Resonances and the extension of dynamics to rigged Hilbert space

Computers and Mathematics with Applications

Volumn 34, Issue 5-6, 1997, Pages 399-425

  Antoniou, I ^a   Dmitrieva, L ^a   Kuperin, Yu ^b   Melnikov, Yu ^a  

^a INTERNATIONAL SOLVAY INSTITUTES   (Belgium)

^b ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

Extended resolvent; Generalized spectral decomposition; Resonance states; Rigged Hilbert space

Indexed keywords

RESONANCE; SPECTRUM ANALYSIS; VECTORS;

GENERALIZED SPECTRAL DECOMPOSITION; RIGGED HILBERT SPACE;

MATHEMATICAL OPERATORS;

EID: 0031224107     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0898-1221(97)00148-x     Document Type: Article

References (49)

1. E. Brändas and N. Elander, Editors, Resonances, Springer Lecture Notes in Physics, Vol 325, Springer-Verlag, Berlin, (1989).
2. P. Lax and R. Phillips, Scattering Theory, Academic Press, New York, (1967).
3. B.S. Pavlov, A factorization of the scattering matrix and series structure of its roots, Math. USSR Izv. 7, 323-350 (1973).
4. B.S. Pavlov, Lax-Phillips Theory and Nonphysical Sheet for the Band Spectrum, Preprint No 36, Inst. Mittag-Leffer, Djursholm, (1991).
5. B.S. Pavlov and S.I. Fedorov, A shift group and harmonic analysis on a Riemann surface of genus one, Leningrad Math. J. 1, 132-168 (1990).
6. I. Prigogine, Nonequilibrium Statistical Mechanics, Wiley, New York, (1962).
7. I. Prigogine, From Being to Becoming, Freeman, New York, (1980).
8. I. Prigogine, Quantum mechanics of dissipative systems, Physics Reports 219, 93-120 (1992).
9. T. Petrosky and I. Prigogine, Alternative formulation of classical and quantum dynamics for non-integrable systems, Physica A 175, 146-209 (1991).
10. I. Antoniou and S. Tasaki, Generalized spectral decomposition of mixing dynamical systems, Int. J. of Quantum Chemistry 46, 425-474 (1993).
11. I. Prigogine, Why irreversibility? The formulation of classical and quantum mechanics for nonintegrable systems, Int. J. of Bifurcation and Chaos 5, 3-16 (1995).
12. T. Petrosky and I. Prigogine, The extension of classical dynamics for unstable Hamiltonian systems, (this issue).
13. T. Petrosky, I. Prigogine and S. Tasaki, Quantum theory of nonintegrable systems, Physica A 173, 175-242 (1991).
14. I. Antoniou and I. Prigogine, Intrinsic irreversibility and integrability of dynamics, Physica A 192, 443-464 (1993).
15. H. Hasegawa and D. Driebe, Transport as a dynamical property of a simple map, Phys. Lett. A 168, 18-24 (1992).
16. H. Hasegawa and D. Driebe, Spectral determination and physical conditions for a class of chaotic piece-wise-linear maps, Phys. Lett. A 176, 193-201 (1993).
17. I. Antoniou and S, Tasaki, Spectral decomposition of the Rényi map, J. Phys. A 26, 73-94 (1993).
18. I. Antoniou and S. Tasaki, Generalized spectral decomposition of the β-adic baker's transformation and intrinsic irreversibility, J. Physica. A 190, 303-329 (1992).
19. O. Bandtlow, I. Antoniou and Z. Suchanecki, Resonances of chaotic systems and Riesz-Predholm operators in rigged Hilbert space, (this issue).
20. I. Gelfand and N. Vilenkin, Generalized Functions, Vol. 4, Academic Press, New York, (1964).
21. A. Böhm, The Rigged Hilbert Space and Quantum Mechanics, Springer Lecture Notes in Physics, Vol. 78, Springer-Verlag, Berlin, (1978).
22. A. Böhm and M. Gadella, Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes in Physics, Vol. 348, Springer-Verlag, Berlin, (1989).
