Many-particle quantum graphs and Bose-Einstein condensation

Abstract

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.

On bound electron pairs in a quantum wire

“…Remark 3.4. Note that Theorem 3.3 can be reformulated in terms of a critical temperature rather than a critical density, see for example [Rue69,BK14].…”

Nonlinear dynamics on branched structures and networks

“…For a complete review on most recent results see the monography [16] In this review there are no proofs of the condensation or of the Gross-Pitaevskii regime for systems on graphs. In fact, it is widely known that no condensation can occur in one-dimensional systems, in the sense that the phase transition that defines the condensation cannot take place: however, in 2015 J. Bolte and Kerner [18] proved condensation for free gas and no condensation for interacting gases in graphs, considered as quasi-dimensional systems, in the sense of the presence of the phase transition. Moreover, in 1996 W. Ketterle and N.J. van Druten [42] gave an evidence of a concentration phenomenon under some aspects analogous to condensation.…”

Particle creation and annihilation at interior boundaries: one-dimensional models

“…The solutions to (30) and (31) can be obtained directly from the scattering states (22). Let k = √ E > 0.…”