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Quasimodes and resonances: sharp lower bounds. (English) Zbl 0952.47013
From the introduction: “The purpose of this paper is to obtain sharp lower bounds of the number of resonances (scattering poles) close to the real axis. We consider a situation where one can construct real quasimodes, that is, a sequence of approximate real “resonances” and corresponding approximate solutions supported in a fixed compact set. Our main result states, lossely speaking, that quasimodes are perturbed resonances near the real axis and that the number of resonances close to the real axis is at least equal to the number of the quasimodes, counting multiplicities.”
In earlier works in the same direction, P. Stefanov and G. Vodev [Commun. Math. Phys. 176, No. 3, 645-659 (1996; Zbl 0851.35032)] and S.-H. Tang and M. Zworski [Math. Res. Lett. 5, No. 3, 261-272 (1998; Zbl 0913.35101)], showed in increasing generality that the existence of a compactly supported approximate eigenfunction generates at least one resonance close to the corresponding approximate real eigenvalue, but the problem of getting at least as many resonances as eigenvalues, remained open in the case when the latter form clusters. The main contribution of the present paper is to solve this problem affirmatively. The proof uses ideas from the earlier works together with a clever improvement. Some (but not all) ideas in these works are reminiscent of the theory of general non-selfadjoint operators. See S. Agmon [“Lectures on elliptic boundary value problems” (1965; Zbl 0142.37401), ch. 16].

MSC:
47A40 Scattering theory of linear operators
35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47F05 General theory of partial differential operators
58J99 Partial differential equations on manifolds; differential operators
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