Models of zero-range interaction for the bosonic trimer at unitarity

Abstract

We present the mathematical construction of the physically relevant quantum Hamiltonians for a three-body systems consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered (the unitarity regime). The subject has several precursors in the mathematical literature. We proceed through an operator-theoretic construction of the self-adjoint extensions of the minimal operator obtained by restricting the free Hamiltonian to wave-functions that vanish in the vicinity of the coincidence hyperplanes: all extensions thus model an interaction precisely supported at the spatial configurations where particles come on top of each other. Among them, we select the physically relevant ones, by implementing in the operator construction the presence of the specific short-scale structure suggested by formal physical arguments that are ubiquitous in the physical literature on zero-range methods. This is done by applying at different stages the self-adjoint extension schemes a la Kre{\uı}n-Višik-Birman and a la von Neumann. We produce a class of canonical models for which we also analyse the structure of the negative bound states. Bosonicity and zero range combined together make such canonical models display the typical Thomas and Efimov spectra, i.e., sequence of energy eigenvalues accumulating to both minus infinity and zero. We also discuss a type of regularisation that prevents such spectral instability while retaining an effective short-scale pattern. Beside the operator qualification, we also present the associated energy quadratic forms. We structured our analysis so as to clarify certain steps of the operator-theoretic construction that are notoriously subtle for the correct identification of a domain of self-adjointness.