vertex boundary conditions using a lattice graph example, and mention known result in both the Neumann and Dirichlet setting.  After that we suggest a way how a wider classes of vertex couplings can be obtained from squeezed Dirichlet networks.  To illustrate the proposed strategy we work out the simplest nontrivial example, a family of bent tubes giving a graph of one vertex and two edges, or a two-parameter family of generalized point interactions on the line.