Geodesic
On the dynamics of the quantum states set for a system with degenerated Hamiltonian
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2011), pp. 200-220.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the sequence of regularizing Cauchy problem as the elliptic regularization of Cauchy problem for Schrodinger equation with discontinuous and degenerated coefficients. The necessary and sufficient condition of the convergence of the regularizing dynamical semigroups sequence are presented. If the convergence is impossible then divergent sequence of the regularizing quantum states is considered as the stochastic process on the measurable space of regularizing parameter endowing with finite additive measure. The expectation of this stochastic process defines the averaging trajectory in the space of quantum states. It was obtained the condition on the finite additive measure such, that averaging trajectory can be defined by its values in two instants with the help of solving the variational problems.
Keywords: finite additive measure, stochastic process, quantum state, dynamical semigroup, observability. 
@article{VSGTU_2011_2_a25,
     author = {V. Zh. Sakbaev},
     title = {On the dynamics of the quantum states set for a system with degenerated {Hamiltonian}},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {200--220},
     publisher = {mathdoc},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2011_2_a25/}
}
TY  - JOUR
AU  - V. Zh. Sakbaev
TI  - On the dynamics of the quantum states set for a system with degenerated Hamiltonian
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2011
SP  - 200
EP  - 220
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2011_2_a25/
LA  - ru
ID  - VSGTU_2011_2_a25
ER  - 
%0 Journal Article
%A V. Zh. Sakbaev
%T On the dynamics of the quantum states set for a system with degenerated Hamiltonian
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2011
%P 200-220
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2011_2_a25/
%G ru
%F VSGTU_2011_2_a25
V. Zh. Sakbaev. On the dynamics of the quantum states set for a system with degenerated Hamiltonian. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2011), pp. 200-220. http://geodesic.mathdoc.fr/item/VSGTU_2011_2_a25/

[1] Pavlotsky I. P., Strianese M., “Irreversibility in classical mechanics as a consequence of Poincaré group”, Inter. J. of Mod. Phys. B., 10:21 (1996), 2675–2685 | DOI | MR

[2] Sakbaev V. Zh., “Spectral aspects of regularization of the Cauchy problem for a degenerate equation”, Proc. Steklov Inst. Math., 261 (2008), 253–261 | DOI | MR | Zbl

[3] Kozlov V. V., “Dynamics of systems with nonintegrable constraints”, Vestn. Mosk. Un-ta. Ser. 1. Matematika. Mekhanika, 1987, no. 5, 76–83 | Zbl

[4] Accardi L., Lu Y. G., Volovich I. V., Quantum theory and its stochastic limit, Springer-Verlag, Berlin, Heidelberg, New York, 2001, 473 pp. | MR

[5] Bogoliubov N. N., On some statistical methods in mathematical physics, Izd-vo AN USSR, Kiev, 1945, 139 pp.

[6] Bratteli O., Robinson D. W., Operator Algebras and Quantum Statistical Mechanics, v. I, $C^*$- and $W^*$-Algebras Symmetry Groups Decomposition of States, Second Edition, Springer-Verlag, Berlin, Heidelberg, New York, 2003, 505 pp. ; Bratelli U., Robinson D., Operatornye algebry i kvantovaya statisticheskaya mekhanika, Mir, M., 1982, 512 pp. | MR | MR

[7] Sakbaev V. Zh., “Set-valued mappings specified by regularization of the Schrödinger equation with degeneration”, Comput. Math. Math. Phys., 46:4 (2006), 651–665 | DOI | MR | Zbl

[8] Dunford N., Schwartz J. T., Linear Operators, v. 1, General Theory, John Willey and Sons, New York, 1988, 858 pp. ; Danford N., Shvarts D., Teoriya operatorov, v. 1, Obschaya teoriya, URSS, M., 2010, 896 pp. | MR

[9] Yosida K., Hewitt E., “Finitely additive measures”, Trans. Am. Math. Soc., 72 (1952), 46–66 | DOI | MR | Zbl

[10] Emch G. G., Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Physics Astronomical Monograph, John Willey and Sons, New York, 1972, 350 pp. ; Emkh Zh., Algebraicheskie metody v statisticheskoi mekhanike i kvantovoi teorii polya, Mir, M., 1976, 423 pp. | Zbl

[11] Glauber R. J., “Optical coherence and photon statistics”, Quantum Optics and Electronics, eds. C. deWitt, A. Blandin, C. Cohen-Tannoudji, Gordon and Breach, New York, 1965, 65–185 ; Glauber R., “Opticheskaya kogerentnost i statistika fotonov”, Kvantovaya optika i kvantovaya radiofizika, Mir, M., 1966, 93–279

[12] Carey A. L., Sukoche F. A., “Dixmier traces and some applications in non-commutative geometry”, Russian Math. Surveys, 61:6 (2006), 1039–1099 | DOI | DOI | MR | Zbl

[13] Srinivas M. D., “Collapse postulate for observables with continuous spectra”, Commun. Math. Phys., 71:2 (1980), 131–158 | DOI | MR | Zbl

[14] Sakbaev V. Zh., “Averaging of quantum dynamical semigroups”, Theoret. and Math. Phys., 164:3 (2010), 1215–1221 | DOI | DOI | MR | Zbl

[15] Varadarajan V. S., “Measures on topological spaces”, Matem. Sb., 55(97):1 (1961), 35–100

[16] Amosov G C., Sakbaev V. Zh., “Stochastic properties of the dynamics of quantum systems”, Vestn. Sam. Gos. Un-ta. Estestvennonauchn. Ser., 2008, no. 8/1(67), 479–494

[17] Sakbaev V. Zh., “On the dynamics of a degenerate quantum system in the space of integrable functions on a finitely additive measure”, Trudy MFTI, 1:4 (2009), 126–147