A sharp lower bound for a resonance-counting function in even dimensions
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 579-604

This paper proves sharp lower bounds on a resonance-counting function for obstacle scattering in even-dimensional Euclidean space without a need for trapping assumptions. Similar lower bounds are proved for some other compactly supported perturbations of -Δ on d , for example, for the Laplacian for certain metric perturbations on d . The proof uses a Poisson formula for resonances, complementary to one proved by Zworski in even dimensions.

L’objet de cette note est de montrer des bornes inférieures optimales pour la fonction de comptage des résonances, dans le cas d’obstacles sur l’espace euclidien en dimension paire ; on ne fait aucune hypothèse de capture du flot de billard extérieur à l’obstacle. Des minorations similaires sont prouvées pour d’autres types de perturbations à support compact sur d . La preuve utilise une formule de Poisson pour les résonances, complémentaire d’une formule montrée par Zworski en dimension paire.

Received : 2015-12-18
Revised : 2016-05-28
Accepted : 2016-06-14
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3092
Classification:  35P25,  58J50,  35P20,  47A40
Keywords: scattering theory, resonance, obstacle, metric
@article{AIF_2017__67_2_579_0,
     author = {Christiansen, T. J.},
     title = {A sharp lower bound for a resonance-counting function in even dimensions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {579-604},
     doi = {10.5802/aif.3092},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_2_579_0}
}
Christiansen, T. J. A sharp lower bound for a resonance-counting function in even dimensions. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 579-604. doi : 10.5802/aif.3092. http://www.numdam.org/item/AIF_2017__67_2_579_0/

[1] Bardos, Claude; Guillot, Jean-Claude; Ralston, James La relation de Poisson pour l’équation des ondes dans un ouvert non borné. Application à la théorie de la diffusion, Comm. Partial Differential Equations, Tome 7 (1982) no. 8, pp. 905-958 | Article

[2] Beale, J. Thomas Purely imaginary scattering frequencies for exterior domains, Duke Math. J., Tome 41 (1974), pp. 607-637 | Article

[3] Borthwick, David; Christiansen, T. J.; Hislop, Peter D.; Perry, Peter A. Resonances for manifolds hyperbolic near infinity: optimal lower bounds on order of growth, Int. Math. Res. Not. IMRN, Tome 2011 (2011) no. 19, pp. 4431-4470 | Article

[4] Burq, Nicolas Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math., Tome 124 (2002) no. 4, pp. 677-735 http://muse.jhu.edu/journals/american_journal_of_mathematics/v124/124.4burq.pdf | Article

[5] Cardoso, Fernando; Vodev, Georgi Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II, Ann. Henri Poincaré, Tome 3 (2002) no. 4, pp. 673-691 | Article

[6] Christiansen, T. J. Lower bounds for resonance counting functions for obstacle scattering in even dimensions (to appear in Amer. J. Math.)

[7] Christiansen, T. J.; Hislop, Peter D. Some remarks on resonances in even-dimensional Euclidean scattering, Trans. Amer. Math. Soc., Tome 368 (2016) no. 2, pp. 1361-1385 | Article

[8] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances (Book in preparation. http://math.mit.edu/~dyatlov/res/res.pdf)

[9] Eckmann, Jean-Pierre; Pillet, Claude-Alain Spectral duality for planar billiards, Comm. Math. Phys., Tome 170 (1995) no. 2, pp. 283-313 http://projecteuclid.org/euclid.cmp/1104273122 | Article

[10] Gohberg, Israel C.; Kreĭn, Mark Grigor’Evich Introduction to the theory of linear nonselfadjoint operators, American Mathematical Society, Providence, R.I., Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18 (1969), xv+378 pages

[11] Govorov, N. V. Riemann’s boundary problem with infinite index, Birkhäuser Verlag, Basel, Operator Theory: Advances and Applications, Tome 67 (1994), xii+252 pages (Edited and with an introduction and an appendix by I. V. Ostrovskiĭ, Translated from the 1986 Russian original by Yu. I. Lyubarskiĭ) | Article

[12] Guillopé, Laurent; Zworski, Maciej Scattering asymptotics for Riemann surfaces, Ann. of Math., Tome 145 (1997) no. 3, pp. 597-660 | Article

[13] Hörmander, Lars The spectral function of an elliptic operator, Acta Math., Tome 121 (1968), pp. 193-218 | Article

[14] Hörmander, Lars The analysis of linear partial differential operators. IV, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 275 (1985), vii+352 pages (Fourier integral operators)

[15] Hörmander, Lars The analysis of linear partial differential operators. I, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 256 (1990), xii+440 pages (Distribution theory and Fourier analysis) | Article

[16] Ivriĭ, Victor Ja. The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen., Tome 14 (1980) no. 2, pp. 25-34 | Article

[17] Jin, Long Scattering resonances of convex obstacles for general boundary conditions, Comm. Math. Phys., Tome 335 (2015) no. 2, pp. 759-807 | Article

[18] Lax, Peter David; Phillips, Ralph Saul Decaying modes for the wave equation in the exterior of an obstacle., Comm. Pure Appl. Math., Tome 22 (1969), pp. 737-787 | Article

