Averaging of quantum dynamical semigroups
- 46 Downloads
- 5 Citations
Abstract
In the framework of the elliptic regularization method, the Cauchy problem for the Schrödinger equation with discontinuous degenerating coefficients is associated with a sequence of regularized Cauchy problems and the corresponding regularized dynamical semigroups. We study a divergent sequence of quantum dynamical semigroups as a random process with values in the space of quantum states defined on a measurable space of regularization parameters with a finitely additive measure. The mathematical expectation of the considered processes determined by the Pettis integral defines a family of averaged dynamical transformations. We investigate the semigroup property and the injectivity and surjectivity of the averaged transformations. We establish the possibility of defining the process by its mathematical expectation at two different instants and propose a procedure for approximating an unknown initial state by solutions of a finite set of variational problems on compact sets.
Keywords
stochastic process finitely additive measure quantum state dynamical semigroup observabilityPreview
Unable to display preview. Download preview PDF.
References
- 1.V. V. Kozlov, Thermal Equilibrium in the Sense of Gibbs and Poincaré [in Russian], RKhD, Moscow (2002).MATHGoogle Scholar
- 2.L. Accardi, Y. G. Lu, and I. V. Volovich, Quantum Theory and its Stochastic Limit, Springer, Berlin (2001).Google Scholar
- 3.N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics [in Russian], Acad. Sci. Ukrainian SSR, Kiev (1945).Google Scholar
- 4.N. N. Bogoliubov, “On some ergodic properties of continuous transformation groups [in Russian],” in: Selected Works, Vol. 1, Naukova Dumka, Kiev (1969), pp. 561–569.Google Scholar
- 5.V. Zh. Sakbaev, Proc. Steklov Inst. Math., 261, 253–261 (2008).CrossRefMathSciNetGoogle Scholar
- 6.O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Texts Monogr. Phys., Vol. 1), Springer, New York (1979).MATHGoogle Scholar
- 7.V. Zh. Sakbaev, Comput. Math. Math. Phys., 46, 651–665 (2006).CrossRefMathSciNetGoogle Scholar
- 8.V. S. Varadarajan, in: Fourteen Papers on Logic, Algebra, Complex Variables, and Topology (Amer. Math. Soc., Transl., II. Ser., Vol. 48), Amer. Math. Soc., Providence, R. I. (1965), pp. 161–228.Google Scholar
- 9.G. G. Amosov and V. Zh. Sakbaev, Trudy Sam. Gos. Univ, 8, No. 1(67), 479–494 (2008).Google Scholar
- 10.R. E. Edwards, Functional Analysis: Theory and Applications, Holt Rinehart and Winston, New York (1965).MATHGoogle Scholar
- 11.V. Zh. Sakbaev, Trudy MFTI, 1, No. 4, 126–147 (2009).Google Scholar