Spectral theory of damped quantum chaotic systems

Stéphane Nonnenmacher[1]

  • [1] Institut de Physique théorique, CEA-Saclay, unité de recherche associée au CNRS, 91191 Gif-sur-Yvette, France

Journées Équations aux dérivées partielles (2011)

  • page 1-23
  • ISSN: 0752-0360

Abstract

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We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on X and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.

How to cite

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Nonnenmacher, Stéphane. "Spectral theory of damped quantum chaotic systems." Journées Équations aux dérivées partielles (2011): 1-23. <http://eudml.org/doc/219807>.

@article{Nonnenmacher2011,
abstract = {We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on $X$ and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.},
affiliation = {Institut de Physique théorique, CEA-Saclay, unité de recherche associée au CNRS, 91191 Gif-sur-Yvette, France},
author = {Nonnenmacher, Stéphane},
journal = {Journées Équations aux dérivées partielles},
keywords = {semiclassical limit; chaotic dynamics; quantum scattering; damped waves},
language = {eng},
month = {6},
pages = {1-23},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Spectral theory of damped quantum chaotic systems},
url = {http://eudml.org/doc/219807},
year = {2011},
}

TY - JOUR
AU - Nonnenmacher, Stéphane
TI - Spectral theory of damped quantum chaotic systems
JO - Journées Équations aux dérivées partielles
DA - 2011/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 23
AB - We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on $X$ and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condition for a spectral gap, depending on the set of minimally damped trajectories.
LA - eng
KW - semiclassical limit; chaotic dynamics; quantum scattering; damped waves
UR - http://eudml.org/doc/219807
ER -

References

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  1. N. Anantharaman, Entropy and the localization of eigenfunctions, Ann. Math. (2) 168, 435–475 (2008) Zbl1175.35036MR2434883
  2. N. Anantharaman, Spectral deviations for the damped wave equation, GAFA 20 (2010) 593–626 Zbl1205.35173MR2720225
  3. N. Anantharaman and S. Nonnenmacher, Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier 57(7), 2465–2523 (2007) Zbl1145.81033MR2394549
  4. M. Asch and G. Lebeau, The Spectrum of the Damped Wave Operator for a Bounded Domain in ℝ 2 , Exper. Math. 12 (2003) 227–241 Zbl1061.35064MR2016708
  5. S. Brooks, On the entropy of quantum limits for 2-dimensional cat maps, Commun. Math. Phys. 293 (2010) 231–255 Zbl1205.37017MR2563805
  6. N. Burq, Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonances au voisinage du réel, Acta Math. 180 (1998) 1–29 Zbl0918.35081MR1618254
  7. N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett. 14 (2007) 35-47 Zbl1122.35015MR2289618
  8. H. Christianson, Semiclassical Non-concentration near Hyperbolic Orbits, J. Funct. Anal. 246 (2007) 145–195; Corrigendum, J. Funct. Anal. 258 (2010) 1060–1065 Zbl1181.58019MR2321040
  9. H. Christianson, Applications of Cutoff Resolvent Estimates to the Wave Equation, Math. Res. Lett. Vol. 16 (2009) 577–590 Zbl1189.58012MR2525026
  10. H. Christianson, Quantum Monodromy and Non-concentration Near a Closed Semi-hyperbolic Orbit, Trans. Amer. Math. Soc. 363 (2011) 3373–3438 Zbl1230.58020MR2775812
  11. C.L. Evans and M. Zworski, Lectures on semiclassical analysis, v.0.75 
  12. M.Hitrik, Eigenfrequencies and expansions for damped wave equations, Meth. Appl. Anal. 10 (2003) 1–22 Zbl1088.58510MR2105039
  13. M. Hitrik and J. Sjöstrand, Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2, Ann. Sci. E.N.S. 41 (2008) 511-571 Zbl1171.35131MR2489633
  14. M. Hitrik and J. Sjöstrand, Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two, preprint 2011, arXiv:1102.0889 Zbl1256.35039
  15. M. Hitrik, J. Sjöstrand, and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for non-selfadjoint operators, Amer. J. Math. 129 (2007) 105-182 Zbl1172.35085MR2288739
  16. A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge UP, 1995 Zbl0878.58019MR1326374
  17. Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990) 505–524. Zbl0714.60019MR1025756
  18. H. Koch and D. Tataru, On the spectrum of hyperbolic semigroups, Comm. Partial Differential Equations, 20, No 5-6, 901-937 (1995) Zbl0823.35108MR1326911
  19. G.Lebeau, Equation des ondes amorties, Algebraie and geometric methods in mathematical physics, (Kaciveli 1993), 73-109, Math. Phys. Stud. 19, Kluwer Acad. Publ., Dordrecht, 1996 Zbl0863.58068MR1385677
  20. A.S. Markus and V.I. Matsaev, Comparison theorems for spectra of linear operators, and spectral asymptotics, Trans. Moscow Math. Soc. (1984) 139–187. Russian original in Trudy Moscow. Obshch. 45 (1982), 133-181 Zbl0557.47009MR704630
  21. S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering, Acta Math 203 (2009) 149–233 Zbl1226.35061MR2570070
  22. S. Nonnenmacher and M. Zworski, Semiclassical Resolvent Estimates in Chaotic Scattering, Appl. Math. Res. eXpr. 2009, Article ID abp003 Zbl1181.81055MR2581379
  23. J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Commun. Pure Appl. Math. 28 (1975) 501-523 Zbl0295.35048MR397184
  24. M. Renardy, On the linear stability of hyperbolic PDEs and viscoelastic flows, Zeit. f. angew. Math. Phys. 45 (1994) 854-865 Zbl0820.76008MR1306936
  25. G. Rivière, Entropy of semiclassical measures in dimension 2, Duke Math. J. 155 (2010) 271-335 Zbl1230.37048MR2736167
  26. G. Rivière, Delocalization of slowly damped eigenmodes on Anosov manifolds, preprint 2011 Zbl1273.58020
  27. E. Schenck, Energy decay for the damped wave equation under a pressure condition, Commun. Math. Phys. 300, 375–410 (2010) Zbl1207.35064MR2728729
  28. E. Schenck, Exponential stabilization without geometric control, preprint 2010 Zbl1244.93144MR2784679
  29. J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36 (2000) 573-611 Zbl0984.35121MR1798488
  30. J. Sjöstrand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J. 137 (2007) 381–459 Zbl1201.35189MR2309150
  31. S. Zelditch, Recent developments in mathematical quantum chaos, in Current Developments in Mathematics, 2009, D.Jerison, B.Mazur, T.Mrowka, W.Schmid, R.Stanley, S-T Yau (eds.), International Press 2009 Zbl1223.37113

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