23. B.S. Pavlov, Extension theory and exactly solvable models, Russ. Math. Survey 42, 127-168 (1987).
24. Yu.A. Kuperin and S.P. Merkuriev, Selfadjoint extensions and scattering theory for several-body systems, Amer. Math. Soc. Transl. 150, 141-176 (1992).
25. Yu.A. Kuperin et al., Algebraic theory of extensions in the quantum few-body problem for particles with internal structure, Sov. J. Nucl. Phys. 48, 224-231 (1988).
26. Yu.A. Kuperin and Yu.B. Melnikov, Two-body resonance scattering and annihilation of composite charged particles, J. Math. Phys. 33, 2795-2807 (1992).
27. A. Lasota and M. Mackey, Probabilistic Properties of Deterministic Systems, Cambridge University Press, Cambridge, U.K., (1985).
28. L.D. Faddeev, Decomposition over the eigenfunctions of the Laplace operator on the fundamental domain of a discrete group on Lobachevsky plane, Proc. Moscow Math. Soc. 17, 323-350 (1967).
29. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8, 477-493 (1957).
30. Ya. Sinai, Dynamical Systems II, Springer-Verlag, Berlin, (1989).
31. I. Cornfeld, S. Fomin and Ya. Sinai, Ergodic Theory, Springer-Verlag, Berlin, (1982).
32. I. Gradshtein and I. Ryzhik, Tables of Integrals, Sums, Series and Products, Fizmatgiz, Moscow, (1963).
33. K. Friedrichs, On the perturbation of continuous spectra, Commun. Pure Appl. Math. 1, 361-406 (1948).
34. P.A.M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, London, (1958).
35. E.C.G. Sudarshan, C. Chiu and V. Gorini, Decaying states as complex energy eigenvectors in generalized quantum mechanics, Phys. Rev. D18, 2914-2929 (1978).
36. G. Parravicini, V, Gorini and E.G.G. Sudarshan, Resonances, scattering theory, and rigged Hilbert spaces, J. Math. Phys. 21, 2208-2226 (1980).
37. L. Horwitz and J.P. Marchand, The decay-scattering systems, Rocky Mnt. J. Math. 1, 225-253 (1971).
38. T. Bailey and W. Schieve, Complex energy eigenstates in quantum decay models, Nuovo Cimento A 47, 221 (1978).
39. T. Petrosky, I. Prigogine and S. Tasaki, Quantum theory of nonintegrable systems, Physica A 173, 175-242 (1991).
40. B.S. Pavlov, The nonphysical sheet for the Friedrichs model, St. Petersburg Math. J. 4, 1245-1256 (1993).
41. S.G. Davison and J.D. Levine, Surface States, Academic Press, London, (1970).
42. E.G. Titchmarsh, Expansions in Eigenfunctions Connected with Differential Operators of Second Order, Volume 2, IIL, Moscow, (1961).
43. N.E. Firsova, The Riemann surface of quasimomentum and the scattering theory for the perturbed Hill operator, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 51, 183-196 (1975).
44. V.A. Marchenko and I.V. Ostrovskii, The characteristics of the Hill operator spectrum, Mat. Sb. (English transl, in Soviet Mathematics: Sbornik) 97, 540-606 (1975).
45. B.A. Dubrovin, V.B. Matveev and S.P. Novikov, Non-linear equations of the Korteweg-de Vries type, finite gap operators and the Abelian manifolds, Uspekhi Mat. Nauk (English transl, in Russian Math. Survey) 31, 55-136 (1976).
46. I. Sch'nol, Mat. Sb. (English transl, in Soviet Mathematics: Sbornik) 42, 273-286 (1957).
47. G.H. Hardy, Divergent Series, 2nd Edition, Clarendon Press, (1956).
48. I. Antoniou and A. Smirnov, An Exactly Solvable Model for Classic Chaotic Scattering (to appear).
49. I. Antoniou and Z. Suchanecki, Quantum systems with diagonal singularity; Their logic and intrinsic irreversibility, (this issue).