Quantum transfer operators and quantum scattering

 Access to full text

 Full (PDF)

1. V. Baladi, Positive Transfer Operators and Decay of Correlations, Book Advanced Series in Nonlinear Dynamics, Vol 16, World Scientific, Singapore (2000) Zbl1012.37015MR1793194
2. R. Blümel and W. P. Reinhardt, Chaos in Atomic Physics, Cambridge University Press, Cambridge, 1997 Zbl0939.81044MR1627898
3. E. B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805–866 Zbl0749.35035MR1174220
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 Zbl0242.58005MR336762
6. H. Christianson, Quantum monodromy and non-concentration near a closed semi-hyperbolic orbit, preprint 2009 MR2625926
7. P. Cvitanović, P. Rosenquist, G. Vattay and H.H. Rugh, A Fredholm determinant for semiclassical quantization, CHAOS 3 (1993) 619–636 Zbl1055.81549MR1256315
8. M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, 1999. Zbl0926.35002MR1735654
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 Zbl0770.58043MR1187738
10. L.C. Evans and M. Zworski, Lectures on Semiclassical Analysis, Version 0.3.2, http://math.berkeley.edu/$\sim$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 MR980393
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 Zbl1020.81937MR1329478
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 Zbl0654.35081MR998698
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 Zbl1088.37010MR2201945
15. M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, 38 (1988) 113–146 Zbl0636.35045MR949013
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 Zbl1223.81127MR2793928
19. S. Nonnenmacher and M. Zworski, Distribution of resonances for open quantum maps, Comm. Math. Phys. 269 (2007) 311–365 Zbl1114.81043MR2274550
20. S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering, Acta Math. 203 (2009) 149–233 Zbl1226.35061MR2570070
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 Zbl1125.37013MR2374466
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 Zbl0591.58025MR807065
23. T. Prosen, General quantum surface-of-section method, J. Phys.A 28 (1995) 4133-4155 Zbl0860.58019MR1352158
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 Zbl0702.35188MR1047116
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. Zbl0877.35090MR1451399
27. J. Sjöstrand and M. Zworski, Quantum monodromy and semiclassical trace formulae, J. Math. Pure Appl. 81 (2002) 1–33 Zbl1038.58033MR1994881
28. J. Sjöstrand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J. 137 (2007) 381–459. Zbl1201.35189MR2309150
29. J. Sjöstrand and M. Zworski, Elementary linear algebra for advanced spectral problems, Annales de l’Institut Fourier 57(2007) 2095–2141 Zbl1140.15009MR2394537
30. H.-J. Stöckmann, Scattering Properties of Chaotic Microwave Billiards, Acta Polonica A 116 (2009) 783–789