The spectrum of the Liouville-von Neumann operator in the Hilbert-Schmidt space

Journal of Mathematical Physics

Volumn 40, Issue 8, 1999, Pages 4106-4118

  Antoniou, I ^a,b   Shkarin, S A ^a,b   Suchanecki, Z ^a,c  

^a INTERNATIONAL SOLVAY INSTITUTES   (Belgium)

^b MOSCOW STATE UNIVERSITY   (Russian Federation)

^c FREE UNIVERSITY OF BRUSSELS   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033241174     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532948     Document Type: Article

References (25)

1. I. Prigogine, Non-Equilibrium Statistical Mechanics (Wiley, New York, 1962).
2. E. E. Prugovecki, Quantum Mechanics in Hilbert Space (Academic, New York, 1981).
3. M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. I: Functional Analysis (Academic, New York, 1980).
4. H. Spohn, "The spectrum of the Liouville-von Neumann operator," J. Math. Phys. 17, 57-60 (1976).
5. L. Bos and B. S. Pavlov, "Absolute Continuity of Convolution of Singular Measures and Localization Problems," in Chapman and Hill/CRC Research Notes in Mathematics Vol. 399: "Generalized Functions, Operator Theory and Dynamical Systems," edited by I. Antoniou and G. Lumer (Chapman and Hill, London, 1999).
6. I. Antoniou, M. Gadella, and Z. Suchanecki, "General properties of the Liouville operator," Int. J. Theor. Phys. 37, 1641-1654 (1998).
7. N. Wiener and A. Wintner, "Fourier-Stieltjes transforms and singular infinite convolution," Am. J. Math. 60, 513-522 (1938).
8. C. Karanikas and S. Koumandos, "Continuous singular measures with absolutely continuous convolution squares on locally compact groups," Ill. J. Math. 35, 490-495 (1991).
9. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); "Dissipative processes in quantum theory," Phys. Rep. 219, 93-120 (1992), and personal communication.
10. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); "Dissipative processes in quantum theory," Phys. Rep. 219, 93-120 (1992), and personal communication.
11. M. Courbage, "On necessary and sufficient conditions for the existence of time and entropy operators in quantum mechanics," Lett. Math. Phys. 4, 425-432 (1980).
12. B. Misra, I. Prigogine, and M. Courbage, "Liapunov variable: entropy and measurements in quantum mechanics," Proc. Natl. Acad. Sci. USA 76, 4768-4772 (1979).
13. W. Pauli, General Principles of Quantum Mechanics (Springer-Verlag, New York, 1980).
14. H. Weidmann, Linear Operators in Hilbert Space (Springer Verlag, New York, 1980).
15. J. Bellisard, "Schroedinger Operators with Almost Periodic Potential: an Overview," Lect. Notes Phys. 153, 356-363 (1982).
16. H. Steinhaus, "A new property of the Cantor set," (in Polish) Wektor 7, 1-3 (1917); English translation: H. Steinhaus, Selected Papers (PWN, Warsaw, 1985).
17. H. Steinhaus, "A new property of the Cantor set," (in Polish) Wektor 7, 1-3 (1917); English translation: H. Steinhaus, Selected Papers (PWN, Warsaw, 1985).
18. P. Billingsley, Probability and Measure (Wiley, New York, 1985).
19. S. Tasaki, Z. Suchanecki, and I. Antoniou, "Ergodic properties of piecewise linear maps on fractal repellers," Phys. Lett. A 179, 103-110 (1993).
20. N. Levenberg, Jr., G. J. Martin, A. L. Shields, and S. Zdravkovska, "Factorizations of Lebesgue measure via convolution," Proc. Am. Math. Soc. 104, 419-430 (1988).
21. P. Exner, Open Quantum Systems and Feynman Integrals (Reidel, The Netherlands, 1985).
22. P. Erdös, "On a family of symmetric Bernoulli convolutions," Am. J. Math. 61, 974-976 (1939).
23. A. Zygmund, Trigonometric Series (Cambridge, U.P., Cambridge, 1968), Vols. I, II.
24. V. S. Varadarajan, "Measures on Topological Spaces," Mat. Sb. 55, 35-103 (1961).
25. I. Antoniou and Y. Melnikov, "Non-Factorizable Extensions of the Liouville Operator to Rigged Hilbert-Schmidt Space," in preparation.