  A rigorous mathematical theory is developed to explain the super-resolution
phenomenon observed in the experiment by F.Lemoult, M.Fink and G.Lerosey
(Acoustic resonators for far-field control of sound on a subwavelength scale,
Phys. Rev. Lett., 107 (2011)). A key ingredient is the calculation of the
resonances and the Green function in the half space with the presence of a
system of Helmholtz resonators in the quasi-stationary regime. By using
boundary integral equations and generalized Rouche's theorem, the existence and
the leading asymptotic of the resonances are rigorously derived. The integral
equation formulation also yields the leading order terms in the asymptotics of
the Green function. The methodology developed in the paper provides an elegant
and systematic way for calculating resonant frequencies for Helmholtz
resonators in assorted space settings as well as in various frequency regimes.
By using the asymptotics of the Green function, the analysis of the imaging
functional of the time-reversal wave fields becomes possible, which clearly
demonstrates the super-resolution property. The result provides the first
mathematical theory of super-resolution in the context of a deterministic
medium and sheds light to the mechanism of super-resolution and super-focusing
for waves in deterministic complex media.
