Abstract
Under a geometric assumption on the region near the end of its neck, we prove an optimal exponential lower bound on the widths of resonances for a general two-dimensional Helmholtz resonator. An extension of the result to the n-dimensional case, \({n \leq 12}\), is also obtained.
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Adams R.A.: Sobolev Spaces. Academic Press, Boston (1975)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Universitext (ISBN 978-0-387-70913-0) (2011)
Brown, R.M., Hislop, P.D., Martinez, A.: Lower Bounds on Eigenfunctions and the first Eigenvalue Gap. Differential Equations with Applications to Mathematical Physics. In: Ames, W.F., Harell, E.M., Herod, J.V. (eds.) Mathematical and Science in Engineering, vol. 192, Academic Press 1993. vol. 22, pp. 269–279 (1971)
Burq N.: Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180, 1–29 (1998)
Burq N.: Lower bounds for shape resonances widths of long range Schrödinger operators. Am. J. Math. 124, 677–735 (2002)
Chazarain, J., Piriou, A.: Introduction à la Théorie des Équations aux Dérivées Partielles Linéaires. Gauthier-Villars, (1981)
Fujiie S., Lahamar-Benbernou A., Martinez A.: Width of shape resonances for non globally analytic potentials. J. Math. Soc. Japan Volume 63(1), 1–78 (2011)
Fernandez C., Lavine R.: Lower bounds for resonance width in potential and obstacle scattering. Comm. Math. Phys. 128, 263–284 (1990)
Harrel E.: General lower bounds for resonances in one dimension. Commun. Math. Phys. 86, 221–225 (1982)
Harrel E., Simon B.: The mathematical theory of resonances whose widths are exponentially small. Duke Math. 47(4), 845–902 (1980)
Helffer, B., Sjöstrand, J.: Résonances en limite semiclassique. Bull. Soc. Math. France, Mémoire 24/25 (1986)
Helffer B., Martinez A.: Comparaison entre les diverses notions de résonances. Helv. Phys. Acta 60, 992–1003 (1987)
Hislop P.D., Martinez A.: Scattering resonances of a Helmholtz resonator. Indiana Univ. Math. J. 40(2), 767–788 (1991)
Le Rousseau J., Lebeau G.: On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations 18(03), 712–747 (2012)
Lebeau G., Robbiano L.: Contrôle exact de l’équation de la chaleur. Comm. Part. Diff. Eq. 20(1&2), 335–356 (1995)
Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Springer-Verlag New-York, UTX Series, ISBN: 0-387-95344-2 (2002)
Martinez A., Nedelec L.: Optimal lower bound of the resonance widths for a Helmholtz tube-shaped resonator. J. Spectr. Theory 2, 203–223 (2012)
Sjöstrand, J.: Lecture on resonances. Preprint (2002)
Sjöstrand J., Zworski M.: Complex scaling and the distribution of scattering poles. J. Amer. Math. Soc. 4, 729–769 (1991)
Watson G.N.: A Treatise on the Theory of Bessel Functions. 2nd edn. Cambridge Univ. Press, Cambridge (1944)
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Communicated by Jens Marklof.
A. Martinez is partly supported by Università di Bologna, Funds for Selected Research Topics and Funds for Agreements with Foreign Universities.
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Martinez, A., Nédélec, L. Optimal Lower Bound of the Resonance Widths for the Helmholtz Resonator. Ann. Henri Poincaré 17, 645–672 (2016). https://doi.org/10.1007/s00023-015-0405-1
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DOI: https://doi.org/10.1007/s00023-015-0405-1