[19] Lax, Peter David; Phillips, Ralph Saul The time delay operator and a related trace formula, Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), Academic Press, New York-London (Adv. in Math. Suppl. Stud.) Tome 3 (1978), pp. 197-215

[20] Lechleiter, Armin; Peters, Stefan Analytical characterization and numerical approximation of interior eigenvalues for impenetrable scatterers from far fields, Inverse Problems, Tome 30 (2014) no. 4, 045006, 22 pages | Article

[21] Levin, B. Ya. Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., Translations of Mathematical Monographs, Tome 5 (1980), xii+523 pages (Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman)

[22] Melrose, Richard B. Scattering theory and the trace of the wave group, J. Funct. Anal., Tome 45 (1982) no. 1, pp. 29-40 | Article

[23] Melrose, Richard B. Polynomial bound on the number of scattering poles, J. Funct. Anal., Tome 53 (1983) no. 3, pp. 287-303 | Article

[24] Melrose, Richard B. Polynomial bound on the distribution of poles in scattering by an obstacle, Journées équations aux dérivées partielles (1984), pp. 1-8 http://eudml.org/doc/93108 | Article

[25] Melrose, Richard B. The trace of the wave group, Microlocal analysis (Boulder, Colo., 1983), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 27 (1984), pp. 127-167 | Article

[26] Melrose, Richard B. Weyl asymptotics for the phase in obstacle scattering, Comm. Partial Differential Equations, Tome 13 (1988) no. 11, pp. 1431-1439 | Article

[27] Petkov, Vesselin; Stoyanov, Latchezar Sojourn times of trapping rays and the behavior of the modified resolvent of the Laplacian, Ann. Inst. H. Poincaré Phys. Théor., Tome 62 (1995) no. 1, pp. 17-45

[28] Petkov, Vesselin; Zworski, Maciej Semi-classical estimates on the scattering determinant, Ann. Henri Poincaré, Tome 2 (2001) no. 4, pp. 675-711 | Article

[29] Popov, Georgi On the contribution of degenerate periodic trajectories to the wave-trace, Comm. Math. Phys., Tome 196 (1998) no. 2, pp. 363-383 | Article

[30] Robert, Didier On the Weyl formula for obstacles, Partial differential equations and mathematical physics (Copenhagen, 1995; Lund, 1995), Birkhäuser Boston, Boston, MA (Progr. Nonlinear Differential Equations Appl.) Tome 21 (1996), pp. 264-285 | Article

[31] Sjöstrand, Johannes; Zworski, Maciej Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., Tome 4 (1991) no. 4, pp. 729-769 | Article

[32] Sjöstrand, Johannes; Zworski, Maciej Lower bounds on the number of scattering poles. II, J. Funct. Anal., Tome 123 (1994) no. 2, pp. 336-367 | Article

[33] Sjöstrand, Johannes; Zworski, Maciej Asymptotic distribution of resonances for convex obstacles, Acta Math., Tome 183 (1999) no. 2, pp. 191-253 | Article

[34] Stefanov, Plamen Quasimodes and resonances: sharp lower bounds, Duke Math. J., Tome 99 (1999) no. 1, pp. 75-92 | Article

[35] Stefanov, Plamen Sharp upper bounds on the number of the scattering poles, J. Funct. Anal., Tome 231 (2006) no. 1, pp. 111-142 | Article

[36] Tang, Siu-Hung Existence of resonances for metric scattering in even dimensions, Lett. Math. Phys., Tome 52 (2000) no. 3, pp. 211-223 | Article

[37] Vodev, Georgi Sharp bounds on the number of scattering poles for perturbations of the Laplacian, Comm. Math. Phys., Tome 146 (1992) no. 1, pp. 205-216 http://projecteuclid.org/euclid.cmp/1104249981 | Article

[38] Vodev, Georgi Sharp bounds on the number of scattering poles in even-dimensional spaces, Duke Math. J., Tome 74 (1994) no. 1, pp. 1-17 | Article

[39] Vodev, Georgi Sharp bounds on the number of scattering poles in the two-dimensional case, Math. Nachr., Tome 170 (1994), pp. 287-297 | Article

[40] Zworski, Maciej Sharp polynomial bounds on the number of scattering poles, Duke Math. J., Tome 59 (1989) no. 2, pp. 311-323 | Article

[41] Zworski, Maciej Sharp polynomial bounds on the number of scattering poles of radial potentials, J. Funct. Anal., Tome 82 (1989) no. 2, pp. 370-403 | Article

[42] Zworski, Maciej Poisson formulae for resonances, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, École Polytech., Palaiseau (1997) (Exp. No. XIII, 14 p.)

[43] Zworski, Maciej Poisson formula for resonances in even dimensions, Asian J. Math., Tome 2 (1998) no. 3, pp. 609-617 | Article

[44] Zworski, Maciej A remark on: “Inverse resonance problem for 2 -symmetric analytic obstacles in the plane” [in Geometric methods in inverse problems and PDE control, 289–321, Springer, New York, 2004] by S. Zelditch, Inverse Probl. Imaging, Tome 1 (2007) no. 1, pp. 225-227 | Article