Adiabatic theorem without a gap condition

Communications in Mathematical Physics

Volumn 203, Issue 2, 1999, Pages 445-463

  Avron, Joseph E ^a   Elgart, Alexander ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033471912     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050620     Document Type: Article

References (58)

1. Arai, A., Hirokawa, M.: On the existence and uniqueness of ground states of a generalized spin-boson model. J. Funct. Anal. 151 (2), 455-503 (1997)
2. Arnold, V.I.: Geometrical Methods in the theory of Ordinary Differential Equations. Berlin-Heidelberg-New-York: Springer, 1983
3. Avron, J. E., Elgart, A.: An adiabatic theorem without a gap condition: Two level system coupled to quantized radiation field. Phys. Rev. A 58, 4300-4306 (1998)
4. Avron, J. E., Howland, J. S., Simon, B.: Adiabatic theorems for dense point spectra. Commun. Math. Phys. 128, 497-507 (1990)
5. Avron, J. E., Seiler, R., Yaffe, L. G.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys. 110, 33-49 (1987), (Erratum: Commun. Math. Phys. 153, 649-650 (1993))
6. Bach, V., Fröhlich, J., Sigal, I. M.: Mathematical theory of nonrelativistic matter and radiation. Lett. Math. Phys. 34, 183-201 (1995)
7. Bach, V., Fröhlich, J., Sigal, I. M.: Quantum electrodynamics of confined non-relativistic particles. Adv. in Math. 137, 299 (1998)
8. Bach, V., Fröhlich, J., Sigal, I. M.: Renormalization group analysis of spectral problems in quantum field theory. Adv. in Math. 137, 205-298 (1998)
9. Bach, V., Fröhlich J., Sigal, I. M., Sofer, A.: Positive commutators and spectrum of non-relativistic QED. To appear
10. Berry, M.V.: Proc. Roy. Soc. Lond. A 392, 45 (1984); The quantum phase: Five years after. In: Geometric phases in physics. Shapere, A. and Wilczek, F., eds., Singapore: World Scientific, 1989
11. Berry, M.V.: Proc. Roy. Soc. Lond. A 392, 45 (1984); The quantum phase: Five years after. In: Geometric phases in physics. Shapere, A. and Wilczek, F., eds., Singapore: World Scientific, 1989
12. Berry, M.V.: Histories of adiabatic transition. Proc. Roy. Soc. Lond. A 429, 61-72 (1990)
13. Berry, M.V., Robbins, J.M.: Chaotic classical and half classical adiabatic reactions: Geometric magnetism and deterministic friction. Proc. Roy. Soc. Lond. A 442, 659-672 (1993); Proc. Roy. Soc. A 392, 45 (1984)
14. Berry, M.V., Robbins, J.M.: Chaotic classical and half classical adiabatic reactions: Geometric magnetism and deterministic friction. Proc. Roy. Soc. Lond. A 442, 659-672 (1993); Proc. Roy. Soc. A 392, 45 (1984)
15. Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of one and two electron atoms. New York: Plenum, 1977
16. Born, M.: The Mechanics of the Atom. New York: Ungar, 1960
17. Born, M., Fock, V.: Beweis des Adiabatensatzes. Z. Phys. 51, 165-169 (1928)
18. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg G.: Atoms and Photons Interactions. New York: Wiley, 1992
19. Combes, J. M.: In: Differential equations with applications to mathematical physics. Boston: Academic Press, 1993; Combes, J. M., Montcho, R.: Remarks on the relation between quantum dynamics and fractal spectra J. Math. Anal. and Appl. 2130, 698-722 (1997)
20. Combes, J. M.: In: Differential equations with applications to mathematical physics. Boston: Academic Press, 1993; Combes, J. M., Montcho, R.: Remarks on the relation between quantum dynamics and fractal spectra J. Math. Anal. and Appl. 2130, 698-722 (1997)
21. Cycon, H. L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Berlin-Heidelberg-New-York: Springer, 1987
22. Davies, E. B., Spohn, H.: Open Quantum Systems with Time-Dependent Hamiltonians and Their Linear Response. J. Stat. Phys. 19, 511-523 (1978)
23. Ehrenfest, P.: Adabatische Invarianten u. Quantentheorie. Ann. d. Phys. 51, 327 (1916)
24. Friedrichs, K. O.: On the perturbation of continuous spectra. Comm. Pure Appl. Math. 1, 361-406 (1948)
25. Friedrichs, K. O.: Special topics in quantum theory. Lecture notes, Courant Institute of Mathematical Science, New York University, (1953); On the adiabatic theorem in quantum theory, Part I. Courant Institute of Mathematical Science, New York University, (1955); On the adiabatic theorem in quantum theory, Part II. Courant Institute of Mathematical Science, New York University, 1956
26. Friedrichs, K. O.: Special topics in quantum theory. Lecture notes, Courant Institute of Mathematical Science, New York University, (1953); On the adiabatic theorem in quantum theory, Part I. Courant Institute of Mathematical Science, New York University, (1955); On the adiabatic theorem in quantum theory, Part II. Courant Institute of Mathematical Science, New York University, 1956
