Quantum graphs with singular two-particle interactions

Jens Bolte, Joachim Kerner

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We construct quantum models of two bosonic particles on a compact metric
graph with singular two-particle interactions. The Hamiltonians are self-adjoint
realisations of Laplacians acting on functions defined on pairs of edges in
such a way that the interaction is provided by boundary conditions. In order
to find such Hamiltonians closed and semi-bounded quadratic forms are
constructed, from which the associated self-adjoint operators are extracted.
We provide a general characterisation of such operators and, furthermore,
produce certain classes of examples. Finally, we show that these operators
possess purely discrete spectra and that the eigenvalues are distributed
following an appropriate Weyl asymptotic law.
Original languageEnglish
Article number045206
JournalJournal of Physics A: Mathematical and Theoretical
Issue number4
Early online date15 Jan 2013
Publication statusPublished - 2013

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