O. G. Smolyanov's research while affiliated with Lomonosov Moscow State University and other places
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Publications (192)
An Erratum to this paper has been published: https://doi.org/10.1134/S1064562422070018
Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non‐compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to gen...
The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large n...
Вводятся меры Вигнера для бесконечномерных открытых квантовых систем; важными примерами таких систем являются объекты квантовой теории управления. Кроме того, предлагается аксиоматическое определение когерентной квантовой обратной связи.
The Lebesgue-Feynman measure on a linear space E is a generalized measure on E which is defined as a linear functional μ on some linear space \(F(E,{\mathbb {C}})\) of functions \(E\to \mathbb {C}\) which is invariant with respect to the shift on any vector h ∈ E:
$$ \mu (f)=\mu (\textbf{S}_{\textit{h}}f) \ \forall \ f\in F(E,{\mathbb{C}}), \forall...
The Lebesgue-Feynman measure on a linear space E is a generalized measure on E which is defined as a linear functional μ on some linear space F(E,ℂ) of functions E → ℂ which is invariant with respect to the shift on any vector. There exists two different approach to definition of Lebesgue-Feynman measure. The first definition is given by Feynman fo...
Обсуждается проблема происхождения квантовых аномалий, для которой сравнительно недавно в монографиях были предложены противоречащие друг другу решения. Предлагаемый в настоящей работе подход является новым; он использует дифференциальные свойства обобщенных мер. Для определения таких мер вводится пространство пробных функций, определенных на локал...
Manifolds of bounded geometry form a class of Riemannian manifolds which includes all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a generally non-compact manifold of bounded geometry. A construction of Chernof...
Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particular, the connection between two definitions of the PDO is considered: one given in terms of the Hamiltonian Feynman measure, and another introduced in this work in terms of the Kolmogorov integral.
В предлагаемой работе исследуются последовательности композиций независимых одинаково распределенных случайных однопараметрических полугрупп линейных преобразований гильбертова пространства и асимптотические свойства распределений таких композиций при стремлении их числа к бесконечности. Для изучения математического ожидания таких композиций примен...
We study sequences of compositions of independent identically distributed random one-parameter semigroups of linear transformations of a Hilbert space and the asymptotic properties of the distributions of such compositions when the number of terms in the composition tends to infinity. To study the expectation of such compositions, we apply the Feyn...
Properties of infinite-dimensional pseudodifferential operators (PDO) are discussed. In particular, the connection between two definitions of PDO is considered: one given in terms of the Hamiltonian Feynman measure and the other introduced in this work in terms of the Kolmogorov integral.
Randomized Hamiltonian mechanics is the Hamiltonian mechanics which is determined by a time-dependent random Hamiltonian function. Corresponding Hamiltonian system is called random Hamiltonian system. The Feynman formulas for the random Hamiltonian systems are obtained. This Feynman formulas describe the solutions of Hamilton equation whose Hamilto...
Hamiltonian mechanics determined by time-dependent random Hamiltonian functions is called randomized Hamiltonian mechanics. The corresponding Hamiltonian systems are called random. Feynman formulas for random Hamiltonian systems are obtained. It is shown that these formulas describe solutions of a Hamiltonian equation whose Hamiltonian is the mean...
Structures and objects used in Hamiltonian secondary quantization are discussed. By the secondary quantization of a Hamiltonian system ℋ, we mean the Schrödinger quantization of another Hamiltonian system ℋ1 for which the Hamiltonian equation is the Schrödinger one obtained by the quantization of the original Hamiltonian system ℋ. The phase space o...
Spaces of test functions and spaces of distributions (generalized measures) on infinite-dimensional spaces are constructed, which, in the finite-dimensional case, coincide with classical spaces \(\mathscr{D}\) and \(\mathscr{D}'\). These distribution spaces contain generalized Feynman measures (but do not contain a generalized Lebesgue measure, whi...
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]).
In this chapter we consider projective limits (in particular, products) of families of topological vector spaces, inductive limits (in particular, topological direct sums) of families of locally convex spaces, including strict inductive limits and inductive limits with compact embeddings, tensor products of locally convex spaces, and nuclear spaces...
The concept of differentiable mapping from a topological vector space to a topological vector space was worked out relatively recently. In the mid 60s of the XX century the number of existing definitions of differentiability of mappings of topological vector spaces was very large and was comparable (if not greater) with the number of papers devoted...
A powerful method of proving a great number of results in the theory of locally convex spaces employs passage from some subsets in such spaces to their polars, which are subsets of the dual spaces. Moreover, in place of properties of the original sets certain properties of their polars are studied and then one returns back, more precisely, to the p...
