Groups of Operators for Evolution Equations of Quantum Many-particle Systems

Gerasimenko, V. I.

doi:10.1007/978-3-7643-9921-4_21

Cited by 5 publications

(19 citation statements)

References 24 publications

(60 reference statements)

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“…On the space L 1 (F H ) one-parameter mapping ( 7) is a bounded strong continuous group of nonlinear operators and it is determined a solution of the Cauchy problem for the von Neumann hierarchy for correlation operators [16].…”

Section: Preliminaries: Dynamics Of Correlationsmentioning

confidence: 99%

“…, s | G(t)) ≡ N (Y | G(t)), s ≥ 1, are generators (12) of the von Neumann hierarchy (10) and ǫ > 0 is a scaling parameter. The rigorous derivation of the hierarchy of evolution equations for marginal correlation operators (16), according to definition (15), consists in its derivation from the von Neumann hierarchy for correlation operators (10) [19] (for marginal density operators see paper [20]).…”

Section: A Nonperturbative Solution Of the Nonlinear Bbgky Hierarchy ...mentioning

confidence: 99%

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On Dynamics of Correlations in Quantum Many-Particle Systems

Gerasimenko

2015

Preprint

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The paper deals with the problem of the rigorous description of the evolution of states of large particle quantum systems by means of correlation operators. A nonperturbative solution of the Cauchy problem of the hierarchy of nonlinear evolution equations for a sequence of marginal correlation operators is constructed. For initial states specified in terms of a one-particle density operator and correlation operators we also develop an approach to the description of the processes of the creation and the propagation of correlations within the framework of a one-particle density operator. Moreover, the mean field asymptotic behavior of constructed marginal correlation operators is established.

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Section: Preliminaries: Dynamics Of Correlationsmentioning

confidence: 99%

Section: A Nonperturbative Solution Of the Nonlinear Bbgky Hierarchy ...mentioning

confidence: 99%

On Dynamics of Correlations in Quantum Many-Particle Systems

Gerasimenko

2015

Preprint

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“…A solution of the Cauchy problem – is given by the following expansion

g_{s} (t, Y) = G (t; Y | g (0)) ≐ \underset{P : Y = \underset{i}{140%true \cup} X_{i}}{140%true \sum} A_{| normalP |} (t, {X_{1}}, \dots, {X_{| normalP |}}) \underset{X_{i} \subset P}{140%true \prod} g_{| X_{i} |} (0, X_{i}), s \geq 1,

where

\underset{P : Y = \underset{i}{140%true \cup} X_{i}}{140%true \sum}

is the sum over all possible partitions P of the set Y ≡ (1, … , s ) into | P | nonempty mutually disjoint subsets X i ⊂ Y , the evolution operator

A_{| normalP |} (t)

is the | P | th‐order cumulant of groups of operators defined by the formula

A_{| normalP |} (t, {X_{1}}, \dots, {X_{| normalP |}}) ≐ \underset{� P^{'} : ({X_{1}}, \dots, {X_{| normalP |}}) = \underset{k}{140%true \cup} Z_{k}}{140%true \sum} {(- 1)}^{| {normalP}^{'} | - 1} (| P^{'} | - 1)! \underset{Z_{}}{140%true \prod}

…”

Section: Preliminary Facts: the Von Neumann Hierarchymentioning

confidence: 99%

A nonperturbative solution of the nonlinear BBGKY hierarchy for marginal correlation operators

Gerasimenko

Polishchuk

2013

Math Methods in App Sciences

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Communicated by M. A. LachowiczWe consider the problem of the rigorous description of nonequilibrium quantum correlations. Within the framework of an alternative approach to the description of the evolution of states of finitely many particles in terms of correlation operators governed by the von Neumann hierarchy, we derive the nonlinear quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for marginal correlation operators that is adopted to the description of quantum infinite-particle systems as well. A nonperturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed.

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“…In many works devoted to the investigation of many-particle quantum systems, the evolution is described by a group of operators acting in the space L 1 .H/ of operators with bounded trace over a Hilbert space H [1][2][3]. However, this space appears to be insufficient, in particular, for the solution of problems whose statement is natural in the class of operators with translation-invariant kernel because these operators do not belong to the space L 1 .H/: In this connection, the problem of properties of groups of operators in operator spaces more general than L 1 .H/ was posed in [3].…”

Section: Introductionmentioning

confidence: 99%

“…However, this space appears to be insufficient, in particular, for the solution of problems whose statement is natural in the class of operators with translation-invariant kernel because these operators do not belong to the space L 1 .H/: In this connection, the problem of properties of groups of operators in operator spaces more general than L 1 .H/ was posed in [3]. In [4], within the framework of the problem posed, the spaces of linear continuous operators with bounded projection trace were constructed, and it was established that these spaces are not complete in the general case.…”

Section: Introductionmentioning

confidence: 99%