Time-periodic schrödinger equations

Topics in the Theory of Schrodinger Operators

Volumn , Issue , 2014, Pages 9-70

  Yajima, Kenji ^a  

^a GAKUSHUIN UNIVERSITY   (Japan)

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EID: 85013599833     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/9789812562470_0002     Document Type: Chapter

References (73)

1. S. Agmon. Spectral properties of Schrodinger operators and scattering theory, Ann. Scuola. Norm. Sup. Pisa, Ser. IV. 2 (1975), 151-218.
2. S. Agmon, I. Herbst and E. Skibsted. Perturbation of embedded eigenvalues in the generalized N-body problem, Commun. Math. Phys. 122 (1989), 411-438.
3. J. Aguilar and J. M. Combes. A class of analytic perturbations for one-body Schrödinger Hamiltonians, Commun. Math. Phys. 22 (1971), 269-279.
4. G. Artbazar and K. Yajima. Lp-boundedness of wave operators for one-dimensional Schrödinger operators, J. Math. Sei. Univ. Tokyo. 7 (2000), 221240-.
5. M. Ben-Artzi and A. Devinatz. Local smoothing and convergence properties of Schröding er type equations, J. Funct. Anal. 101 (1991), 231-254.
6. E. Balslev and J. M. Combes. Spectral properties of Schrödinger operators with dilatation-analytic interactions, Commun. Math. Phys. 22 (1971), 280294-.
7. P. Constantin and J. C. Saut. Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J. 38 (1989), 791-810.
8. O. Costin, R. D. Costin, J. L. Lebowitz, and A. Rokhlenko. Evolution of a model quantum system under time-periodic forcing: Conditions for complete ionization, Comm. Math. Phys. 221 (2001), 1-26.
9. H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon. Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics, Springer Study Edition (Springer-Verlag, Berlin, 1987).
10. N. Burq, F. Planchon, J. G. Stalker and A. Shadi Tahvildar-Zadeh. Strichartz estimates for the wave and Schrödinger equations with inverse-square potential, reprint, 2002.
11. V. Enss and K. Veselic. Bound states and propagating states for time-dependent Hamiltonians, Ann. Inst. H. Poincare Sect. A (N.S.). 39 (1983), 159-191.
12. J. Derezinski and C. Gérard. Scattering theory of classical and quantum N-particle systems, Texts and Monographs in Physics (Springer Verlag, Berlin-Heidelberg, 1997).
13. A. Galtbayar, A. Jensen and K. Yajima. Local time-decay of solutions to Schrödinger equations with time-periodic potentials, To appear in J. Stat. Phys. (2003).
14. J. Ginibre and G. Velo. Smoothing properties and retarded estimates for some dispersive evolution equations, Commun. Math. Phys. 144 (1992), 163-188.
15. L. Hörmander. The analysis of linear partial differential operators I, Distribution theory and Fourier analysis. Second edition, Springer Study Edition (Springer-Verlag, Berlin, 1990).
16. T. Hoshiro. Mourre’s method and smoothing pfoperties of dispersive equations, Commun. Math. Phys. 202 (1999), 255-265.
17. J. Howland. Stationary theory for time dependent Hamiltonians, Math. Ann. 207 (1974), 315-335.
18. J. Howland. Scattering theory for Hamiltonians periodic in time, Indiana Univ. Math. J. 28 (1979), 471-494.
19. W. Hunziker and I. M. Sigal. The quantum N-body problem, J. Math. Phys. 41 (2000), 3448-3510
20. H. Isozaki and H. Kitada. Scattering matrices for two-body Schrodinger operators, Sei. Papers College Arts Sei. Univ. Tokyo. 35 (1986), 81-107.
21. A. Jensen and T. Kato. Spectral properties of Schródinger operators and time-decay of the wave functions, Duke. Math. J. 46 (1979), 583-611.
22. J.-L. Journé, A. Soffere and CD. Sogge. Decay estimates for Schródinger operators, Comm. Pure Appi. Math. 44 (1991), 573-604.
23. T. Kato, Perturbation theory for linear operators (Springer Verlg. New-York, 1966).
24. T. Kato. Wave operators and similarities for some non-self adjoint operators, Math. Ann. 162 (1965/1966), 7.
25. T. Kato. On nonlinear Schródinger equations. II. Hs-solutions and unconditional well-posedness, J. d’Anal. Math. 67 (1995), 281-306.
26. T. Kato and K. Yajima. Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481-496.
27. M. Keel and T. Tao. Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980.
28. C. E. Kenig, G. Ponce and L. Vega. Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69.
29. E. Kirr and M. I. Weinstein. Parametric excited Hamiltonian partial differential equations, SIAM J. Math. Anal. 33 (2001), 16-52.
30. H. Kitada and K. Yajima. Bound states and scattering states for time periodic Hamiltonians, Ann. Inst. H. Poincaré Sect. A (N.S.). 39 (1983), 145-157.
31. H. Kitada and K. Yajima. A scattering theory for time-dependent long-range potentials, Duke Math. J. 49 (1982), 341-376.
32. E. L. Korotyaev. Scattering theory for three-particle systems with time-periodic pair interactions, (Russian), Teoret. Mat. Fiz. 62 (1985), 242-252.
33. E. L. Korotyaev. Eigenj unctions of the monodromy operator of the Schródinger operator with a potential that is periodic with respect to time (Russian), Mat. Sb. (N.S.) 124(166) (1984), 431-446.
34. S. T. Kuroda. Scattering theory for differential operators. I. Operator theory; II. Self-adjoint elliptic operators. J. Math. Soc. Japan 25 (1973), 75-104 and 222-234.
