ANALYTIC CONTINUATION OF THE RESOLVENT OF THE LAPLACIAN AND THE DYNAMICAL ZETA FUNCTION

Analysis and PDE

Volumn 3, Issue 4, 2010, Pages 427-489

  Petkov, Vesselin ^a   Stoyanov, Luchezar ^b  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

Author keywords

open billiard; periodic rays; zeta function

Indexed keywords

EID: 79952958227     PISSN: 21575045     EISSN: 1948206X     Source Type: Journal    
DOI: 10.2140/apde.2010.3.427     Document Type: Article

References (40)

1. [Bowen 1973] R. Bowen, “Symbolic dynamics for hyperbolic flows”, Amer. J. Math. 95 (1973), 429–460. MR 49 #4041 Zbl 0282.58009
2. [Bowen 1975] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics 470, Springer, Berlin, 1975. MR 56 #1364 Zbl 0308.28010
3. [Burq 1993] N. Burq, Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 55, Soc. Math. France, Paris, 1993. MR 95d:93007 Zbl 0930.93007
4. [Cvitanović and Eckhardt 1989] P. Cvitanović and B. Eckhardt, “Periodic-orbit quantization of chaotic systems”, Phys. Rev. Lett. 63:8 (1989), 823–826. MR 90k:58128
5. [Dolgopyat 1998] D. Dolgopyat, “On decay of correlations in Anosov flows”, Ann. of Math. (2) 147:2 (1998), 357–390. MR 99g:58073 Zbl 0911.58029
6. [Gérard 1988] C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes, Mém. Soc. Math. France (N.S.) 31, Soc. Math. France, Paris, 1988. MR 91e:35168 Zbl 0654.35081
7. [Guillarmou and Naud 2009] C. Guillarmou and F. Naud, “Wave decay on convex co-compact hyperbolic manifolds”, Comm. Math. Phys. 287:2 (2009), 489–511. MR 2009m:58060 Zbl 05610591
8. [Guillemin and Sternberg 1977] V. Guillemin and S. Sternberg, Geometric asymptotics, Mathematical Surveys 14, American Mathematical Society, Providence, R.I., 1977. MR 58 #24404 Zbl 0364.53011
9. [Hasselblatt 1994] B. Hasselblatt, “Regularity of the Anosov splitting and of horospheric foliations”, Ergodic Theory Dynam. Systems 14:4 (1994), 645–666. MR 95j:58130 Zbl 0821.58032
10. [Hasselblatt 1997] B. Hasselblatt, “Regularity of the Anosov splitting. II”, Ergodic Theory Dynam. Systems 17:1 (1997), 169–172. MR 98d:58135 Zbl 0872.58045
11. [Hörmander 1985a] L. Hörmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren der Mathematischen Wissenschaften 274, Springer, Berlin, 1985. MR 87d:35002a
12. [Hörmander 1985b] L. Hörmander, The analysis of linear partial differential operators, IV: Fourier interior operators, Grundlehren der Mathematischen Wissenschaften 275, Springer, Berlin, 1985. MR 87d:35002a
13. [Ikawa 1982] M. Ikawa, “Decay of solutions of the wave equation in the exterior of two convex obstacles”, Osaka J. Math. 19:3 (1982), 459–509. MR 84e:35018 Zbl 0498.35008
14. [Ikawa 1987] M. Ikawa, “Precise informations on the poles of the scattering matrix for two strictly convex obstacles”, J. Math. Kyoto Univ. 27:1 (1987), 69–102. MR 88e:35143 Zbl 0637.35068
15. [Ikawa 1988] M. Ikawa, “Decay of solutions of the wave equation in the exterior of several convex bodies”, Ann. Inst. Fourier (Grenoble) 38:2 (1988), 113–146. MR 90a:35028 Zbl 0636.35045
16. [Ikawa 1990] M. Ikawa, “Singular perturbations of symbolic flows and the poles of the zeta function”, Osaka J. Math. 27 (1990), 281–300.
17. [Ikawa 1994] M. Ikawa, “On zeta function and scattering poles for several convex bodies”, in Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1994), exposé II-14, École Polytech., Palaiseau, 1994. MR 96h:58173 Zbl 0872. 58048
18. [Ikawa 2000] M. Ikawa, “On scattering by several convex bodies”, J. Korean Math. Soc. 37:6 (2000), 991–1005. MR 2001k: 58061 Zbl 0967.37015
