Lie-series approach to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics

Journal of Mathematical Physics

Volumn 39, Issue 4, 1998, Pages 1887-1909

  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: 0032371987     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532301     Document Type: Article

References (47)

1. M. Kummer and R. Gompa, J. Math. Phys. 29, 1405 (1988).
2. R. Gompa, "Approximations to the quantum mechanical time evolution," Ph.D. thesis, University of Toledo, Toledo, OH, 1987.
3. M. Born and P. Jordan, Elementare Quantenmechanik (Springer, Berlin, 1930). Secs. 38, 41.
4. This normal form approach differs from that used, e.g., by J. Bellisard and M. Vittot, Ann. Inst. Henri Poincaré 52, 175 (1990) and by M. Degli Esposti, S. Graffi, and J. Herczinski, Ann. Phys. (N.Y.) 209, 364 (1991). These authors use a transformation based on a single exponential operator, rather than on a product of such operators, to reduce the pertinent Hamiltonian operators to normal form.
5. This normal form approach differs from that used, e.g., by J. Bellisard and M. Vittot, Ann. Inst. Henri Poincaré 52, 175 (1990) and by M. Degli Esposti, S. Graffi, and J. Herczinski, Ann. Phys. (N.Y.) 209, 364 (1991). These authors use a transformation based on a single exponential operator, rather than on a product of such operators, to reduce the pertinent Hamiltonian operators to normal form.
6. J. Moser, Mem. Am. Math. Soc. 81 (1968).
7. R. Churchill, M. Kummer, and D. L. Rod, J. Diff. Eqns. 49, 359 (1983).
8. A. Baider and R. Churchill. Proc. R. Soc. Edinburgh Sect. A 108, 27 (1988).
9. P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems (Springer-Verlag, New York, 1988), especially Appendix 7.
10. J. M. Finn, in Local and Global Methods of Nonlinear Dynamics, Lecture Notes in Physics 252, edited by A. W. Sáenz, W. W. Zachary, and R. Cawley (Springer-Verlag, Berlin, 1986), p. 63.
11. V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961); Proc. Natl. Acad. Sci. USA 48, 199 (1962); Commun. Pure Appl. Math. 26, 1 (1969).
12. V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961); Proc. Natl. Acad. Sci. USA 48, 199 (1962); Commun. Pure Appl. Math. 26, 1 (1969).
13. V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961); Proc. Natl. Acad. Sci. USA 48, 199 (1962); Commun. Pure Appl. Math. 26, 1 (1969).
14. M. Kummer, Nuovo Cimento B 1, 123 (1971).
15. 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.
16. See, however, the interesting article by M. Kummer, in Essays in Classical and Quantum Mechanics (Festschrift for A. W. Sáenz), edited by J. A. Ellison and H. Überall (Gordon and Breach, Reading, PA, 1991), p. 139.
17. A. W. Sáenz, J. Math. Phys. 37, 2182 (1996).
18. β〉." In Lemma A2, p. 2203, the condition f ∈. ℋ should read f ∈. ℋ.
19. Here and henceforth, all equations involving t (resp., ε) hold for t ∈ R (resp.. ε≥0). in the absence of an explicit statement to the contrary.
20. The relevant definitions are given in Sec. III.
21. +.
22. The obvious formal power series defining the exponentials in (2.11) are analogous to the Lie series occurring in classical mechanics. See, e.g., Refs. 8 and 9.
23. i (i≥=0) in the formal series on the Ihs is an extension of the corresponding coefficient on the rhs. For a review of formal power series with coefficients in ℐ(ℋ), based on the properties of the pseudo-algebra of unbounded linear operators, see Ref. 2, Chap. 1, Sec. 1.1. The basic rules of calculation with unbounded operators in Hilbert space are given, e.g., by F. Riesz B. Sz. Nagy, Functional Analysis (F. Ungar Publishing, New York, 1955), p. 299. In the rigorous discussions in Sec. IV, we will avoid using formal power series.
24. (i).
25. M. Reed and B. Simon, Methods of Mathematical Physics, Vol. II (Academic, New York, 1975), Example 3, p. 266.
26. See, for example, V. I. Arnold, Russ. Math. Surveys 18, 85 (1963) and G. Gallavotti, The Elements of Mechanics (Springer-Verlag, New York, 1983), Sec. 5.12.
27. See, for example, V. I. Arnold, Russ. Math. Surveys 18, 85 (1963) and G. Gallavotti, The Elements of Mechanics (Springer-Verlag, New York, 1983), Sec. 5.12.
28. C. L. Siegel and J. K. Moser, Lectures in Celestial Mechanics (Springer-Verlag, Berlin, 1971), pp. 260, 261.
29. If A is an operator on ℋ, its domain is denoted by D(A) and its restriction to a subset ℒ⊂D(A) of ℋ by A|ℒ. If A,B are operators on ℋ, then we write A = B iff A and B have the same domain and range; we say that A = B on ℒ⊂. ℋ if ℒ⊂D(A)∩D(B) and A|ℒ=B|ℒ; and A⊂B (resp., A⊃B) means that A is a restriction (resp., an extension) of B.
30. 22 p. 201.
31. mn), etc., will always be Ω, even if this is not stated explicitly.
32. M. H. Stone, Linear Transformations in Hilbert Space, A.M.S. Colloq. Pub., Vol. XV (A.M.S., New York, 1932), pp. 88-93.
33. J. von Neumann, Math. Ann. 102, 49 (1929); J. f. Math. 161, 208 (1929).
34. J. von Neumann, Math. Ann. 102, 49 (1929); J. f. Math. 161, 208 (1929).
35. See, for example, Ref. 28, p. 88 and R. G. Cooke, Infinite Matrices and Sequence Spaces (Macmillan and Co., Ltd., London, 1950), pp. 8-9.
36. 0) invariant.
37. The notations Lemma 2(2). Lemma 4(1), etc., denote assertion (2) of Lemma 2, assertion (1) of Lemma 4, etc. The lemmas cited in the text are those stated in Sec. IV of the present paper, unless a statement to the contrary is made.
38. The symbol "const" appearing in a relation involving the unsummed indices m or n should be understood to be a constant which is independent of these indices.
39. Whenever inequalities (3.4a) or (3.4b) are invoked in an argument in the text, allusion is being made to the fact that the entries of the matrix in ℳ considered in the argument satisfy an inequality of one of these respective forms.
40. v) ∈Ω) are as specified in Definition 2.
41. i.
42. Recall that two algebras A,A′ over the same field F are isomorphic: if there exists a bijective map h:A→A′ such that h(ab) = h(a)h(b), h(a + b) = h(a) + h(b), and h(αa) = αh(a) for all a, b∈A and all α∈eF.
43. (i) (i≥1) is self-adjoint.
44. s)ℬ is hermitian.
45. See, for example, M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, (Academic, New York, 1972), pp. 270, 271.
46. n〉) ∈. ℋ.
47. This result is a slight variant of Proposition 10.6, p. 186 of V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, Dordrecht, 1981).