Long-time approximation to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics

Journal of Mathematical Physics

Volumn 37, Issue 5, 1996, Pages 2182-2205

  Saenz A W ^a,b  

^a CATHOLIC UNIVERSITY OF AMERICA   (United States)

^b NAVAL RESEARCH LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030528513     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531799     Document Type: Article

References (51)

1. E. P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, J. J. Griffin, Trans. (Academic, New York, 1959), pp. 42, 43.
2. G. Lochak, C. R. Acad. Sci. 259, 3183 (1964).
3. G. Lochak, J. Phys. (Paris) 30, 482 (1969).
4. K. M. Case, Suppl. Prog. Theor. Phys. (Kyoto) 37, 998 (1966).
5. R. Yee Yen Lee, "On a new perturbation method," Ph.D thesis, University of Michigan, 1964.
6. N. N. Bogoliubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan, New Delhi, 1961).
7. A. Ben Lemlih, "An extension of the method of averaging to partial differential equations," Ph.D. thesis, University of New Mexico, Albuquerque, NM, 1986.
8. A. Ben Lemlih and J. A. Ellison, Phys. Rev. Lett. 55, 1950 (1985).
9. M. Kummer, Nuovo Cim. B 1, 123 (1971).
10. J. Moser, Mem. Am. Math. Soc. 81, 1 (1968).
11. N. D. Antoniou, "On the convergence of perturbation theories that avoid secular terms in quantum mechanics," Ph.D. thesis, University of Toledo, Toledo, OH (1972).
12. R. Gompa, "Approximations to the quantum mechanical time evolution," Ph.D. thesis, University of Toledo, Toledo, OH, 1987.
13. M. Kummer and R. Gompa, J. Math. Phys. 29, 1405 (1988).
14. R. Gompa, J. Math. Phys. 32, 959 (1991).
15. V. Bargmann, Comm. Pure Appl. Math. 14, 187 (1951); Proc. Natl. Acad. Sci. USA 48, 199 (1962); Comm. Pure Appl. Math. 26, 1 (1969).
16. V. Bargmann, Comm. Pure Appl. Math. 14, 187 (1951); Proc. Natl. Acad. Sci. USA 48, 199 (1962); Comm. Pure Appl. Math. 26, 1 (1969).
17. V. Bargmann, Comm. Pure Appl. Math. 14, 187 (1951); Proc. Natl. Acad. Sci. USA 48, 199 (1962); Comm. Pure Appl. Math. 26, 1 (1969).
18. R. Churchill, M. Kummer, and D. L. Rod, J. Diff. Eq. 49, 359 (1983).
19. A. Baider and R. Churchill, Proc. R. Soc. Edin. A 108, 27 (1988).
20. See M. Born and P. Jordan, Elementare Quantemmechanik (Springer, Berlin, 1930), § 38, 41.
21. M. Kummer, in Essays in Classical and Quantum Dynamics (A Festschrift for A. W. Sáenz), edited by J. A. Ellison and H. Überall (Gordon and Breach, Reading, PA, 1991), p. 139.
22. In what follows, all equations involving t (resp., ε) hold for t ∈ R (resp., ε≥0), in the absence of an explicit statement to the contrary.
23. A. W. Sáenz, in Ref. 19, p. 163.
24. v) which they leave invariant. The operator equations in this section hold on this set.
25. εs with compact resolvents and simple spectra, see Ref. 11.
26. This will be proved in Sec. IV on the basis of the assumptions on V in Sec. III.
27. + .
28. (i) are Hermitian.
29. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1975), Vol. II, Example 3, p. 266.
30. See, e.g., V. I. Arnold, Russ. Math. Surv. 18, 85 (1963) and G. Gallavotti, The Elements of Mechanics (Springer-Verlag, New York, 1983), § 5.12.
31. See, e.g., V. I. Arnold, Russ. Math. Surv. 18, 85 (1963) and G. Gallavotti, The Elements of Mechanics (Springer-Verlag, New York, 1983), § 5.12.
32. C. L. Siegel and J. K. Moser, Lectures in Celestial Mechanics (Springer-Verlag, Berlin, 1971), pp. 260, 261.
33. The domain of an operator A is denoted by D(A).
34. ∞-vectors for A.
35. M. H. Stone, Linear Transformations in Hilbert Space, A. M. S. Colloq. Pub., Vol. XV (A.M.S., New York, 1932), Theorem 3.1, pp. 88, 89.
36. See, e.g., J. von Neumann, Math. Ann. 102, 49 (1929); J. Math. 161, 208 (1929).
37. See, e.g., J. von Neumann, Math. Ann. 102, 49 (1929); J. Math. 161, 208 (1929).
38. In the main text, all operators associated with matrices have this property wrt Φ. Hence the last qualifier will be usually omitted.
39. The notations Lemma 3(1), Lemma 4(2), etc., denote assertion (1) of Lemma 3, assertion (2) of Lemma 4, etc.
40. Henceforth, all constants in equations involving the free indices m,n should be understood to be independent of m,n even if this is not stated explicitly.
41. See the above volume by Reed and Simon, Ref. 27, Eq. (X.28), p. 175.
42. When any of the estimates (4.13) or (4.16) is mentioned henceforth in the text, it should be understood to hold for all p ∈ N, even if this is not stated explicitly.
43. +.
44. See the theorem on p. 121 of J. Dixmier, General Topology (Springer-Verlag, New York, 1984).
45. (s)) is densely defined.
46. 0), even if this is not stated explicitly.
47. See M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I (Academic, New York, 1972), Theorem VIII.11, p. 269.
48. See, e.g., the last cited volume of Reed and Simon, Ref. 43, pp. 270, 271.
49. See, e.g., V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, Dordrecht, the Netherlands, 1981), Proposition 10.6, p. 186.
50. Unfortunately, the crucial third sentence of this theorem is confusing. But what it was meant to state is readily reconstructed from the proof of the theorem (Ref. 32, especially p. 92), whose arguments imply a result stronger than our Lemma A4.
51. Here and henceforth, summations over α,β range over Γ unless otherwise stated.