Quantum transfer operators and quantum scattering
Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 7, 18 p.
@article{SEDP_2009-2010____A7_0,
     author = {Nonnenmacher, St\'ephane},
     title = {Quantum transfer operators and quantum scattering},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2009-2010},
     note = {talk:7},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2009-2010____A7_0}
}
Nonnenmacher, Stéphane. Quantum transfer operators and quantum scattering. Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 7, 18 p. http://www.numdam.org/item/SEDP_2009-2010____A7_0/

[1] V. Baladi, Positive Transfer Operators and Decay of Correlations, Book Advanced Series in Nonlinear Dynamics, Vol 16, World Scientific, Singapore (2000) | MR 1793194 | Zbl 1012.37015

[2] R. Blümel and W. P. Reinhardt, Chaos in Atomic Physics, Cambridge University Press, Cambridge, 1997 | MR 1627898 | Zbl 0939.81044

[3] E. B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805–866 | MR 1174220 | Zbl 0749.35035

[4] F. Borgonovi, I. Guarneri and D. L. Shepelyansky, Statistics of quantum lifetimes in a classically chaotic system, Phys. Rev. A 43 (1991) 4517–4520

[5] R. Bowen, One-dimensional hyperbolic sets for flows, J. Diff. Equ. 12 (1972) 173–179 | MR 336762 | Zbl 0242.58005

[6] H. Christianson, Quantum monodromy and non-concentration near a closed semi-hyperbolic orbit, preprint 2009 | MR 2625926

[7] P. Cvitanović, P. Rosenquist, G. Vattay and H.H. Rugh, A Fredholm determinant for semiclassical quantization, CHAOS 3 (1993) 619–636 | MR 1256315 | Zbl 1055.81549

[8] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, 1999. | MR 1735654 | Zbl 0926.35002

[9] E. Doron and U. Smilansky, Semiclassical quantization of chaotic billiards: a scattering theory approach, Nonlinearity 5 (1992) 1055–1084; C. Rouvinez and U. Smilansky, A scattering approach to the quantization of Hamiltonians in two dimensions – application to the wedge billiard, J. Phys. A 28 (1995) 77–104 | MR 1187738 | Zbl 0770.58043

[10] L.C. Evans and M. Zworski, Lectures on Semiclassical Analysis, Version 0.3.2, http://math.berkeley.edu/zworski/semiclassical.pdf

[11] P. Gaspard and S. A. Rice, Semiclassical quantization of the scattering from a classical chaotic repeller, J. Chem. Phys. 90(4) 2242–2254 | MR 980393

[12] B. Georgeot and R. E. Prange, Fredholm theory for quasiclassical scattering, Phys. Rev. Lett. 74 (1995) 4110-4113; A. M. Ozorio de Almeida and R. O. Vallejos, Decomposition of Resonant Scatterers by Surfaces of Section, Ann. Phys. (NY) 278 (1999) 86–108 | MR 1329478 | Zbl 1020.81937

[13] C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes. Mémoires de la Société Mathématique de France Sér. 2, 31(1988) 1–146 | Numdam | MR 998698 | Zbl 0654.35081

[14] S. Gouëzel and C. Liverani,Banach spaces adapted to Anosov systems, Ergod. Th. Dyn. Sys. 26 (2006) 189–217; V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier, 57 (2007) 127–154 | MR 2201945 | Zbl 1088.37010

[15] M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, 38 (1988) 113–146 | Numdam | MR 949013 | Zbl 0636.35045

[16] J.P. Keating, M. Novaes, S.D. Prado and M. Sieber, Semiclassical structure of quantum fractal eigenstates, Phys. Rev. Lett. 97 (2006) 150406; S. Nonnenmacher and M. Rubin, Resonant eigenstates for a quantized chaotic system, Nonlinearity 20 (2007) 1387–1420.

[17] K. Nakamura and T. Harayama, Quantum Chaos and Quantum Dots, Oxford University Press, Oxford, 2004

[18] S. Nonnenmacher, J. Sjöstrand and M. Zworski, From open quantum systems to open quantum maps, preprint, arXiv:1004.3361 | MR 2793928

[19] S. Nonnenmacher and M. Zworski, Distribution of resonances for open quantum maps, Comm. Math. Phys. 269 (2007) 311–365 | MR 2274550 | Zbl 1114.81043

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

[21] V. Petkov and L. Stoyanov, Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function, C. R. Acad. Sci. Paris, Ser.I, 345 (2007) 567–572 | MR 2374466 | Zbl 1125.37013

[22] M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985) 413–426; D. Ruelle, Resonances for Axiom A flows, J. Diff. Geom. 25 (1987) 99–116 | MR 807065 | Zbl 0591.58025

[23] T. Prosen, General quantum surface-of-section method, J. Phys.A 28 (1995) 4133-4155 | MR 1352158 | Zbl 0860.58019

[24] H. Schomerus and J. Tworzydlo, Quantum-to-classical crossover of quasi-bound states in open quantum systems, Phys. Rev. Lett. 93 (2004) 154102

[25] J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990) 1–57 | MR 1047116 | Zbl 0702.35188

[26] J. Sjöstrand, A trace formula and review of some estimates for resonances, in Microlocal analysis and spectral theory (Lucca, 1996), 377–437, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 490, Kluwer Acad. Publ., Dordrecht, 1997. | MR 1451399 | Zbl 0877.35090

[27] J. Sjöstrand and M. Zworski, Quantum monodromy and semiclassical trace formulae, J. Math. Pure Appl. 81 (2002) 1–33 | MR 1994881 | Zbl 1038.58033

[28] J. Sjöstrand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J. 137 (2007) 381–459. | MR 2309150 | Zbl pre05154881

[29] J. Sjöstrand and M. Zworski, Elementary linear algebra for advanced spectral problems, Annales de l’Institut Fourier 57(2007) 2095–2141 | Numdam | MR 2394537 | Zbl 1140.15009

[30] H.-J. Stöckmann, Scattering Properties of Chaotic Microwave Billiards, Acta Polonica A 116 (2009) 783–789