In this chapter we present basic concepts and examples connected with topological vector spaces.
In this chapter we give a brief account of measure theory on linear spaces. We assume some acquaintance with basics of the Lebesgue theory of measure and integral (see, for example, Chapters 2 and 3 in [72]). We present the fundamental facts of the theory of Gaussian measures, discuss weak convergence of measures and the Fourier transform of measur...
The Hudson–Parthasarathy equation and the Itô–Schrödinger equation (known also as the Belavkin equation) describe a Markov approximation of the dynamics of open quantum systems. The former is a stochastic version of the classical Heisenberg equation, while the latter is a stochastic version of the classical Schrödinger equation (but this analogy is...
This book gives a compact exposition of the fundamentals of the theory of locally convex topological vector spaces. Furthermore it contains a survey of the most important results of a more subtle nature, which cannot be regarded as basic, but knowledge which is useful for understanding applications. Finally, the book explores some of such applicati...
We call a Lebesgue-Feynman measure (LFM) any generalized measure (distribution in the sense of Sobolev and Schwartz) on a locally convex topological vector space E which is translation invariant. In the present paper, we investigate transformations of the LFM generated by transformations of the domain and also discuss the connections of these trans...
Вводится и исследуется вероятностная интерполяция различных методов квантования. При этом используется развиваемый в настоящей работе метод нахождения математического ожидания случайных неограниченных операторов в гильбертовом пространстве, основанный на усреднении - с помощью формул Фейнмана - порождаемых ими случайных однопараметрических полугруп...
Representations of Schrödinger semigroups and groups by Feynman iterations are studied. The compactness, rather than convergence, of the sequence of Feynman iterations is considered. Approximations of solutions of the Cauchy problem for the Schrödinger equation by Feynman iterations are investigated. The Cauchy problem for the Schrödinger equation...
In the paper, an asymptotic expansion of path integrals of functionals having exponential form with polynomials in the exponent is constructed. The definition of the path integral in the sense of analytic continuation is considered.
Applications of transformations of Feynman path integrals and Feynman pseudomeasures to explain arising quantum anomalies are considered. A contradiction in the literature is also explained.
Representations of regularized determinants of elements of one-parameter operator semigroups whose generators are second-order elliptic differential operators by Lagrangian functional integrals are obtained. Such semigroups describe solutions of inverse Kolmogorov equations for diffusion processes. For self-adjoint elliptic operators, these semigro...
Representations of solutions of Lindblad equations by randomized Feynman integrals over trajectories are obtained by averaging similar representations for solutions of stochastic Schrödinger equations (Schrödinger–Belavkin equations). An approach based on the application of Chernoff’s theorem is applied. First, (randomized) Feynman formulas approxi...
We introduce and study probabilistic interpolations of various quantization methods. To do this, we develop a method for finding the expectations of unbounded random operators on a Hilbert space by averaging (with the help of Feynman formulae) the random one-parameter semigroups generated by these operators (the usual method for finding the expecta...
Extensions of locally convex topological spaces are considered such that finite cylindrical measures which are not countably additive on their initial domains turn out to be countably additive on the extensions. Extensions of certain transformations of the initial spaces with respect to which the initial measures are invariant or quasi-invariant to...
We propose a method for finding the mathematical expectation of random unbounded operators in a Hilbert space. The method is based on averaging random one-parameter semigroups by means of the Feynman-Chernoff formula. We also consider an application of this method to the description of various operations that assign quantum Hamiltonians to the clas...
In this note we describe algorithms for obtaining formulae for transformations of measures on infinite dimensional topological vector spaces or manifolds, generated by transformations of the domains of the measures and by transformations of the range.
In this paper, we obtain a representation for the regularized traces (1) of evenorder differential opera� tors in terms of Hamiltonian functional integrals and the corresponding Feynman formulas. We also present an example showing how such a representation can be applied to obtain formulas not containing functional integrals. A Feynman formula (2)...
This note is devoted to representation of some evolution semigroups. The
semigroups are generated by pseudo-differential operators, which are obtained
by different (parametrized by a number $\tau$) procedures of quantization from
a certain class of functions (or symbols) defined on the phase space. This
class contains functions which are second ord...
We determine the rate with which finitely multiple approximations in the Feynman formula converge to the exact expression for the equilibrium density operator of a harmonic oscillator in the linear τ-quantization. We obtain an explicit analytic expression for a finitely multiple approximation of the equilibrium density operator and the related Wign...
This paper considers Hamiltonian structures related to quantum mechanics. We do not discuss structures used in the quantization of classical Hamil� tonian or Lagrangian systems (which was first done by Heisenberg and Feynman); on the contrary, we are mainly interested in structures which make it possible to consider quantum systems as classical (al...
