Acta Mathematica

Quantum decay rates in chaotic scattering

Stéphane Nonnenmacher and Maciej Zworski

Full-text: Open access


We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.

Article information

Acta Math., Volume 203, Number 2 (2009), 149-233.

Received: 13 August 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2009 © Institut Mittag-Leffler


Nonnenmacher, Stéphane; Zworski, Maciej. Quantum decay rates in chaotic scattering. Acta Math. 203 (2009), no. 2, 149--233. doi:10.1007/s11511-009-0041-z.

Export citation


  • A nantharaman, N., Entropy and the localization of eigenfunctions. Ann. of Math., 168 (2008), 435–475.
  • A nantharaman, N. & N onnenmacher, S., Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Ann. Inst. Fourier (Grenoble), 57 (2007), 2465–2523.
  • B indel, D. & Z worski, M., Theory and computation of resonances in 1D scattering.
  • B owen, R. & R uelle, D., The ergodic theory of Axiom A flows. Invent. Math., 29 (1975), 181–202.
  • B urq, N., Contrôle de l’équation des plaques en présence d’obstacles strictement convexes. Mém. Soc. Math. France, 55 (1993).
  • B urq, N. & Z worski, M., Geometric control in the presence of a black box. J. Amer. Math. Soc., 17 (2004), 443–471.
  • C hristianson, H., Cutoff resolvent estimates and the semilinear Schrödinger equation. Proc. Amer. Math. Soc., 136:10 (2008), 3513–3520.
  • — Dispersive estimates for manifolds with one trapped orbit. Comm. Partial Differential Equations, 33 (2008), 1147–1174.
  • D atchev, K., Local smoothing for scattering manifolds with hyperbolic trapped sets. Comm. Math. Phys., 286 (2009), 837–850.
  • D encker, N., S jöstrand, J. & Z worski, M., Pseudospectra of semiclassical (pseudo-) differential operators. Comm. Pure Appl. Math., 57 (2004), 384–415.
  • D imassi, M. & S jöstrand, J., Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999.
  • D oi, S., Smoothing effects of Schrödinger evolution groups on Riemannian manifolds. Duke Math. J., 82 (1996), 679–706.
  • E vans, L. C. & Z worski, M., Lectures on Semiclassical Analysis.
  • G aspard, P. & R ice, S. A., Semiclassical quantization of the scattering from a classically chaotic repellor. J. Chem. Phys., 90:4 (1989), 2242–2254.
  • G érard, C. & S jöstrand, J., Semiclassical resonances generated by a closed trajectory of hyperbolic type. Comm. Math. Phys., 108 (1987), 391–421.
  • 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.
  • H örmander, L., The Analysis of Linear Partial Differential Operators. I, II. Grundlehren der Mathematischen Wissenschaften, 256, 257. Springer, Berlin–Heidelberg, 1983.
  • I kawa, M., Decay of solutions of the wave equation in the exterior of several convex bodies. Ann. Inst. Fourier (Grenoble), 38:2 (1988), 113–146.
  • K atok, A. & H asselblatt, B., Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.
  • K eating, J. P., N ovaes, M., P rado, S. D. & S ieber, M., Semiclassical structure of chaotic resonance eigenfunctions. Phys. Rev. Lett., 97:15 (2006), 150406.
  • K lopp, F. & Z worski, M., Generic simplicity of resonances. Helv. Phys. Acta, 68 (1995), 531–538.
  • L in, K. K., Numerical study of quantum resonances in chaotic scattering. J. Comput. Phys., 176 (2002), 295–329.
  • L in, K. K. & Z worski, M., Quantum resonances in chaotic scattering. Chem. Phys. Lett., 355 (2002), 201–205.
  • L u, W. T., S ridhar, S. & Z worski, M., Fractal Weyl laws for chaotic open systems. Phys. Rev. Lett., 91:15 (2003), 154101.
  • M artinez, A., Resonance free domains for non globally analytic potentials. Ann. Henri Poincaré, 3 (2002), 739–756.
  • M orita, 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.
  • N akamura, S., S tefanov, P. & Z worski, M., Resonance expansions of propagators in the presence of potential barriers. J. Funct. Anal., 205 (2003), 180–205.
  • N aud, F., Classical and quantum lifetimes on some non-compact Riemann surfaces. J. Phys. A, 38:49 (2005), 10721–10729.
  • N onnenmacher, S. & R ubin, M., Resonant eigenstates for a quantized chaotic system. Nonlinearity, 20:6 (2007), 1387–1420.
  • N onnenmacher, S. & Z worski, M., Fractal Weyl laws in discrete models of chaotic scattering. J. Phys. A, 38:49 (2005), 10683–10702.
  • — Distribution of resonances for open quantum maps. Comm. Math. Phys., 269 (2007), 311–365.
  • — Semiclassical resolvent estimates in chaotic scattering. Appl. Math. Res. Express, 2009 (2009), 1–13.
  • P esin, Y. B. & S adovskaya, V., Multifractal analysis of conformal Axiom A flows. Comm. Math. Phys., 216 (2001), 277–312.
  • P etkov, V. & S toyanov, L., Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. C. R. Math. Acad. Sci. Paris, 345 (2007), 567–572.
  • R uelle, D., Thermodynamic Formalism. Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley, Reading, MA, 1978.
  • S chomerus, H. & T worzydło, J., Quantum-to-classical crossover of quasibound states in open quantum systems. Phys. Rev. Lett., 93:15 (2004), 154102.
  • S hubin, M. A. & S jöstrand, 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.
  • S jöstrand, J., Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J., 60 (1990), 1–57.
  • — 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.
  • — Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations. Preprint, 2008. arXiv:0802.3584 [math.SP].
  • S jöstrand, J. & Z worski, M., Quantum monodromy and semi-classical trace formulae. J. Math. Pures Appl., 81 (2002), 1–33.
  • — Fractal upper bounds on the density of semiclassical resonances. Duke Math. J., 137 (2007), 381–459.
  • T ang, S. H. & Z worski, M., From quasimodes to reasonances. Math. Res. Lett., 5 (1998), 261–272.
  • W alters, P., An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79. Springer, New York, 1982.
  • W irzba, A., Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems. Phys. Rep., 309 (1999).
  • W ojtkowski, M. P., Design of hyperbolic billiards. Comm. Math. Phys., 273 (2007), 283–304.
  • Z worski, M., Resonances in physics and geometry. Notices Amer. Math. Soc., 46 (1999), 319–328.