General adiabatic evolution with a gap condition

Communications in Mathematical Physics

Volumn 275, Issue 1, 2007, Pages 139-162

  Joye, Alain ^a  

^a UNIV GRENOBLE ALPES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34547843619     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-007-0299-y     Document Type: Article

References (42)

1. Abou-Salem, W.: On the quasi-static evolution of nonequilibrium steady states. To appear in Ann. Henri Poincaré, DOI: 10.1007/s00023-006-0316-z, 2007
2. Abou-Salem, W.: "Nonequilibrium quantum statistical mechanics and thermodynamics". PhD Thesis, ETH 16187, (2005)
3. Abou-Salem W. and Fröhlich J. (2005). Adiabatic Theorems and Reversible Isothermal Processes. Lett. Math. Phys. 72: 153-163
4. Abou-Salem, W., Fröhlich, J.: Adiabatic Theorems for Quantum Resonances. To appear in Commun. Math. Phys., DOI: 10.1007/s00220-007-0198-z, 2007
5. Avron J.E. and Elgart A. (1999). Adiabatic Theorem without a Gap Condition. Commun. Math. Phys. 203: 445-463
6. Avron J.E., Howland J.S. and Simon B. (1990). Adiabatic theorems for dense point spectrum. Commun. Math. Phys. 128: 497-507
7. Avron J.E., Seiler R. and Yaffe L.G. (1987). Adiabatic Theorems and Applications to the Quantum Hall Effect. Commun. Math. Phys. 110: 33-49
8. Baumgärtel, H.: "Analytic Perturbation Theory for Matrices and Operators". Operator Theory: Advances and Applications. Vol. 15, Basel-Boston: Birkhäuser, 1985
9. Berry M.V. (1990). Histories of adiabatic quantum transitions. Proc. Roy. Soc. Lond. A 429: 61-72
10. Born M. and Fock V. (1928). Beweis des Adiabatensatzes. Zeit. f. Physik 51: 165-180
11. Bornemann, F.: "Homogeneization in Time of Singularly Perturbed Mechanical Systems". Lecture Notes in Mathematics 1687, Heidelberg: Springer, 1998
12. Boutet de Monvel A. and Nenciu G. (1990). "On the Theory of Adiabatic Invariants for Linear Hamiltonian Systems". Comptes Rendus Acad. Sci. Paris A 310: 807-810
13. Dimassi, M., Sjöstrand, J.: "Spectral asymptotics in the semi-classical limit". London Math. Soc. Lecture Note Ser. 268, Cambridge: Cambridge University Press, 1999
14. Fedoryuk, M.: "Méthodes Asymptotiques pour les Equations différentielles ordinaires linéaires". Moscou: Mir, 1987
15. Guérin S., Joye A., Jauslin H. and Monti F. (1999). Adiabatic Evolution for Systems with Infinitely many Eigenvalue Crossings. J. Math. Phys. 40: 5456-5472
16. Hagedorn G. and Joye A. (2002). Elementary Exponential Error Estimates for the Adiabatic Approximation. J. Math. Anal. Appl. 267: 235-246
17. Hagedorn, G., Joye, A.: Recent Results on Non-Adiabatic Transitions in Quantum Mechanics. In: Recent Advances in Differential Equations and Mathematical Physics. AMS Contemporary Mathematics Series 412, Ed. by N. Chernov, Y. Karpeshina, I. Knowles, R. Lewis, R. Weikard. Providence, RI: Amer. Math. Soc., 2006, p. 183-198
18. Joye A. (1995). An Adiabatic Theorem for Singularly Perturbed Hamiltonians. Ann. Inst. H. Poincaré Phys. Théor. 63: 231-250
19. Joye A. (1997). Exponential Asymptotics in a Singular Limit for n-Level Scattering Systems. SIAM J. Math. Anal. 28: 669-703
20. Joye A., Kunz H. and Pfister C.-E. (1991). Exponential Decay and Geometric Aspect of Transition Probabilities in the Adiabatic Limit. Ann. Phys. 208: 299-332
21. Joye A. and Pfister C.-E. (1991). Exponentially small adiabatic invariant for the Schrödinger equation. Commun. Math. Phys. 140: 15-41
22. Joye A. and Pfister C.-E. (1993). Superadiabatic evolution and adiabatic transition probability between two non-degenerate levels isolated in the spectrum. J. Math. Phys. 34: 454-479
23. Joye A. and Pfister C.-E. (1995). Semiclassical Asymptotics Beyond all Orders for Simple Scattering Systems. SIAM J. Math. Anal. 26: 944-977
24. Joye, A., Pfister, C.-E.: Exponential Estimates in Adiabatic Quantum Evolution. In: XIIth International Congress of Mathematical Physics, ICMP '97, De Wit, Bracken, Gould, Pearce, eds, Cambridge, MA: International Press, 1999, p. 309-315
25. Kato T. (1950). On the Adiabatic Theorem of Quantum Mechanics. J. Phys. Soc. Japan 5: 435-439
26. Kato T. (1980). Perturbation Theory for Linear Operators. Springer-Verlag, Berlin-Heidelberg-New York
27. Krein, S.G.: Linear Differential Equations in Banach Space. Providence, RI: Amer. Math. Soc. 1971
28. Martin Ph.A. and Nenciu G. (1995). Semi-classical Inelastic S-Matrix for One-dimensional N-States Systems. Rev. Math. Phys. 7: 193-242
29. Martinez A. (2002). An Introduction to Semiclassical and Microlocal Analysis. Springer-Verlag UTX Series, New York
30. Nenciu G. (1980). On the adiabatic theorem of quantum mechanics. J. Phys. A 13: L15-L18
31. Nenciu G. (1993). Linear adiabatic theory. Exponential estimates. Commun. Math. Phys. 152: 479-496
32. Nenciu G. and Rasche G. (1992). On the adiabatic theorem for nonself-adjoint Hamiltonians. J. Phys. A 25: 5741-5751
33. Kvitsinsky A. and Putterman S. (1991). Adiabatic evolution of an irreversible two level system. J. Math. Phys. 32: 1403-1407
34. Reed, M., Simon, B.: "Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness". London-New York: Academic Press, 1975
35. Sarandy M.S. and Lidar D.A. (2005). Adiabatic approximation in open quantum systems. Phys. Rev. A 71: 012331
36. Sarandy M.S. and Lidar D.A. (2005). Adiabatic Quantum Computation in Open Quantum Systems. Phys. Rev. Lett. 95: 250503
37. Sarandy M.S. and Lidar D.A. (2006). Abelian and non-Abelian geometric phases in adiabatic open quantum systems. Phys. Rev. A 73: 062101
38. Stengle G. (1971). A new basis for asymptotic solution of differential equations containing one or several parameters. Trans. AMS 156: 1-43
39. Teufel S. (2001). A note on the adiabatic theorem without gap condition. Lett. Math. Phys. 58: 261-266
40. Teufel, S.: "Adiabatic perturbation theory in quantum dynamics". Lecture Notes in Mathematics 1821, Berlin-Heidelberg-New York: Springer-Verlag, 2003
41. Thunstrom P., Aberg J. and Sjoqvist E. (2005). Adiabatic Approximation for weakly open systems. Phys. Rev. A 72: 022328
42. Turrittin, H.L.: Asymptotic expansions of solutions of systems of ordinary differential equations containing a parameter. In: "Contributions to the theory of nonlinear oscillations. Volume II", Lefschetz, S. ed., Annals of mathematics studies (029), Princeton, NJ: University Press, 1965