  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.
