# D. P. SankovichRussian Academy of Sciences | RAS · Steklov Mathematical Institute

D. P. Sankovich

PhD

## About

62

Publications

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148

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Introduction

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April 1984 - present

## Publications

Publications (62)

We study, using the Bogolyubov approximation, the thermodynamic behaviour of a superstable Bose system whose energy operator in the second-quantized form contains a nonlinear expression in the occupation numbers operators. We prove that for all values of the chemical potential satisfying µ > λ(0), where λ(0) ≤ 0 is the lowest energy value, the syst...

A weakly interacting Bose gas on a simple cubic lattice is considered. We prove the existence of the standard or zero-mode Bose condensation at sufficiently low temperature. This result is valid for sufficiently small interaction potential and small values of chemical potential. Our method exploits infra-red bound for the suitable two-point Bogolyu...

A weakly interacting Bose gas on a simple cubic lattice is considered. We prove the existence of the standard or zero-mode Bose condensation at sufficiently low temperature. This result is valid for sufficiently small interaction potential and small values of chemical potential. Our method exploits infrared bound for the suitable two-point Bogolyub...

The upper bound on the isothermal compressibility for lattice bosons in the uniform Bose-Hubbard model is derived.

Quantum systems of particles obeying Bose statistics and moving in d-dimensional lattices are studied. The original Bose–Hubbard Hamiltonian is considered, together with model systems related to this Hamiltonian: the Bose–Hubbard model with exchange interaction of infinite radius and the Bose–Hubbard model with infinite interaction potential. Rigor...

Universal Journal of Physics an Applications, 8, 42-47- 2014.
http://www.hrpub.org/download/20131215/UJPA8-18401511.pdf
Within the Bogolyubov approximating Hamiltonian approach the limit pressure is found in a system of atoms with internal three-level structure (spin-1 system). Moreover, the emergence of a Bose–Einstein condensation is proved for...

We use infrared bounds method to study the existence of a Bose
condensation phase transition in a three-dimensional lattice model of
interacting bosons. Upper bound (local Gaussian domination) is presented
on the Bogolyubov inner product of creation and annihilation bosonic
operators in momentum space. We focus on the situation with a
non-negative...

A generating thermodynamic functional for the one-dimensional, one-component hydrodynamic type system with the Hamiltonian density au2+bu4is determined. Correlation functions of this model are considered.

The Bogolyubov model of liquid helium is considered. The validity of substituting a c-number for the k = 0 mode operator â0 is established rigorously. The domain of stability of the Bogolyubov's Hamiltonian is found. We derive sufficient conditions which ensure an appearance of the Bose condensate in the model. For some temperatures and some positi...

The equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. We calculate the superfluid density in the Bogolyubov model of imperfect Bose gas. The model superstable Hamiltonian is considered. We examine the case of some pair potential and find the estimate for temperature and density in the λ-point.

Application of the functional integration methods in equilibrium statistical mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. We consider some problems related to int...

A model of the non-ideal Bose gas is considered. We prove the existence of condensate in the model at sufficiently low temperature. The method of majorizing estimates for the Duhamel Two Point Functions is used. The equation for the critical temperature and the upper bound for the one-particle excitations energy are obtained.

We use the Bogolyubov approximating Hamiltonian method to rigorous study the equilibrium properties of imperfect Bose gases.
We calculate the pressure of the mean field Bose gas model. This model in external potential is considered.
Key wordsBose-condensation-approximating Hamiltonian method

The Bogolyubov model of liquid helium is considered. We derive sufficient conditions which ensure an appearance of the Bose
condensate in the model. For some temperatures and some positive values of the chemical potential there is the gapless Bogolyubov
spectrum of elementary excitations, leading to the proper microscopic interpretation of the supe...

The Bogolyubov model of liquid helium is considered. The validity of substituting a c-number for the k = 0 mode operator ^ a0 is established rigorously. The domain of stability of the Bogolyubov's Hamiltonian is found. We derive sufcient conditions which ensure the appearance of the Bose condensate in the model. For some temperatures and some posit...

We analyze the approximating Hamiltonian method for Bose sys-tems. Within the framework of this method, the pressure for the mean field model of an imperfect boson gas is calculated. The problem is considered by the systematic application of the Bogolyubov–Ginibre approximation.

For a Bose atom system whose energy operator is diagonal in the so-called number operators and its ground state has an internal two-level structure with negative energies, exact expressions for the limit free canonical energy and pressure are obtained. The existence of non-conventional Bose-Einstein condensation has been also proved.

In the framework of the Bogolyubov approximation and using the Bogolyubov inequalities, we give a simple proof of the coexistence of two non-conventional Bose–Einstein condensates in the case of some superstable Bose system whose atoms have an internal two-level structure, and their energy operators in the second quantized form depend on the number...

We study, using the Bogolyubov approximation, the thermodynamic behaviour of a superstable Bose system whose energy operator in the second-quantized form contains a nonlinear expression in the occupation numbers operators. We prove that for all values of the chemical potential satisfying $\mu > \lambda(0)$, where $\lambda (0)\leq 0$ is the lowest e...

A model of a nonideal Bose gas with a repulsive interaction is considered. It is proved that there is not macroscopic occupation of the ground state in the thermodynamic limit in this model, but nevertheless the generalized condensation occurs.

