Feynman formulas for nonlinear evolution equations

Transformations of measures, generalized measures, and functions generated by evolution differential equations on a Hilbert space E are studied. In particular, by using Feynman formulas, a procedure for averaging nonlinear random flows is described and an analogue of the law of large number for such flows is established (see [1, 2]).     

Self-adjoint approximations of the degenerate Schrödinger operator

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.     

On the properties of ambiguity and irreversibility of dynamical maps of the initial data space of Cauchy problem

We investigate the Cauchy problems for evolutionary differential equations which possess the following properties: the solutions of considered problems admit the arising on the bounded time interval of singularities such that destroying of existence or uniqueness of solution and unbounded growth of norm of solution in Cauchy problem Banach space. The opportunity of continuation (probably of many-valued continuation) of the dynamical maps of the space of initial data by the procedure of passage to the limit for the sequences of approximating Cauchy problems is studied.     

On analogs of spectral decomposition of a quantum state

The set of quantum states in a Hilbert space is considered. The structure of the set of extreme points of the set of states is investigated and an arbitrary state is represented as the Pettis integral over a finitely additive measure on the set of vector states, which is a generalization of the spectral decomposition of a normal state.     

On the law of large numbers for compositions of independent random semigroups

We study random linear operators in Banach spaces and random one-parameter semigroups of such operators. For compositions of independent random semigroups of linear operators in the Hilbert space we obtain sufficient conditions for fulfilment of the law of large numbers and give examples of its violation.     

On the variational description of the trajectories of averaging quantum dynamical maps

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.