arXiv subjects

Vsevolod Sakbaev

Publications and source records attributed to Vsevolod Sakbaev.

5 recordsLinked to original sources

arXiv2025

Measures and Trajectory Properties in Oscillator Systems

This paper investigates the properties of trajectories in harmonic oscillator systems equipped with a point, absolutely continuous, or singular measure. As demonstrated in [30], infinite-dimensional linear flows of countable oscillator systems exhibit a new class of trajectory behavior. Specifically, these trajectories are non-periodic, and their projections onto any four-dimensional symplectic subspace fail to be dense in the corresponding projection of the invariant torus. Such trajectories do not arise in finite-dimensional systems, are non-generic for countable oscillator systems, but become generic in the continual case. We prove that for a countable harmonic oscillator system, every point on a non-degenerate invariant torus is a non-wandering point of the flow. In contrast, for a continual system with an absolutely continuous measure, all points on such a torus are wandering. Furthermore, for continual systems with a singular measure, we establish sufficient conditions on the measure and torus that rule out the existence of both transitive trajectories and non-wandering points. As an application, we exhibit a class of singular Bernoulli measures satisfying these conditions.

math.DS↗

arXiv2024

On the extension of singular linear infinite-dimensional Hamiltonian flows

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the flows of completely integrable Hamiltonian systems are investigated. These construction gives the opportunity to describe Hamiltonian flows in the phase space by means of unitary groups in the space of functions that are quadratically integrable by the invariant measure. Invariant measures are applied to the study of model linear Hamiltonian systems that admit features of the type of unlimited increase in kinetic energy over a finite time. Due to this approach solutions of Hamilton equations that admit singularities can be described by means of the phase flow in the extended phase space and by the corresponding Koopman representation of the unitary

math-ph↗

arXiv2024

Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space

We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.

quant-ph↗

arXiv2021

Measures on a Hilbert space that are invariant with respect to shifts and orthogonal transformations

A finitely-additive measure $λ$ on an infinite-dimensional real Hilbert space $E$ which is invariant with respect to shifts and orthogonal mappings has been defined. This measure can be considered as the analog of the Lebesgue measure in the sense of its invariance with respect to the above transformations. The constructed measure is defined on the ring $\cal R$ of subsets of the Hilbert space generated by measurable rectangles. A measurable rectangle is an infinite-dimensional parallelepiped such that the product of the lengths of its edges converges unconditionally. The shift and rotation-invariant measure is obtained as a continuation of a family of shift-invariant measures $λ_{\cal E}$, where each measure $λ_{\cal E}$ is defined on the ring ${\cal R}_{\cal E}$ of measurable rectangles with edges collinear to the vectors of some orthonormal basis $\cal E$ in the space $E$. An equivalence relation is introduced on the set of orthonormal bases in terms of the transition matrix from one orthonormal basis to another. The equivalence relation allows to glue measures defined on the subset rings corresponding to different bases into the one measure $λ$ defined on the unique ring $\cal R$. The obtained measure $λ$ is invariant with respect to shifts and rotations. The decomposition of the measure $λ$ into the sum of mutually singular shift-invariant measures is obtained. The paper describes the structure of the space $\cal H$ of numerical functions square integrable with respect to the constructed shift and rotation-invariant measure $λ$. The decomposition of the space $\cal H$ into the orthogonal sum of subspaces corresponding to all possible equivalence classes of bases is obtained. Unitary groups acting by means of orthogonal transformations of the argument in the space $\cal H$ of square integrable functions are investigated.

math.FA↗

arXiv2015

Averaging of one-parameter semigroups and passage to the limit in the space of pseudomeasures

The sequence of one-parameter semigroups arising as the approximation of initial-boundary value problem with singularities is the object of investigation of this paper. The set of limit points of the sequence of approximating semigroups is studied. The set of limit points of the map with values in a linear topological space is presented as the set of mean values of this map by measures on the domain of definition of the map. One to one correspondence betwin the semigroups generated by any approximating initial-boundary value problems and the pseudomeasures on the space of maps of time semiaxe into the coordinate space is studied. The linear space of pseudomesures endowed with the structure of Banach space and with the structure of the linear topological space such that the convergence of semigroup sequence is equivalent to the convergence of the sequence of corresponding pseudomeasures. The desctiption of a limit point of the sequence of approximating semigroups is obtained by a measure on the topological vector space of corresponding pseudomeasures. The trajectories the limit one-parameter family of transformations of the space of initial data is described by the mean value of the random pseudomeasure.

quant-ph↗