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Quantum decay rates in chaotic scattering. (English) Zbl 1226.35061
This long and rather technical paper starts with a section in which the authors have formulated their principal results in the form of five theorems. Their proofs are scattered within the next eight sections and the appendix.
The main subject of their studies is the distribution of quantum resonances associated with hyperbolic classical flows in two and higher dimensions.
The main result is that there exist gaps between the resonances and the real axis. In dimension higher than two the proof relies on the notion of topological pressure associated with the classical hyperbolic flow.

MSC:
35P25 Scattering theory for PDEs
35B34 Resonance in context of PDEs
35B45 A priori estimates in context of PDEs
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[1] Anantharaman, N., Entropy and the localization of eigenfunctions. Ann. of Math., 168 (2008), 435–475. · Zbl 1175.35036
[2] Anantharaman, N. & Nonnenmacher, S., Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Ann. Inst. Fourier (Grenoble), 57 (2007), 2465–2523. · Zbl 1145.81033
[3] Bindel, D. & Zworski, M., Theory and computation of resonances in 1D scattering. http://www.cs.cornell.edu/\(\sim\)bindel/cims/resonant1d . · Zbl 1161.81430
[4] Bowen, R. & Ruelle, D., The ergodic theory of Axiom A flows. Invent. Math., 29 (1975), 181–202. · Zbl 0311.58010
[5] Burq, N., Contrôle de l’équation des plaques en présence d’obstacles strictement convexes. Mém. Soc. Math. France, 55 (1993). · Zbl 0930.93007
[6] Burq, N. & Zworski, M., Geometric control in the presence of a black box. J. Amer. Math. Soc., 17 (2004), 443–471. · Zbl 1050.35058
[7] Christianson, H., Cutoff resolvent estimates and the semilinear Schrödinger equation. Proc. Amer. Math. Soc., 136:10 (2008), 3513–3520. · Zbl 1156.35085
[8] – Dispersive estimates for manifolds with one trapped orbit. Comm. Partial Differential Equations, 33 (2008), 1147–1174. · Zbl 1152.58024
[9] Datchev, K., Local smoothing for scattering manifolds with hyperbolic trapped sets. Comm. Math. Phys., 286 (2009), 837–850. · Zbl 1189.58016
[10] Dencker, N., Sjöstrand, J. & Zworski, M., Pseudospectra of semiclassical (pseudo-) differential operators. Comm. Pure Appl. Math., 57 (2004), 384–415. · Zbl 1054.35035
[11] Dimassi, M. & Sjöstrand, J., Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999. · Zbl 0926.35002
[12] Doi, S., Smoothing effects of Schrödinger evolution groups on Riemannian manifolds. Duke Math. J., 82 (1996), 679–706. · Zbl 0870.58101
[13] Evans, L. C. & Zworski, M., Lectures on Semiclassical Analysis. http://math.berkeley.edu/\(\sim\)zworski/semiclassical.pdf .
[14] Gaspard, P. & Rice, S. A., Semiclassical quantization of the scattering from a classically chaotic repellor. J. Chem. Phys., 90:4 (1989), 2242–2254.
[15] Gérard, C. & Sjöstrand, J., Semiclassical resonances generated by a closed trajectory of hyperbolic type. Comm. Math. Phys., 108 (1987), 391–421. · Zbl 0637.35027
[16] Gérard, P., Mesures semi-classiques et ondes de Bloch, in Séminaire sur les Équations aux Dérivées Partielles (1990–1991), Exp. No. XVI. École Polytech., Palaiseau, 1991.
[17] Hörmander, L., The Analysis of Linear Partial Differential Operators. I, II. Grundlehren der Mathematischen Wissenschaften, 256, 257. Springer, Berlin–Heidelberg, 1983.
[18] Ikawa, M., Decay of solutions of the wave equation in the exterior of several convex bodies. Ann. Inst. Fourier (Grenoble), 38:2 (1988), 113–146. · Zbl 0636.35045
[19] Katok, A. & Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. · Zbl 0878.58020
[20] Keating, J. P., Novaes, M., Prado, S. D. & Sieber, M., Semiclassical structure of chaotic resonance eigenfunctions. Phys. Rev. Lett., 97:15 (2006), 150406.
