From quasimodes to resonances

Stefanov and Vodev [11], [12], obtained a remarkable result which says that for scattering by compactly supported perturbations in odd dimensional Euclidean space, existence of localized quasimodes implies existence of resonances rapidly converging to the real axis. The purpose of this note is to extend this result to all dimensions and to a wide class of non-compactly supported perturbations. Our method also gives lower bounds for the number of resonances in small neighbourhoods of the real axis. In fact, any quasi-mode construction will immediately provide a linear lower bound. A finer analysis of quasimodes should provide examples for which our main theorem gives the optimal lower bounds, r. A typical example is the Helmholtz resonator (see Fig.1) for which using the results of Babich-Buldyrev [1], Ralston [8], Lazutkin [4] and Popov [6] we obtain at least a linear growth of resonances in small neighbourhoods of the real axis. We could also consider metric perturbations – see [2] and references given there for construction of quasimodes for manifolds. The argument of Stefanov and Vodev is an application of the PhragménLindelöf principle and of an a priori bound on the meromorphic continuation of the resolvent, R(λ):