We construct an infinite-dimensional linear space [Formula: see text] of vector-valued distributions (generalized functions), or sequences, f*(x)=(f n (x)) finite from the left (i.e. f n (x)=0 for n<n 0 (f*)) whose components f n (x) belong to the linear span [Formula: see text] generated by the distributions δ (m-1) (x-c k ), P((x-c k ) -m ), x m-...
We consider the behaviour of a proto-Solar (or proto-Saturnian) nebula in the presence of a gravitating point mass whose gravitaional atraction dominates its evolution. We assume that a proportion of the quantum particles which constitute the nebula are in an appropriate atomic elliptic state and investigate the related diffusion from Nelson’s theo...
A method for obtaining the infinite system of Bogolyubov equations without passing to the thermodynamical limit is applied to derive quantum versions of Bogolyubov-type equations. An infinite-dimensional analogue of the quantum mechanics representation is obtained. The Wigner measure generated by a density operator s defined as the image of the E-c...
Feynman and Feynman-Kac formulas are obtained for a heat-type equation with respect to complex-valued functions defined on the product of a half-line or an interval and an infinite-dimensional space over the field of adic numbers. Facts related to analysis of complex-valued functions of adic arguments and the terminology of the theory of topologica...
Researchers conducted a study to demonstrate two types of surface measures on trajectories in Riemannian submanifolds of Euclidean spaces. It was precisely two types of surface measures existed on trajectories in Riemannian manifolds under natural assumptions. The construction of a surface measure was regarded as a new method for constructing diffu...
In this paper, we obtain Feynman formulas for solutions to equations describing the diffusion of particles with mass depending on the particle position and to Schrödinger-type equations describing the evolution of quantum particles with similar properties. Such particles (to be more precise, quasi-particles) arise in, e.g., models of semiconductors...
In recent paper [5], Gough and James introduces a model for quantum feedback networks. For such a model they took a family of boson fields propagating along edges of a graph and assumed that each vertex is assigned some open quantum system. They derived a formula making it possible to remove edges in the limit of zero time delay (and described boun...
The Hamiltonian Feynman formula is proved for a certain class of Feller semigroups. The generators of these semigroups are PDOs whose symbols are continuous functions of a variable and are smooth and negative with respect to a variable for each fixed variable and bounded with respect to a variable for each fixed variable. A strongly continuous cont...
Feynman formulas for Schr̈odinger groups and semigroups generated by some Hamiltonians describing the one-dimensional quantum dynamics and the diffusion of a quiasi-particle with variable mass in an electromagnetic field has been reported. A Feynman-Kacformula is a representation of such a solution in terms of an integral with respect to a measure...
35A8
35C15 Integral representations of solutions of PDE
35C20 Asymptotic expansions
35R15 Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) (See also 46Gxx, 58D25)
35R60 Partial differential equations with randomness (See also 60H15)
46L45 Decomposition theory for C*-algebras
4...
An infinite system of equations with respect to time-dependent finite-dimensional probability distributions equivalent to the Liouville equation with respect to functions of a real argument taking values in the space of probability measures on the phase space of an infinite-dimensional Hamiltonian system has been reported. The Liouville infinite-di...
The mathematical theory of quantum feedback networks has recently been developed by Gough and James \cite{QFN1} for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, that their feedback reduction formula for the coefficients of the closed-loop quantum stochastic differential equation can be formu...
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...
The difference between models suggested in this paper and Kolmogorov models is that the former allow transformations of the probability space depending on the results of preceding measurements or on the type of the experiment (the so-called context). The existence of such transformations agrees with intuition, according to which the results of prec...
The Poincaré model, which was analyzed dealt with an ideal gas in a rectangular parallelepiped consisting of equal noninteracting classical free particles. The density of particles in the configuration space, or, equivalently, the density of the ideal gas was identified with the density of the probability distribution of free particle positions in...
We obtain Feynman formulas in the momentum space and Feynman-Kac formulas in the momentum and phase spaces for a p-adic analog
of the heat equation in which the role of the Laplace operator is played by the Vladimirov operator. We also present the Feynman
and Feynman-Kac formulas in the configuration space that have been proved in our previous pape...
The generalization of Wiener-Segal-Fock representation useful in the investigation of the so called asymptotic superselection rules to present some differential operators of the second order in the generalized Wiener-Segal-Fock space are considered with the Feynman formulae. The Wiener-Segal-Fock representation coincides with the usual Schrödinger...
