Spectral theory of damped quantum chaotic systems
Journées équations aux dérivées partielles (2011), article no. 9, 23 p.

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.

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     author = {Nonnenmacher, St\'ephane},
     title = {Spectral theory of damped quantum chaotic systems},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {9},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2011},
     doi = {10.5802/jedp.81},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.81/}
}
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Nonnenmacher, Stéphane. Spectral theory of damped quantum chaotic systems. Journées équations aux dérivées partielles (2011), article  no. 9, 23 p. doi : 10.5802/jedp.81. http://archive.numdam.org/articles/10.5802/jedp.81/

[Anan08] N. Anantharaman, Entropy and the localization of eigenfunctions, Ann. Math. (2) 168, 435–475 (2008) | MR 2434883 | Zbl 1175.35036

[Anan10] N. Anantharaman, Spectral deviations for the damped wave equation, GAFA 20 (2010) 593–626 | MR 2720225 | Zbl 1205.35173

[AN1] N. Anantharaman and S. Nonnenmacher, Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier 57(7), 2465–2523 (2007) | EuDML 10304 | Numdam | MR 2394549 | Zbl 1145.81033

[AschLeb] M. Asch and G. Lebeau, The Spectrum of the Damped Wave Operator for a Bounded Domain in 2 , Exper. Math. 12 (2003) 227–241 | EuDML 51262 | MR 2016708 | Zbl 1061.35064

[Bro10] S. Brooks, On the entropy of quantum limits for 2-dimensional cat maps, Commun. Math. Phys. 293 (2010) 231–255 | MR 2563805 | Zbl 1205.37017

[Bur98] 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 | MR 1618254 | Zbl 0918.35081

[BuHi07] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett. 14 (2007) 35-47 | MR 2289618 | Zbl 1122.35015

[Chris07] H. Christianson, Semiclassical Non-concentration near Hyperbolic Orbits, J. Funct. Anal. 246 (2007) 145–195; Corrigendum, J. Funct. Anal. 258 (2010) 1060–1065 | MR 2321040 | Zbl 1181.58019

[Chris09] H. Christianson, Applications of Cutoff Resolvent Estimates to the Wave Equation, Math. Res. Lett. Vol. 16 (2009) 577–590 | MR 2525026 | Zbl 1189.58012

[Chris11] H. Christianson, Quantum Monodromy and Non-concentration Near a Closed Semi-hyperbolic Orbit, Trans. Amer. Math. Soc. 363 (2011) 3373–3438 | MR 2775812

[EvZw] C.L. Evans and M. Zworski, Lectures on semiclassical analysis, v.0.75

[Hit03] M.Hitrik, Eigenfrequencies and expansions for damped wave equations, Meth. Appl. Anal. 10 (2003) 1–22 | MR 2105039 | Zbl 1088.58510

[HitSjo08] 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 | Numdam | MR 2489633 | Zbl 1171.35131

[HitSjo11] M. Hitrik and J. Sjöstrand, Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two, preprint 2011, arXiv:1102.0889

[HSVN07] M. Hitrik, J. Sjöstrand, and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for non-selfadjoint operators, Amer. J. Math. 129 (2007) 105-182 | MR 2288739 | Zbl 1172.35085

[KatHas95] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge UP, 1995 | MR 1326374 | Zbl 0878.58019

[Kif90] Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc. 321 (1990) 505–524. | MR 1025756 | Zbl 0714.60019

[KoTa94] H. Koch and D. Tataru, On the spectrum of hyperbolic semigroups, Comm. Partial Differential Equations, 20, No 5-6, 901-937 (1995) | MR 1326911 | Zbl 0823.35108

[Leb93] 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 | MR 1385677 | Zbl 0863.58068

[MarMat84] 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 | MR 704630 | Zbl 0557.47009

[NZ2] S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering, Acta Math 203 (2009) 149–233 | MR 2570070

[NZ3] S. Nonnenmacher and M. Zworski, Semiclassical Resolvent Estimates in Chaotic Scattering, Appl. Math. Res. eXpr. 2009, Article ID abp003 | MR 2581379 | Zbl 1181.81055

[RauTay75] J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Commun. Pure Appl. Math. 28 (1975) 501-523 | MR 397184 | Zbl 0295.35048

[Ren94] M. Renardy, On the linear stability of hyperbolic PDEs and viscoelastic flows, Zeit. f. angew. Math. Phys. 45 (1994) 854-865 | MR 1306936 | Zbl 0820.76008

[Riv10] G. Rivière, Entropy of semiclassical measures in dimension 2, Duke Math. J. 155 (2010) 271-335 | MR 2736167

[Riv11] G. Rivière, Delocalization of slowly damped eigenmodes on Anosov manifolds, preprint 2011

[Sche10] E. Schenck, Energy decay for the damped wave equation under a pressure condition, Commun. Math. Phys. 300, 375–410 (2010) | MR 2728729 | Zbl 1207.35064

[Sche11] E. Schenck, Exponential stabilization without geometric control, preprint 2010 | MR 2784679

[Sjo00] J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36 (2000) 573-611 | MR 1798488 | Zbl 0984.35121

[SjoZwo07] J. Sjöstrand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J. 137 (2007) 381–459 | MR 2309150 | Zbl 1201.35189

[Zel09] 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

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