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Infinite Dimensional Analysis Quantum Probability and Related Topics

STOCHASTIC PROPERTIES OF DEGENERATED QUANTUM SYSTEMS

Vsevolod Zhanovich Sakbaev

Published Mar 1, 2010• 17 citations
Publisher

Abstract

We study Schrödinger equation with degenerated symmetric but not self-adjoint Hamiltonian. The above properties of the quantum Hamiltonian arise in the description of the asymptotic behavior of the regularizing self-adjoint Hamiltonians sequence. A quantum dynamical semigroup corresponding to degenerated Hamiltonian is defined by means of the passage to the limit for the sequence of the regularizing dynamical semigroups. These semigroups are generated by the regularizing self-adjoint Hamiltonians. The necessary and sufficient conditions are obtained for the convergence of the regularizing semigroups sequence. The description of the divergent sequence of semigroups requires the extension of the stochastic process concept. We extend the stochastic process concept onto the family of measurable functions defined on the space endowed with finite additive measure. The above extension makes it possible to describe the structure of the partial limits set of the regularizing semigroups sequence.

DOI
10.1142/s0219025710003948

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We study Schrödinger equation with degenerated symmetric but not self-adjoint Hamiltonian. The above properties of the quantum Hamiltonian arise in the description of the asymptotic behavior of the regularizing self-adjoint Hamiltonians sequence. A quantum dynamical semigroup corresponding to degenerated Hamiltonian is defined by means of the passage to the limit for the sequence of the regularizing dynamical semigroups. These semigroups are generated by the regularizing self-adjoint Hamiltonians. The necessary and sufficient conditions are obtained for the convergence of the regularizing semigroups sequence. The description of the divergent sequence of semigroups requires the extension of the stochastic process concept. We extend the stochastic process concept onto the family of measurable functions defined on the space endowed with finite additive measure. The above extension makes it possible to describe the structure of the partial limits set of the regularizing semigroups sequence.
We study the dynamics of quantum system with degenerated Hamiltonian. To this end we consider the approximating sequence of regularized Hamiltonians and corresponding sequence of dynamical semigroups acting in the space of quantum states. The limit points set of the sequence of regularized semigroups is obtained as the result of averaging by finitely additive measure on the set of regularizing parameters. We establish that the family of averaging dynamical maps does not possess the semigroup property and the injectivity property. We define the functionals on the space of maps of the time interval into the quantum states space such that the maximum points of this functionals coincide with the trajectories of the family of averaging dynamical maps.
The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C *-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C *-algebras are investigated and it is shown that pure states could evolve into mixed states.
The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.
We consider the problem of constructing quantum dynamics for symmetric Hamiltonian operators that have no self-adjoint extensions. For an earlier studied model, it was found that an elliptic self-adjoint regularization of a symmetric Hamiltonian operator allows one to construct quantum dynamics for vector states on certain C *-subalgebras of the algebra of bounded operators in a Hilbert space. In the present study, we prove that one can extend the dynamics to arbitrary states on these C *-subalgebras while preserving the continuity and convexity. We show that the obtained extension of the dynamics of the set of states on C *-subalgebras is the limit of a sequence of regularized dynamics under removal of the elliptic regularization. We also analyze the properties of the limit dynamics of the set of states on the C *-subalgebras.
Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.
We investigate a model that associates random walks in a finite-dimensional Euclidean coordinate space of a classical system with random quantum walks, i.e. random transformations of the set of states of a quantum system arising from quantization of a classical system. As is known, the convolution semigroup of Gaussian measures on a coordinate space admits a representation by a semigroup of self-adjoint contractions in the space of square-integrable functions described by the heat equation. We obtain a representation of the convolution semigroup of Gaussian measures on a coordinate space by a quantum dynamic semigroup in the space of nuclear operators. We give a description of the quantum dynamic semigroup by solutions of the Cauchy problem for a degenerate diffusion equation. We establish the generalized convergence in distribution of a sequence of quantum random walks to an operator-valued random process with values in the Abelian algebra of shift operators by a vector with a normal distribution.