Quantum graphs with singular two-particle interactions

TL;DR

This paper develops quantum models for two particles on compact metric graphs with singular interactions, characterizing the Hamiltonians via boundary conditions and analyzing their spectral properties.

Contribution

It introduces a general framework for constructing self-adjoint Hamiltonians with singular two-particle interactions on quantum graphs, including classes of examples and spectral analysis.

Findings

Hamiltonians are self-adjoint and semi-bounded

Operators have purely discrete spectra

Eigenvalues follow Weyl asymptotic law

Abstract

We construct quantum models of two 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. We then consider identical particles and project to the bosonic and fermionic subspaces. Finally, we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate Weyl asymptotic law.