35. E. H. Lieb. The stability of matter: from atoms to stars, Bull. Amer. Math. Soc. (N.S.). 22 (1990), 1-49.
36. K. Masuda. Theory of evolution equations (Kinokuniya-Shoten, Tokyo, 1975), in Japanese.
37. M. Merkli and I. M. Sigal. A time-dependent theory of quantum resonances, Comm. Math. Phys. 201 (1999), 549-576.
38. J. Moeller and E. Skibsted. Spectral theory for time-periodic many-body problem, preprint, Aarhus University (2003).
39. P. D. Miller, A. Soffer, and I. M. Weinstein. Metastability of breather modes of time dependent potentials, Nonlinearity. 13 (2000), 507-568.
40. M. Murata. Asymptotic expansions in time for solutions of Schròdinger-type equations, J. Funct. Anal. 49 (1982), 10-56.
41. S. Nakamura. Asymptotic completeness for three-body Schródinger equations with time-periodic potentials, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), 379-402.
42. S. Nakamura. Structure of the scattering operator for time-periodic Schródinger equation, J. Math. Soc. Japan. 38 (1986), 261-273.
43. S. Nakamura. Integral kernels of the scattering operator for time-periodic Schrodinger equation, J. Func. Anal. 76 (1988), 176-192.
44. R. O’Neil. Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963), 129-142.
45. A. Pazy. Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44 (Springer-Verlag, New York, 1983).
46. G. Ponce. Smoothing propeties of solutions to the Benjamin-Ono equation, Analysis and partial deifferential equations, 667-679, Lect. Notes in Pure and Appl. Math. 122 (Dekker, New York, 1990).
47. M. Reed and B. Simon. Methods of modern mathematical physics. I: Functional analysis, Second edition (Academic Press, New York, 1980).
48. M. Reed and B. Simon. Methods of modern mathematical physics. II: Fourier Analysis, Selfadjointness (Academic Press, New York-London, 1975).
49. M. Reed and B. Simon. Methods of modern mathematical physics. Ill: Scattering theory (Academic Press, New York-London, 1979).
50. M. Reed and B. Simon. Methods of modern mathematical physics. IV: Analysis of Operators (Academic Press, New York-London, 1978).
51. I. Rodnianski and W. Schlag. Time decay for solutions of Schrodinger equations with rough and time dependent potentials, preprint, 2001.
52. I. Rodnianski, Schlag and A. Soffer. Dispersive analysis of charge transfer models, preprint.
53. B. Simon. Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math. (2) 97 (1973), 247-274.
54. P. Sjolin. Regularity of solutions to the Schrodinger equations, Duke Math. J. 55 (1987), 699-715.
55. J. Sjostrand. Quantum resonances and trapped trajectories, Long time behaviour of classical and quantum systems (Bologna, 1999), 33-61, Ser. Concr. Appl. Math. 1, World Sci. Publishing, River Edge, NJ (2001).
56. R. S. Strichartz. Restrictions of Fourier transforms to a quadratic surface and decay of solutions of wave equations, Duke Math. J. 44 (1977), 704-714.
57. A. Soffer and M. I. Weinstein. Nonautonomous Hamiltonians, J. Statist. Phys. 93 (1998), 359-391.
58. A. Soffer and M. I. Weinstein. Time dependent resonance theory, Geom. Funct. Anal. 8 (1998), 1086-1128.
59. A. Soffer and M. I. Weinstein. Ionization and scattering for short-lived potentials, Lett. Math. Phys. 48 (1999), 339-352.
60. A. Soffer and M. I. Weinstein. Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), 9-74.
61. H. Tanabe. Equations of evolution, Monographs and Studies in Mathematics, 6 (Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979).
62. D. Tataru. Unique continuation for solutions to PDE’s; between Hormander’s theorem and Holmgren’s theorem, Commun. in PDE. 20 (1995), 855-884.
63. L. Vega. Schrodinger equations: Pointwise convergence to the initial data, Proc. A. M. S. 102 (1988), 874-878.
64. R. Weder. Lp-Lp estimates for the Schrddinger equation on the line and inverse scattering for the nonlinear Schrddinger equation with a potential, J. Funct. Anal. 170 (2000), 37-68.
65. K. Yajima. Scattering theory for Schrddinger equations with potentials periodic in time, J. Math. Soc. Japan. 29 (1977), 729-743.
66. K. Yajima. Resonances for the AC-Stark effect, Commun. Math. Phys. 78 (1982), 331-352.
67. K. Yajima. Exponential decay of quasi-stationary states of time-periodic Schrodinger equation with short range potentials, Sci. Papers College Arts Sci. Univ. Tokyo. 40 (1990), 27-36.
68. K. Yajima. Existence of solutions for Schrddinger evolution equation, Commun. Math. Phys. 110 (1987), 415-426.
69. K. Yajima. The Wk’p-continuity of wave operators for Schrodinger operators, J. Math. Soc. Japan. 47 (1995), 551-581.
70. K. Yajima. The Wk’p-continuity of wave operators for Schrodinger operators, III, Even dimensional case m > 4, J. Math. Sci. Univ. Tokyo 2 (1995), 311-346.
71. K. Yajima. Lp-boundedness of wave operators for two-dimensional Schrddinger operators, Commun. Math. Phys. 208 (1999), 125-152.
72. K. Yokoyama. Mourre theory for time-periodic systems, Nagoya Math. J. 149 (1998), 193-210.
73. M. Zworski. Quantum resonances and partial differential equations, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 243-252, (Higher Ed. Press, Beijing, 2002).