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.
Highlights
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∗
Models of particles with point interactions are ubiquitously used in physics as an idealized description whenever the range of the interparticle interactions is much shorter than other relevant length scales
They were introduced in the early days of quantum mechanics as models of nuclear interactions [2,3,14,35,38], but have proved useful in other branches of physics, like polarons and cold atomic gases [40]
Summary
Models of particles with point interactions are ubiquitously used in physics as an idealized description whenever the range of the interparticle interactions is much shorter than other relevant length scales. The system under consideration is stable for m = 1 This latter case is of particular importance, in view of constructing a model of a gas of spin 1/2 fermions close to the unitary limit, where the scattering length becomes much larger than the range of the interactions. We note that stability in other spin sectors is still an open problem, whose solution would be of great interest because of the relevance of the model for cold atomic gases (see [40] and references there).
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