It is well known that solutions of initial‐boundary value problems for classical evolution equations with a pseudodifferential operator on the right-hand side can be represented by integrals over the trajectory space in the configuration, momentum, or phase space. In the first case, integration is performed with respect to the Wiener measure or Fey...
A class of the Schroedinger type equations, with respect to functions defined on the product of [0∞) (``time'') and of the p-adic field (``space''), is considered. In the equation a Vladimirov operator substitutes the standard Lalacian. The main aim of the paper is to get representations of solutions of Cauchy problems for such equations by integra...
In this paper, we show that the composition of the generalized Lévy Laplacian of order p (which is gener- ated by the corresponding Cesàro mean) and a certain linear transformation of the domain of the function to which it is applied is proportional to the generalized Lévy Laplacian of smaller or larger order; this makes it possible to transform th...
The solutions and the role of problems posed by D. A. Raikov in the sixties are discussed.
In the present note we consider a class of second order parabolic equations with position dependent coefficients; such equations describe a diffusion of (quasi) particles with a variable mass. We represent a solution of Cauchy-Dirichlet problem for such class of equations in a bounded domain in the form of a limit of finite dimensional integrals of...
Diffusion on Riemannian Manifolds and integration with respect to anticommuting variables is presented. Solutions to the Cauchy problem is represented for the heat equations and Schrödinger equations on a Riemannian manifold in terms of path integrals in superspace with even part being a Riemannian manifold. A Feynman formula is a representation of...
Feynman-Kac formulas for heat conduction equations with Vladimirov’s operator (acting here as a Laplace operator) are proved;
it is assumed that unknown functions are determined on the product of the real axis and a space over the field of p-adic numbers and take real or complex values. Similar formulas may be obtained for Schrödinger type equation...
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 spac...
Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there exist numerous approximations for such equations, which are also called master equations.
In the paper, we...
The integrable function determines a generalized function from the space, as the generalized functions are often associated to measurable functions that are not locally integrable but admit a so-called regularization. The dimension of the space of all regularizations equals the codimension of the natural domain of definition. The regularization of...
A representation of a solution to the boundary value Cauchy-Dirichlet problem for a class of second-order parabolic equations with coordinate-dependent coefficient is obtained with the help of a limit of finite-dimensional integrals of elementary functions depending on the coefficients of the equation and the initial conditions. Such representation...
A study was conducted to evaluate the transformations of smooth complex-valued measures on a locally convex space, which are analytic continuations of transformations of smooth measures generated by transformations of this space. The study found that the notion of the logarithmic derivative of a function of a real argument taking values in the spac...
An infinite family of Lévy-type Laplacians is introduced and investigated (its elements are both the classical Lévy Laplacians
and some of their modifications), and a relationship between these operators and some quantum random processes is described.
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 L...
The convergence of the initial state of a quantum system to the microcanonical distribution was considered. It was proved that for an initial states, the difference between a state of the system at an arbitrary moment and the microcanonical state tends to zero in the weak topology of the state space. Two interpretations of a quantum system as an in...
Descriptive characterizations of the point, the continuous, and the residual spectra of operators in Banach spaces are put forward. In particular, necessary and sufficient conditions for three disjoint subsets of the complex plane to be the point spectrum, the continuous spectrum, and the residual spectrum of a linear continuous operator in a separ...
As is known, there are everywhere discontinuous infinitely Fréchet differentiable functions on the real locally convex spaces and of finitely supported infinitely differentiable functions and, respectively, of generalized functions. In this paper the relationship between the complex differentiability and continuity of a function on a complex locall...
In this paper, we show that both the surface measures on the paths in a (compact) Remaining subminimal of Euclidean space that are generated by the Wiener measure on the paths in Euclidean space and representations of solutions to the Cauchy‐Neumann and Cauchy‐Directly problems for the heat equation on domains of Euclidean spaces (and Remaining man...
A quantum Poincaré model (realizing behavior of ideal gas of noninteracting quantum Boltzmann particles) is introduced. We use the fact that the evolution of the Wigner function corresponding to a quantum system with a quadratic Hamiltonian coincides with the evolution of a probability distribution on a phase space of the Hamiltonian system, the qu...
The Feynman formulas to solve Cauchy problems for the Schrodinger equation and heat equation with Levy Laplacian on the infinite-dimensional manifold of mappings from a closed real interval to a Riemannian manifold are obtained. The aim of the problem is to reduce the derivation of Feynman formulas for equations on a manifold to the derivation of s...
A Kolmogorov probability model with transformable probability spaces has been constructed and studied which describe measurements of quantum observables that cannot be measured simultaneously and Einstein-Podolsky-Rosen (EPR) experiments, including Greenberger-Horne-Zeilinger (GHZ) paradox. The transition from a situation in which a standard Kolmog...
