Quantum graphs with singular twoparticle interactions
Abstract
We construct quantum models of two particles on a compact metric graph with singular twoparticle interactions. The Hamiltonians are selfadjoint realizations 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 semibounded quadratic forms are constructed, from which the associated selfadjoint operators are extracted. We provide a general characterization 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 February 2013
 DOI:
 10.1088/17518113/46/4/045206
 arXiv:
 arXiv:1112.4751
 Bibcode:
 2013JPhA...46d5206B
 Keywords:

 Mathematical Physics
 EPrint:
 J. Phys. A: Math. Theor. 46 (2013) 045206