Hamiltonian for a Bose gas with Contact Interactions

Abstract

We study the Hamiltonian for a three-dimensional Bose gas of $N \geq 3$ spinless particles interacting via zero-range (also known as contact) interactions. Such interactions are encoded by (singular) boundary conditions imposed on the coincidence hyperplanes, i.e., when the coordinates of two particles coincide. It is well known that imposing the same kind of boundary conditions as in the two-body problem with a point interaction leads to a Hamiltonian unbounded from below (and thus unstable). This is due to the fact that the interaction becomes overly strong and attractive when the coordinates of three or more particles coincide. In order to avoid such instability, we develop a suggestion originally formulated by Minlos and Faddeev in 1962, introducing slightly modified boundary conditions that weaken the strength of the interaction between two particles $i$ and $j$ in two scenarios: (a) a third particle approaches the common position of $i$ and $j$; (b) another distinct pair of particles approach each other. In all other cases, the usual boundary condition is restored. Using a quadratic form approach, we construct a class of Hamiltonians characterized by such modified boundary conditions, that are self-adjoint and bounded from below. We also compare our approach with the one developed years ago by Albeverio, H{\o}egh-Krohn and Streit using the theory of Dirichlet forms (J. Math. Phys., 18, 907--917, 1977). In particular, we show that the $N$-body Hamiltonian defined by Albeverio et al. is a special case of our class of Hamiltonians. Furthermore, we also introduce a Dirichlet form by considering a more general weight function, and we prove that the corresponding $N$-body Hamiltonians essentially coincide with those constructed via our method.