19. [Katok and Hasselblatt 1995] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encycl. Math. Appl. 54, Cambridge Univ. Press, Cambridge, 1995. MR 96c:58055 Zbl 0878.58020
20. [Lin 2002] K. K. Lin, “Numerical study of quantum resonances in chaotic scattering”, J. Comput. Phys. 176:2 (2002), 295–329. MR 1894769 Zbl 1021.81021
21. [Lin and Zworski 2002] K. Lin and M. Zworski, “Quantum resonances in chaotic scattering”, Chem. Phys. Lett. 355 (2002), 201–205.
22. [Lin et al. 2003] K. Lin, S. Sridhar, and M. Zworski, “Fractal law for chaotic open systems”, Phys. Rev. Lett. 91 (2003), 154.
23. [Nonnenmacher and Zworski 2009] S. Nonnenmacher and M. Zworski, “Quantum decay rates in chaotic scattering”, Acta Math. 203:2 (2009), 149–233. MR 2570070
24. [Parry and Pollicott 1990] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188, Soc. Math. France, 1990. MR 92f:58141 Zbl 0726.58003
25. [Patterson and Perry 2001] S. J. Patterson and P. A. Perry, “The divisor of Selberg’s zeta function for Kleinian groups”, Duke Math. J. 106:2 (2001), 321–390. MR 2002a:11103 Zbl 1012.11083
26. [Petkov and Stoyanov 1992] V. M. Petkov and L. N. Stoyanov, Geometry of reflecting rays and inverse spectral problems, Wiley, Chichester, 1992. MR 93i:58161 Zbl 0761.35077
27. [Petkov and Stoyanov 2009] V. Petkov and L. Stoyanov, “Correlations for pairs of periodic trajectories for open billiards”, Nonlinearity 22:11 (2009), 2657–2679. MR 2550690 Zbl 1185.37094
28. [Sinai 1970] Y. Sinai, “Dynamical systems with elastic reflections”, Russian Math. Surveys 25 (1970), 137–190.
29. [Sinai 1979] Y. Sinai, “Development of Krylov’s ideas”, pp. xxviii+283 pp. (1 plate) in Works on the foundations of statistical physics by N. S. Krylov, Princeton Univ. Press, Princeton, N.J., 1979. MR 82e:82001
30. [Sjöstrand 1990] J. Sjöstrand, “Geometric bounds on the density of resonances for semiclassical problems”, Duke Math. J. 60:1 (1990), 1–57. MR 91e:35166 Zbl 0702.35188
31. [Stefanov and Vodev 1995] P. Stefanov and G. Vodev, “Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body”, Duke Math. J. 78:3 (1995), 677–714. MR 96e:35034 Zbl 0846.35139
32. [Stoyanov 1999] L. Stoyanov, “Exponential instability for a class of dispersing billiards”, Ergodic Theory Dynam. Systems 19:1 (1999), 201–226. MR 99m:58149 Zbl 0923.58028
33. [Stoyanov 2001] L. Stoyanov, “Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows”, Amer. J. Math. 123:4 (2001), 715–759. MR 2002f:37061 Zbl 0994.37018
34. [Stoyanov 2005] L. Stoyanov, “On the Ruelle–Perron–Frobenius theorem”, Asymptot. Anal. 43:1–2 (2005), 131–150. MR 2007d:37023 Zbl 1083.37023
35. [Stoyanov 2007] L. Stoyanov, “Spectra of Ruelle transfer operators for Axiom A flows on basic sets”, preprint, 2007. arXiv 0810.1126v3
36. [Stoyanov 2009] L. Stoyanov, “Non-integrability of open billiard flows and Dolgopyat type estimates”, preprint, 2009. arXiv 0911.5000v3
37. [Tang and Zworski 2000] S.-H. Tang and M. Zworski, “Resonance expansions of scattered waves”, Comm. Pure Appl. Math. 53:10 (2000), 1305–1334. MR 2001f:35306 Zbl 1032.35148
38. [Titchmarsh 1968] E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, 1968.
39. [Vainberg 1989] B. R. Vainberg, Asymptotic methods in equations of mathematical physics, Gordon and Breach, New York, 1989. MR 91h:35081 Zbl 0743.35001
40. [Wirzba 1999] A. Wirzba, “Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems”, Phys. Rep. 309:1–2 (1999), 116. MR 99k:81069