Ricerc@Sapienza

We study Winter or delta-shell model at finite volume (length),
describing a small resonating cavity
weakly-coupled to a large one.
For generic values of the coupling,
a resonance of the usual model corresponds,
in the finite-volume case,
to a compression of the spectral lines;
for specific values of the coupling,
a resonance corresponds instead to a
degenerate or a quasi-degenerate doublet.
A secular term of the form g^3 N
occurs in the perturbative expansion
of the momenta (or of the energies)
of the particle at third order in g,
where g is the coupling
among the cavities and N is the
ratio of the length of the large cavity
over the length of the small one.
These secular terms, which tend
to spoil the convergence of the
perturbative series in the large
volume case, N >> 1, are resummed
to all orders in g by means of standard
multi-scale methods.
The resulting improved perturbative expansions
provide a rather complete analytic description
of resonance dynamics at finite volume.