Quantum graphs with two-particle contact interactions

TL;DR

This paper develops models of many-particle quantum graphs with singular two-particle contact interactions, analyzing their spectral properties and extending to N bosons, including the Lieb-Liniger model on a graph.

Contribution

It introduces a rigorous construction of multi-particle quantum graph models with contact interactions, including hardcore and delta types, and extends to the Lieb-Liniger model for N bosons.

Findings

Spectra are discrete with proven Weyl laws.

Models are constructed for two distinguishable particles and identical bosons.

Extension to N bosons implementing the Lieb-Liniger model.

Abstract

We construct models of many-particle quantum graphs with singular two-particle contact interactions, which can be either hardcore- or delta-interactions. Self-adjoint realisations of the two-particle Laplacian including such interactions are obtained via their associated quadratic forms. We prove discreteness of spectra as well as Weyl laws for the asymptotic eigenvalue counts. These constructions are first performed for two distinguishable particles and then for two identicle bosons. Furthermore, we extend the models to N bosons with two-particle interactions, thus implementing the Lieb-Liniger model on a graph.