Adiabatic Theorems for Generators of Contracting Evolutions

Communications in Mathematical Physics

Volumn 314, Issue 1, 2012, Pages 163-191

  Avron, J E ^a   Fraas, M ^a   Graf, G M ^b   Grech, P ^b  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84864463467     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-012-1504-1     Document Type: Article

References (35)

1. Abou Salem W. K.: On the quasi-static evolution of nonequilibrium steady states. Ann. H. Poincaré 8, 569-596 (2007).
2. Avron J. E., Elgart A.: Adiabatic theorem without a gap condition. Commun. Math. Phys. 203, 445-463 (1999).
3. Avron J. E., Fraas M., Graf G. M., Grech P.: Landau-Zener tunneling for dephasing lindblad evolutions. Commun. Math. Phys. 305(3), 633-639 (2011).
4. Avron J. E., Fraas M., Graf G. M., Grech P.: Optimal time-schedule for adiabatic evolution. Phys. Rev. A 82, 040304 (2010).
5. Avron J. E., Fraas M., Graf G. M., Kenneth O.: Geometry of quantum transport for dephasing lindbladians. New J. Phys. 13, 053042 (2011).
6. Avron J. E., Seiler R., Yaffe L. G.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys. 110, 33-49 (1987).
7. Berry M. V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45-57 (1984).
8. Berry M. V.: Histories of adiabatic quantum transitions. Proc. R. Soc. Lond. A 429, 61-72 (1990).
9. Berry M. V., Robbins J. M.: Chaotic classical and half-classical adiabatic reactions: geometric magnetism and deterministic friction. Proc. R. Soc. Lond. A 442, 659-672 (1993).
10. Bornemann F.: Homogeneization in Time of Singularly Perturbed Mechanical Systems. Lecture Notes in Math. 1687. Springer, Berlin-Heidelberg (1998).
11. Bratteli O., Robinson D. W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer-Verlag, Berlin-Heidelberg (1979).
12. Cycon H. L., Froese R. G., Kirsch W., Simon B.: Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer-Verlag, Berlin-Heidelberg New York (1987).
13. Davies, E. B.: Quantum theory of open systems. London: Academic Press [Harcourt Brace Jovanovich Publishers], 1976.
14. Gorini V., Kossakowski A., Sudarshan E. C. G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17(5), 821-825 (1976).
15. Hagedorn, G. A., Joye, A.: Recent results on non-adiabatic transitions in quantum mechanics. In Recent advances in differential equations and mathematical physics, Volume 412 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2006, pp. 183-198.
16. Hille, E., Phillips, R. S.: Functional analysis and semi-groups. Providence, R. I.: Amer. Math. Soc., 1997.
17. Horowitz J., Jarzynski C.: Exact formula for currents in strongly pumped diffusive systems. J. Stat. Phys. 136, 917-925 (2009).
18. Joye A.: General adiabatic evolution with a gap condition. Commun. Math. Phys. 275, 139-162 (2007).
19. Kato T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Japan 5, 435-439 (1950).
20. Kato T.: Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan 5(2), 208-234 (1953).
21. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Berlin, Heidelberg: Springer, 1995.
22. Krein, S. G.: Linear differential equations in Banach space. Providence, RI: Amer. Mathe. Soc., 1972.
23. Lindblad G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119-130 (1976).
24. Lumer G., Phillips R. S.: Dissipative operators in a Banach space. Pacific J. Math. 11, 679-698 (1961).
25. Maes C., Netočný K., Thomas S. R.: General no-go condition for stochastic pumping. J. Chem. Phys. 132(23), 234116 (2010).
26. Mandal D., Jarzynski C.: A proof by graphical construction of the no-pumping theorem of stochastic pumps. J. Stat. Mech. Th. Exp. 2011(10), P10006 (2011).
27. Nenciu G.: Linear adiabatic theory. exponential estimates. Commun. Math. Phys. 152, 479-496 (1993).
28. Nenciu G., Rasche G.: On the adiabatic theorem for nonself-adjoint Hamiltonians. J. Phys. A. 25, 5741-5751 (1992).
29. Parrondo J. M. R.: Reversible ratchets as Brownian particles in an adiabatically changing periodic potential. Phys. Rev. E. 57(6), 7297-7300 (1998).
30. Paz J. P., Zurek W. H.: Quantum limit of decoherence: Environment induced superselection of energy eigenstates. Phys. Rev. Lett. 82, 5181 (1999).
31. Rahav S., Horowitz J., Jarzynski Ch.: Directed flow in nonadiabatic stochastic pumps. Phys. Rev. Lett. 101(14), 140602 (2008).
32. Reed, M., Simon, B.: Fourier Analysis and Self-Adjointness, Volume2 of Methods of Modern Mathematical Physics. London: Academic Press, 1975.
33. Simon, B.: Trace ideals and their applications. Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1979.
34. Stroock D. W.: An introduction to Markov processes, Volume 230 of Graduate Texts in Mathematics. Springer-Verlag, Berlin (2005).
35. Teufel S.: A note on the adiabatic theorem without gap condition. Lett. Math. Phys. 58(3), 255-261 (2001).