A many-body problem with point interactions on two-dimensional manifolds

Abstract

A non-perturbative renormalization of a many-body problem, where nonrelativistic bosons living on a two-dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean-field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan-Symanzik equation) for the principal operator of the model is derived and the β function is exactly calculated for the general case, which includes all particle numbers.

From Short-Range to Contact Interactions in Two-dimensional Many-Body Quantum Systems

“…The well-known Lieb-Liniger model with repulsive δ-interactions is derived from a trapped 3d Bose gas with two-body potentials in [22]. TMS Hamiltonians like H in Theorem 1.1 have also been described as resolvent limits of N -body Hamiltonians, where the regularized two-body contact interaction is an integral operator, rather than a potential, and the regularization is achieved by an ultraviolet cutoff [6,7,9] or a reversed heat flow [8]. In these cases the convergence is easier to establish than in the case studied here.…”

From Short-Range to Contact Interactions in Two-dimensional Many-Body Quantum Systems

“…The well-known Lieb-Liniger model with repulsive δ-interactions is derived from a trapped 3d Bose gas with two-body potentials in [19]. TMS Hamiltonians like H in Theorem 1.1 have also been described as resolvent limits of N -body Hamiltonians, where the regularized two-body contact interaction is an integral operator, rather than a potential, and the regularization is achieved by an ultraviolet cutoff [5,6,8] or a reversed heat flow [7]. In these cases the convergence is easier to establish than in the case studied here.…”

One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials

“…Green's function approach is rather useful since it includes all the information about the spectrum of the Hamiltonian. The method we use here has been constructed in the nonrelativistic version of the model on two-and three-dimensional manifolds [28,29] and in the nonrelativistic manybody version of it in [30]. A one-dimensional nonrelativistic many-body version of the model (1.2), where the particles are interacting through the two-body Dirac delta potentials, is known as the Lieb-Liniger model [31] and has been studied in great detail in the literature [32][33][34][35].…”