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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
##### Citations:
Zbl 0851.35032; Zbl 0913.35101; Zbl 0142.37401
Full Text:
##### References:
 [1] Fernando Cardoso and Georgi Popov, Rayleigh quasimodes in linear elasticity , Comm. Partial Differential Equations 17 (1992), no. 7-8, 1327-1367. · Zbl 0795.35067 [2] Yves Colin de Verdière, Quasi-modes sur les variétés Riemanniennes , Invent. Math. 43 (1977), no. 1, 15-52. · Zbl 0449.53040 [3] Peter D. Lax and Ralph S. Phillips, Scattering theory , Pure and Applied Mathematics, vol. 26, Academic Press Inc., Boston, MA, 1989, 2d ed. · Zbl 0697.35004 [4] Vladimir F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 24, Springer-Verlag, Berlin, 1993. · Zbl 0814.58001 [5] R. Melrose, “Polynomial bound on the distribution of poles in scattering by an obstacle” , Journeés “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, France, 1984), 3, Soc. Math. France, Paris, 1984. · Zbl 0535.35067 [6] Georgi S. Popov, Quasimodes for the Laplace operator and glancing hypersurfaces , Microlocal analysis and nonlinear waves (Minneapolis, MN, 1988-1989), IMA Vol. Math. Appl., vol. 30, Springer, New York, 1991, pp. 167-178. · Zbl 0794.35030 [7] G. Popov, Effective stability invariant tori and quasimodes with exponentially small error , Université de Nantes, preprint, 1997. [8] J. V. Ralston, Approximate eigenfunctions of the Laplacian , J. Differential Geometry 12 (1977), no. 1, 87-100. · Zbl 0385.58012 [9] J. Sjöstrand, A trace formula and review of some estimates for resonances , Microlocal analysis and spectral theory (Lucca, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 490, Kluwer Acad. Publ., Dordrecht, 1997, pp. 377-437. · Zbl 0877.35090 [10] Johannes Sjöstrand and Georgi Vodev, Asymptotics of the number of Rayleigh resonances , Math. Ann. 309 (1997), no. 2, 287-306. · Zbl 0890.35098 [11] Johannes Sjöstrand and Maciej Zworski, Complex scaling and the distribution of scattering poles , J. Amer. Math. Soc. 4 (1991), no. 4, 729-769. JSTOR: · Zbl 0752.35046 [12] Johannes Sjöstrand and Maciej Zworski, Distribution of scattering poles near the real axis , Comm. Partial Differential Equations 17 (1992), no. 5-6, 1021-1035. · Zbl 0766.35031 [13] P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body , Duke Math. J. 78 (1995), no. 3, 677-714. · Zbl 0846.35139 [14] P. Stefanov and G. Vodev, Neumann resonances in linear elasticity for an arbitrary body , Comm. Math. Phys. 176 (1996), no. 3, 645-659. · Zbl 0851.35032 [15] Siu-Hung Tang and Maciej Zworski, From quasimodes to reasonances , Math. Res. Lett. 5 (1998), no. 3, 261-272. · Zbl 0913.35101 [16] Georgi Vodev, Sharp bounds on the number of scattering poles for perturbations of the Laplacian , Comm. Math. Phys. 146 (1992), no. 1, 205-216. · Zbl 0766.35032 [17] Maciej Zworski, Sharp polynomial bounds on the number of scattering poles , Duke Math. J. 59 (1989), no. 2, 311-323. · Zbl 0705.35099 [18] M. Zworski, 1990, untitled work.
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