Sjöstrand, Johannes Asymptotic distribution of eigenfrequencies for damped wave equations. (English) Zbl 0984.35121 Publ. Res. Inst. Math. Sci. 36, No. 5, 573-611 (2000). The author deals with the eigenfrequencies associated to a damped wave equation. He derives Weyl asymptotics for the distribution of the real parts of the eigenfrequencies and shows that up to a set of denstiy 0, the eigenfrequencies are confined to a band determined by the Birkhoff limits of the damping coefficient. The author proves that certain averages of the imaginary parts converge to the average of the damping coefficient. Reviewer: Messoud Efendiev (Berlin) Cited in 1 ReviewCited in 35 Documents MSC: 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35P15 Estimates of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation Keywords:Birkhoff limits; Weyl asymptotics; damping coefficient × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bony, J. F., Majoration du nombre de resonances dans des domames de taille h, Preprint, (2000). [2] Dimassi, M. and Sjostrand, J., Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes 268, Cambridge Univ. Press, 1999. · Zbl 0926.35002 [3] Freitas, P., Spectral sequences for quadratic pencils and the inverse problem for the damped wave equation, J. Math. Pures et AppL, 78 (1999), 965-980. · Zbl 0956.47006 · doi:10.1016/S0021-7824(99)00135-X [4] Gohberg, I. C. and Krein, M. G., Introduction to the theory of linear non-self adjoint operators, Amer. Math. Soc., Providence, RI 1969. · Zbl 0181.13503 [5] Hormander, L., Fourier integral operators I, Acta Math., 127 (1971), 79-183. [6] Lebeau, G., Equation des ondes amorties, Algebraic and geometric methods in mathe- matical physics, (Kaciveli, 1993), 73-109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. [7] Rauch, J. and Taylor, M., Decay of solutions to nondissipative hyperbolic systems on compact manifolds, CPAM, 28 (1975), 501-523. · Zbl 0295.35048 · doi:10.1002/cpa.3160280405 [8] Sjostrand, J., Density of resonances for strictly convex analytic obstacles, Can. J. Math., 48 (2) (1996), 397-447. · Zbl 0863.35072 · doi:10.4153/CJM-1996-022-9 [9] , A trace formula and review of some estimates for resonances, p.377-437 in Microlocal Analysis and Spectral Theory, NATO ASI Series C, vol. 490, Kluwer 1997. See also Resonances for bottles and trace formulae. Math. Nachr., to appear. [10] Sjostrand, J. and Zworski, M., Asymptotic distribution of resonances for convex ob- stacles, Acta Math., 183 (1999), 191-253. Note added in ptoof: Theorem 0.1 is due to A. S. Marks, V. I. Matsaev, Trans. Moscow Math. Soc., 184 (1), 139-187. See also S. Markus, Introduction to the spectral theory of polynomial operator pencils, AMS 1988. We thank M. Solomjak, D. Yafaev and S. Markus for this information. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.