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We consider Winter (or delta-shell) model at finite volume, describing asmall resonating cavity weakly coupled to a large one, for small andintermediate volumes (lengths). By defining N as the ratio of the length of thelarge cavity over the small one, we study the symmetric case N=1, in which thetwo cavities actually have the same length, as well as the cases N=2,3,4.  By increasing N in the above range, the transition from a simple quantumoscillating system to a system having a resonance spectrum is investigated. Wefind that each resonant state is represented, at finite volume, by a cluster ofstates, each one resonating in a specific coupling region, centered around astate resonating at very small couplings.  We derive high-energy expansions for the particle momenta in the above Ncases, which (approximately) resum their perturbative series to all orders inthe coupling among the cavities. These new expansions converge rather quicklywith the order, provide, surprisingly, a uniform approximation in the couplingand also work, again surprisingly, at low energies.  We construct a first resummation scheme having a clear physical picture,which is based on a function-series expansion, as well as a second scheme basedon a recursion equation. The two schemes coincide at leading order, while theydiffer from next-to-leading order on. In particular, the recursive schemerealizes an approximate resummation of the function-series expansion generatedwithin the first scheme.

Winter (or delta-shell) Model at Small and Intermediate Volumes