A class of polynomial potentials for which infinite dimensional Schrödinger equations have solutions that can be represented in terms of sequential Feynman path integrals in the configuration space is studied. Cauchy problem for the Schrödinger equations in terms of limits of sequences of integral over Cartesian powers of the configuration or phase...
The information entropy of a Gibbs state in problems of both classical and quantum statistical mechanics is discussed. It is generally accepted that the information entropy of both quantum and classical systems coincides with the thermodynamic entropy of an equilibrium state. Each state of a classical Hamiltonian system with a Borel probability mea...
We obtain a Hamilton-Jacobi-Bellman equation and a Hamilton-Jacobi-Bellman-Itô stochastic equation for a controllable quantum system. The phase space is the state space of the quantum system; it is assumed that the objective of control is minimizing a certain functional on this space. As is known, the classical optimal control problem consists of t...
Several Feynman-type formulas for solutions to the Schrödinger and heat equations with respect to functions on infinite-dimensional manifolds of mappings from a real interval to a compact Riemannian manifold were obtained. A Feynman formula is a representation of a solution to an evolution pseudo differential equations as a limit of integrals over...
In this paper we consider several Kolmogorov type models describing some quantum correlations. The main ideas beyond most
of these models comes back to pioneering works of Accardi (see [1, 10, 15] and references therein); because of that we call
some of them the Accardi models. Namely we discuss, in our frame, the Kolmogorov type Accardi model of t...
We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton–Jacobi–Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the...
Citations
... for sufficiently smooth compactly supported f. See also [257,318,319] for other applications of the Chernoff product formula to pathwise approximations of random processes. Note that this result does not recover the first order of the weak Euler-Maruyama scheme. ...
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![[object Object]](https://i1.rgstatic.net/ii/profile.image/502943045439489-1496922374717_Q64/Ivan-Remizov-2.jpg)
... Quantum random walks as compositions of i.i.d. random quantum dynamical semigroups were studied in Gough et al. (2022). ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272593861214215-1442002849208_Q64/John-Gough-4.jpg)
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... Then, an analysis of minimal reachable values of the objective functionals for various values of the parameters and intensity of the interaction with the environment has been made using DAA available in the Python library SciPy (for DAA implementation see [97] and for discussion [98][99][100][101][102][103]). A suitable choice of a convenient parametrization of quantum states can be useful for the analysis and numerical simulations [104]. To essentially speed up the computation for piecewise constant controls we use products of matrix exponents instead of solving differential evolution equations. ...
... In Kolokoltsov (2020), continuous time random walks were applied to modeling of the continuous quantum measurements yielding the new fractional in time evolution quantum filtering equation and thus new fractional equation for open quantum systems. The ambiguity of quantization procedure was considered as a source of randomness of quantum Hamiltonian in Gough et al. (2021). Quantum random walks as compositions of i.i.d. ...
... Nowadays, a large number of directions, papers, and books are devoted to quantum control. Various types of quantum systems and controls are considered, for example, infinite-dimensional open quantum systems with coherent quantum feedback [23], Floquet engineering [14,15] for control of quantum systems using time-periodic external fields. The first experimental realization of Kraus maps studied in [24] for an open single qubit was done in [25]. ...
... The condition of strong continuity in space H and the description of continuity subspaces for these unitary groups are obtained in Section 7. These results are important for extending the procedure of the averaging of random orthogonal mappings to the infinitedimensional case, and for obtaining the differential equation describing the mean values of the compositions of independent random orthogonal mappings [43]. ...
... The generalized measure is defined not as an additive function of the set, but as a linear functional on a suitable space of test functions. The use of a generalized shift-invariant measure allowed to define the Hilbert space of square integrable functions in which the unitary Fourier transform operates Smolyanov and Shamarov (2020). ...
... Shift-invariant measures on a separable Hilbert space were studied in [10], [11] without assuming the countable additivity and Borel measurability properties. Shift-invariant measures on a locally convex space of numerical sequences were studied in [5], [12], [13] without the local finiteness and σ-finiteness properties. ...
... Then, according to theorem 3 in [23], we obtain (12). ...
... The studying of a random processes in infinite dimension Banach spaces and its description by a partial differential equation for a functions on the Banach space are the important topics of contemporary mathematics (see [4,8,9]). To the investigation of the above topics and to construct the quantum theory of infinite dimension Hamiltonian systems the analogs of the Lebesgue measure on the infinite dimension linear space are introduced in the works [1,10,13,17]. ...