[21] Klopp, F. & Zworski, M., Generic simplicity of resonances. Helv. Phys. Acta, 68 (1995), 531–538. · Zbl 0844.47040
[22] Lin, K. K., Numerical study of quantum resonances in chaotic scattering. J. Comput. Phys., 176 (2002), 295–329. · Zbl 1021.81021
[23] Lin, K. K. & Zworski, M., Quantum resonances in chaotic scattering. Chem. Phys. Lett., 355 (2002), 201–205.
[24] Lu, W. T., Sridhar, S. & Zworski, M., Fractal Weyl laws for chaotic open systems. Phys. Rev. Lett., 91:15 (2003), 154101.
[25] Martinez, A., Resonance free domains for non globally analytic potentials. Ann. Henri Poincaré, 3 (2002), 739–756. · Zbl 1026.81012
[26] Morita, T., Periodic orbits of a dynamical system in a compound central field and a perturbed billiards system. Ergodic Theory Dynam. Systems, 14 (1994), 599–619. · Zbl 0810.58014
[27] Nakamura, S., Stefanov, P. & Zworski, M., Resonance expansions of propagators in the presence of potential barriers. J. Funct. Anal., 205 (2003), 180–205. · Zbl 1037.35064
[28] Naud, F., Classical and quantum lifetimes on some non-compact Riemann surfaces. J. Phys. A, 38:49 (2005), 10721–10729. · Zbl 1082.81026
[29] Nonnenmacher, S. & Rubin, M., Resonant eigenstates for a quantized chaotic system. Nonlinearity, 20:6 (2007), 1387–1420. · Zbl 1138.81021
[30] Nonnenmacher, S. & Zworski, M., Fractal Weyl laws in discrete models of chaotic scattering. J. Phys. A, 38:49 (2005), 10683–10702. · Zbl 1082.81079
[31] – Distribution of resonances for open quantum maps. Comm. Math. Phys., 269 (2007), 311–365. · Zbl 1114.81043
[32] – Semiclassical resolvent estimates in chaotic scattering. Appl. Math. Res. Express, 2009 (2009), 1–13. · Zbl 1181.81055
[33] Pesin, Y. B. & Sadovskaya, V., Multifractal analysis of conformal Axiom A flows. Comm. Math. Phys., 216 (2001), 277–312. · Zbl 0992.37023
[34] Petkov, V. & Stoyanov, L., Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. C. R. Math. Acad. Sci. Paris, 345 (2007), 567–572. · Zbl 1125.37013
[35] Ruelle, D., Thermodynamic Formalism. Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley, Reading, MA, 1978. · Zbl 0401.28016
[36] Schomerus, H. & Tworzydło, J., Quantum-to-classical crossover of quasibound states in open quantum systems. Phys. Rev. Lett., 93:15 (2004), 154102.
[37] Shubin, M. A. & Sjöstrand, J., Appendix to Weak Bloch property and weight estimates for elliptic operators, in Séminaire sur les Équations aux Dérivées Partielles (1989–1990), Exp. No. V. École Polytech., Palaiseau, 1990.
[38] Sjöstrand, J., Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J., 60 (1990), 1–57. · Zbl 0702.35188
[39] – A trace formula and review of some estimates for resonances, in Microlocal Analysis and Spectral Theory (Lucca, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 490, pp. 377–437. Kluwer Acad. Publ., Dordrecht, 1997.
[40] – Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations. Preprint, 2008. arXiv:0802.3584 [math.SP].
[41] Sjöstrand, J. & Zworski, M., Quantum monodromy and semi-classical trace formulae. J. Math. Pures Appl., 81 (2002), 1–33. · Zbl 1038.58033
[42] – Fractal upper bounds on the density of semiclassical resonances. Duke Math. J., 137 (2007), 381–459. · Zbl 1201.35189
[43] Tang, S. H. & Zworski, M., From quasimodes to reasonances. Math. Res. Lett., 5 (1998), 261–272. · Zbl 0913.35101
[44] Walters, P., An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79. Springer, New York, 1982. · Zbl 0475.28009
[45] Wirzba, A., Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems. Phys. Rep., 309 (1999).
[46] Wojtkowski, M. P., Design of hyperbolic billiards. Comm. Math. Phys., 273 (2007), 283–304. · Zbl 1137.37019
[47] Zworski, M., Resonances in physics and geometry. Notices Amer. Math. Soc., 46 (1999), 319–328. · Zbl 1177.58021
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