27. Friedrichs, K. O.: Special topics in quantum theory. Lecture notes, Courant Institute of Mathematical Science, New York University, (1953); On the adiabatic theorem in quantum theory, Part I. Courant Institute of Mathematical Science, New York University, (1955); On the adiabatic theorem in quantum theory, Part II. Courant Institute of Mathematical Science, New York University, 1956
28. Garrido, L. M.: Generalized adiabatic invariance. J. Math. Phys. 5, 355-362 (1964)
29. Galindo, A., Pascual, P.: Quantum mechanics. Berlin-Heidelberg-New-York: Springer-Verlag, 1991
30. Golin, S., Knauf, A., Marmi, S.: The Hannay angles: Geometry, Adiabaticity and an example. Commun. Math. Phys. 123, 95-122 (1989)
31. Guarneri, I.: On the dynamical meaning of spectral dimensions. Ann. Inst. H. Poincaré. To appear
32. Hagedorn, G.: Adiabatic Expansions near Eigenvalue Crossings. Ann. Phys. 196, 278-295 (1989)
33. Huebner, Spohn, H.: Ann. Inst. H. Poincaré Phys. Theor. 62 no. 3, 289 (1995)
34. Jakšić, V., Segert, J.: On the Landau Zener formula for two-level systems. J. Math. Phys. 34, 2807-2820 (1993)
35. Jarzinski, C.: Multiple-time-scale approach to ergodic adiabatic systems: Another look. Phys. Rev. Lett. 71, 839 (1993)
36. Joye, A., Pfister, C.E.: Exponential Estimates in Adiabatic Quantum Evolution. Proceeding of the XII ICMP, Brisbane Australia (1997); Quantum Adiabatic Evolution. In: On Three Levels. Fannes, M., Maes, C., Verbure, A., editors, London: Plenum, 1994
37. Joye, A., Pfister, C.E.: Exponential Estimates in Adiabatic Quantum Evolution. Proceeding of the XII ICMP, Brisbane Australia (1997); Quantum Adiabatic Evolution. In: On Three Levels. Fannes, M., Maes, C., Verbure, A., editors, London: Plenum, 1994
38. Kato, T.: Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan. 5, 208-234 1953)
39. Kato, T.: On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap. 5, 435-439 (1958)
40. Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New-York: Springer, 1966
41. Klein, M., Seiler, R.: Power law corrections to the Kubo formula vanish in quantum Hall systems. Commun. Math. Phys. 128, 141 (1990)
42. Last, Y.: Quantum Dynamics and Decomposition of Singular Continuous Spectra. J. Funct. Anal. 142, 406-445 (1996)
43. Lennard, A.: Adiabatic Invariance to All Orders. Ann. Phys. 6, 261-276 (1959)
44. Lochak, P., Meunier, C.: Multiphase Averaging for Classical systems. Berlin-Heidelberg-New-York: Springer, 1988
45. Martinez, A.: Precise exponential estimates in adiabatic theory. J. Math. Phys. 35, 3889-3915 (1994)
46. Martinez, A., Nakamura, S.: Adiabatic limit and scattering. C.R. Acd. Sci. Paris. 318, 1153-1158 (1994)
47. Narnhofer, H., Thirring, W.: Adiabatic theorem in quantum statistical mechanics. Phys. Rev. A 26, 364 (1982)
48. Nenciu, G.: On the adiabatic theorem of quantum mechanics. J. Phys. A 13, L15-L18 (1980)
49. Nenciu, G.: Linear Adiabatic Theory: Exponential Estimates. Commun. Math. Phys. 152, 479-496 (1993)
50. Ott, E.: Goodness of ergodic adiabatic invariants. Phys. Rev. Lett. 42, 1628-1631 (1979); and Brown R., Ott, E., Grebogi, C.: Goodness of ergodic adiabatic invariants. J. Stat. Phys. 49, 511-550 (1987)
51. Ott, E.: Goodness of ergodic adiabatic invariants. Phys. Rev. Lett. 42, 1628-1631 (1979); and Brown R., Ott, E., Grebogi, C.: Goodness of ergodic adiabatic invariants. J. Stat. Phys. 49, 511-550 (1987)
52. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness. London: Academic Press, 1975
53. Riemann, B.: Ueber der Darstellbarkeit einer Function durch eine trigonometrishe Reihe In: Math. Werke, Leipzig: Teubner, 1876, pp. 213-253; Lebesgue, H.: Sur les Séries Trigonométriques. Ann. Sci. Ecole Norm. Sup. 20. 453-485 (1903)
54. Riemann, B.: Ueber der Darstellbarkeit einer Function durch eine trigonometrishe Reihe In: Math. Werke, Leipzig: Teubner, 1876, pp. 213-253; Lebesgue, H.: Sur les Séries Trigonométriques. Ann. Sci. Ecole Norm. Sup. 20. 453-485 (1903)
55. Simon, B.: The theory of resonances for dilation analytic potentials and the foundations of time-dependent perturbation theory. Ann. of Math. 97, 247-274 (1973)
56. Spohn, H.: Ground state(s) of the spin-boson Hamiltonian. Comm. Math. Phys. 123, 277-304 (1989)
57. Thouless, D.J.: Topological Quantum Numbers in Nonrelativistic Physics. Singapore: World Scientific, (1998)
58. Yosida, K.: Functional Analysis. Berlin: Springer-Verlag, 1968