Stability of a Fermionic N + 1 Particle System with Point Interactions

Abstract

Abstract:We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m * . The value of m * is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.

High density limit of the Fermi polaron with infinite mass

“…In this appendix we prove (13) by applying Theorem 6.3 and Corollary 6.4 of [8] to the present case.…”

“…We remark that the construction of a semi-bounded Hamiltonian for the Fermi polaron with an impurity of finite mass is much more involved compared to (3), since it is not a simple generalization of a one-body operator. The problem was solved in two [6][7][8][9] and partially in three space dimensions [5,11,13]. Rigorous results concerning the ground state energy of these models mostly adressed the question of stability and the existence of a lower bound to the Hamiltonian that is uniform in the particle number N .…”

On a direct description of pseudorelativistic Nelson Hamiltonians

“…The construction sketched above is in some respect analogous to the one used in setting up zero-range Hamiltonians and the technical tools employed here are in fact inspired by previous works on many-body point interactions, in particular [CDF + 15] and [MS17].…”

Triviality of a model of particles with point interactions in the thermodynamic limit

“…Of particular interest is the unitary limit of infinite scattering length, where one has scale invariance due to the lack of any intrinsic length scale (see, e.g., [3,4,11,12,25]). Despite some effort [5][6][7]9,21], it remains an open problem to establish the existence of a many-particle model with two-body point interactions. Such a model is known B Thomas Moser thomas.moser@ist.ac.at Robert Seiringer robert.seiringer@ist.ac.at 1 Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria to be unstable in the case of bosons (a fact known as Thomas effect [3,5,24], closely related to the Efimov effect [8,22,26]) and hence can only exist for fermionic particles.…”

“…It has the advantage of being manifestly well defined, via a non-negative Dirichlet form. As already noted above, in general it is notoriously hard to define many-body systems with point interactions, see [5][6][7]9,21], due to the inherent instability problems. The model under consideration here was studied in [10], where it was shown to satisfy a Lieb-Thirring inequality, i.e., the energy can be bounded from below by a semiclassical expression of the form C ρ(x) 5/3 dx, with ρ the particle density and C a positive constant.…”