Many-particle quantum graphs and Bose-Einstein condensation

In this paper we propose quantum graphs as one-dimensional models with a complex topology to study Bose-Einstein condensation and phase transitions in a rigorous way. We fist investigate non-interacting many-particle systems on quantum graphs and provide a complete classification of systems that exhibit Bose-Einstein condensation. We then consider models of interacting particles that can be regarded as a generalisation of the well-known Tonks-Girardeau gas. Here our principal result is that no phase transitions occur in bosonic systems with repulsive hardcore interactions, indicating an absence of Bose-Einstein condensation.