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

TL;DR

This paper constructs and analyzes quantum Hamiltonians for a three-boson system with zero-range interactions at unitarity, revealing Efimov and Thomas spectra and proposing regularization methods to stabilize the spectrum.

Contribution

It provides a rigorous operator-theoretic construction of self-adjoint Hamiltonians for the bosonic trimer at unitarity, incorporating physically relevant short-scale structures.

Findings

Reveals Efimov and Thomas spectral patterns in the models.

Develops a regularization method to prevent spectral instability.

Provides energy quadratic forms and clarifies domain issues in the operator construction.

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{\u\i}n-Vi\v{s}ik-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.