We consider two simple model systems describing effective repulsion in a nonideal Bose gas. The interaction Hamiltonians in these systems can be analytically represented as functions of the occupation number operators for modes with nonzero momenta (p0). One of these models contains an interaction term corresponding to repulsion of bosons with the...

The upper- and lower-bound estimates for the thermodynamic averages of the occupation number operators are obtained for the case of a nonlinear photon system in thermal equilibrium with the thermostat. Two-sided inequalities for the internal energy of such a system are constructed from the obtained estimates.

We prove the local Gaussian dominance condition for a Bose system whose Hamiltonian is diagonal with respect to the particle number operators. The proof is based on obtaining an upper bound estimate for the Bogoliubov inner product of the Bose creation and annihilation operators.

For a non-interacting many particle Bose system whose energy operator is diagonal in the number of occupation operators [Formula: see text] upper bounds on the thermal averages [Formula: see text] are obtained. These bounds lead to the proof of Bose–Einstein condensation for finite values of the inverse temperature β and for chemical potential μ=0....

Application of the functional integration methods in equilibrium statistical mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. This measure arises in the Bogolyubov T-...

We investigate some properties of the Bogoliubov measure that appear in statistical equilibrium theory for quantum systems and establish the nondifferentiability of the Bogoliubov trajectories in the corresponding function space. We prove a theorem on the quadratic variation of trajectories and study the properties implied by this theorem for the s...

We consider problems related to integration with respect to the Bogoliubov measure in the space of continuous functions and calculate some functional integrals with respect to this measure. Approximate formulas that are exact for functional polynomials of a given degree and also some formulas that are exact for integrable functionals belonging to a...

We describe recent work3 on the equilibrium thermodynamics of a lattice boson gas, in three dimensions, with the onsite repulsion and nearest neighbor sites atraction. For this system the existence of Bose — condensation is proven and an equation for the critical temperature is obtained. Moreover, upper and lower bounds for the static structure fac...

A quantum system of nonlinear oscillators is considered. Within the framework of Berezin's functional integral we prove the gaussian domination at finite temperature for some values of the chemical potential. Upper and lower bounds for the average number of particles with momentum p are derived.

A Davies model of an imperfect boson gas is considered. The model includes not only a convex, but also a concave type of an interaction function which depends on a dencity operator. A sufficient condition of the Bose–Einstein condensation is proved. An exact value of the critical temperature is obtained.

We consider a polynomial generalization of the Huang-Davies model in the nonideal Bose gas theory. We prove that the Gaussian
dominance condition is fulfilled for all values of the chemical potential. We show that the lower bound for the critical temperature
in the Huang-Davies model obtained by the infrared bound method coincides with the exact va...

We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure
defined in the corresponding space of continuous functions. This measure arises in the Bogoliubov T-product approach and is
non-Wiener.

Using the so-called method of infrared bounds and a Roepstorff's
inequality we obtain a lower bound for the amount of condensate and
derive an upper bound for the anomalous average lim {δ
-> 0} lim V->&infin (1/V)|< adag
0 >δ |2 of Huang-Davies (HD)
model. A lower bound for the static structure factor is also obtained.
We generalize the HD model an...

This is a study of the equilibrium thermodynamics of a lattice boson gas with on-site repulsion and nearest-neighbor site attraction. For this system, the existence of a Bose condensate is proved and an equation for the lower estimate of the critical temperature is obtained. Moreover, the upper and lower bounds for the structure factor are derived....

CONTENTS §1. Introduction §2. Theory of open systems §3. Kinetic theory and hydrodynamics §4. Quantum statistical mechanics References

For the υ-dimensional lattice boson gas with the onsite repulsion and nearest neighbour sites attraction, we prove the existence of phase transition for ν≥3 and sufficiently large density when temperature is sufficiently small. This phase transition is connected with some type of Bose-Einstein condensation (ρ-condensation.4).

A quantum system of coupled anharmonic oscillators on v-dimensional hypercubic lattice is considered. The upper bound on the pair correlator is obtained. For the representation of creation-annihilation operators this system corresponds to a lattice boson gas model with delta-type repulsion. We prove the existence of Bose condensation at sufficientl...

The Gaussian domination for the special model of the quantum nonlinear oscillator is proved. From this condition the upper bound for the thermal expectation follows.

The paper is concerned with the nonequilibrium thermodynamic properties of a system known as the nonlinear Schroedinger model, which belongs to a class of completely integrable infinite-dimensional Hamiltonian systems in classical mechanics. Hydrodynamic equations are established for this model on the basis of equations of motion, with the approxim...

An application of the Weyl quantization scheme to problems of quantum statistical mechanics is considered. A closed kinetic equation for the distribution function of a ``small'' subsystem interacting with a thermostat is obtained. On the basis of this equation we consider the kinetics in the polaron model. The developed technique is a generalizatio...

Within the framework of the Bogolyubov’s approach to the time-ordered operator product the Wiener functional integral is constructed.

Problems of integration with respect to a special Gaussian measure (the Bogolyubov measure) that arises in the statistical equilibrium theory for quantum systems are considered. It is shown that the Gibbs equilibrium means of Bose operators can be represented as functional integrals with respect to this measure. Certain functional integrals with re...

## Projects

Project (1)