O. G. Smolyanov

Chernoff’s Theorem and the Construction of Semigroups

The classical Chernoff Theorem is a statement about the convergence of discrete-time approximations of semigroups using a family of contraction operators strongly differentiable at t = 0. By Chernoffs construction, different families may yield the same semigroup. Such families are called Chernoff equivalent, i.e., their derivatives at t = 0 coincide. This is outlined in Section 2. Moreover Chernoffs Theorem still holds if the families used as input of the construction are only asymptotically contractive in a sense made precise in the notion of properness (see Definition 2).

Feynman, Wigner, and Hamiltonian Structures Describing the Dynamics of Open Quantum Systems

Gough, J. E., S. Ratiu, T., G. Smolyanov, O. (2014). Feynman, Wigner, and Hamiltonian Structures Describing the Dynamics of Open Quantum Systems. Doklady Mathematics, 89 (1), 68-71.

Hamiltonian and Feynman aspects of secondary quantization

Reference EPFL-ARTICLE-189128doi:10.1134/S1064562413030174View record in Web of Science Record created on 2013-10-01, modified on 2017-09-27

Feynman Formulas Generated by Self-Adjoint Extensions of the Laplacian

A Feynman formula is a representation of the Schrodinger group or the Schrodinger semigroup by using limits of integrals over Cartesian powers of some space E . Here, H is a classical Hamiltonian and is a quantum mechanical Hamiltonian, which is a self-adjoint extension of a (pseudo)differential operator with the symbol H . If E is the phase or configuration space of a classical Hamiltonian system, we have a Feynman formula in the corresponding space. The multiple integrals in Feynman formulas approximate integrals with respect to some measures or pseudomeasures on the set of E -valued functions defined on a real interval (such functions are called trajectories in E ). A

Feynman and Feynman-Kac Formulas for Evolution Equations with Vladimirov Operator

A Feynman formula is a representation of a solution to the Cauchy problem for an evolution differential or pseudodifferential equation in terms of a limit of integrals over the Cartesian degrees of some space E . A Feynman‐Kac formula is a representation of a solution to the same problem in terms of a path integral. We assume that, on the path space, a countably additive measure or a pseudomeasure (of the type of the Feynman measure; see [2, 3]) is defined, and the multiple integrals in the Feynman formulas coincide with integrals of finite multiplicity approximating integrals with respect to this measure or pseudomeasure. In this paper, we obtain Feynman and Feynman‐ Kac formulas for solutions to the Cauchy problems for the heat equation with respect to complex-valued functions on the product of the real half-line and the p -adic line � � ; the role of the Laplace operator in these equations is played by the Vladimirov operator. Similar formulas can be obtained for Schrodinger-type equations and for the case of a multidimensional space over � � . Such equations may be useful in constructing mathematical models of processes on scales characterized by Planck length and time and phenomenological models in chemistry, continuum mechanics, and psychology (see [1, 4‐6] and the references therein).

Generalized Lévy Laplacians and Cesàro means

In this paper, we show that the composition of the generalized Levy Laplacian of order p (which is gener- ated by the corresponding Cesaro mean) and a certain linear transformation of the domain of the function to which it is applied is proportional to the generalized Levy Laplacian of smaller or larger order; this makes it possible to transform the generalized Levy Laplacian of any order into the classical Levy Laplacian. The con- struction is based on an expression of generalized Cesaro means in terms of ordinary ones, which is also suggested in this paper. By virtue of results of (5), this implies that all results on the classical Levy Laplacian have analogues for the generalized Levy Laplacian. In particular, we consider a relationship between the gen- eralized Levy Laplacian and quantum random pro- cesses.

Classical and nonclassical Levy Laplacians

A general method for defining and studying operators introduced by Paul Levy referred as classical Levy Laplacians and their modifications referred as nonclassical Laplacians, which makes its possible to extend results on Levy Laplacians to nonclassical Laplacians, has been discussed. An infinite family of Laplacians, whose elements are classical Laplacians and the non-classical Laplacians related to the Yang-Mills equations, has been defined and the relationship between these Laplacians and quantum random process has been described. A quantum random process is a function defined on a part of the real line and taking values in some space of operators. Volterra Laplacians have also been considered and analogies between them and Levy Laplacians have been found.

Randomized Hamiltonian Feynman integrals and Schrödinger-Itô stochastic equations

In this paper, we consider stochastic Schrodinger equations with two-dimensional white noise. Such equations are used to describe the evolution of an open quantum system undergoing a process of continuous measurement. Representations are obtained for solutions of such equations using a generalization to the stochastic case of the classical construction of Feynman path integrals over trajectories in the phase space.

Lévy--Laplace Operators in Functional Rigged Hilbert Spaces

New results on analytical properties of the Levy–Laplace operator (the Levy Laplacian) and the semigroups generated by this operator are established in this paper. To this end, we use the methods developed earlier for the Levy–Volterra operators (all necessary definitions are given below); the corresponding results are similar in both cases. The analogy arises because Hilbert space is replaced by a rigged Hilbert space; in fact, Hilbert space corresponds to a degenerate case (cf. [1, 2]): it is just in this case that the Levy Laplacian possesses “unusual” properties (cf. [3, 4]). Our approach develops the techniques proposed in [1, 5–8].

Hamiltonian Feynman integrals for equations with the Vladimirov operator

209 In his Nobel lecture [1], Feynman mentioned that he discovered its integral while developing in [2] an idea of Dirac [3], who conjectured that the integral kernel of an evolution operator transforming a wave function during a short time interval is similar to the complex exponential for a classical action. Feynman strengthened Dirac’s conjecture by suggesting that these integral kernels must be simply proportional to such exponentials; it is this assumption that made it possible to represent the solution of the Schrödinger equation by using the Feynman integral over an infi nite dimensional affine manifold consisting of func tions of time which take values in the configuration space of the initial classical Lagrangian system. Feyn man defined his integral as the limit of a sequence of effectively computable usual integrals over finite pow ers of the configuration space. The representation of the solution thus obtained is now known as the Feyn man formula in the configuration space, and the inte gral itself is called the Feynman integral (along paths in the configuration space). Since this integral involves an action functional in Lagrangian form, it can be called the Lagrangian Feynman integral.