Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9702061

Quantum stochastic differential equation is unitary equivalent to a symmetric boundary value problem in Fock space

Alexander Mikhailovich Chebotarev

Published Feb 27, 1997• 0 citations
arXivHTMLPDF

Abstract

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of Schrödinger equations in Fock space: the strong resolvent limit is unitary equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitary equivalent to the symmetric (or Stratonovich) form of QSDE. We prove that QSDE is unitary equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps of the phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.}

DOI
10.48550/arxiv.quant-ph/9702061

Related papers

Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space

Alexander Mikhailovich Chebotarev

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Apr 1, 1998•17 citations

The Quantum Stochastic Differential Equation Is Unitarily Equivalent to a Symmetric Boundary Value Problem for the Schrödinger Equation

Alexander Mikhailovich Chebotarev

We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schrödinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the ``master equation'') is derived from the boundary value problem for the Schrödinger equation.

arXiv:funct-an/9706006•Jun 19, 1997•0 citations

Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space

Alexander Mikhailovich Chebotarev, Dmitry Victorov

By using F. A. Berezin's canonical transformation method [5], we derive a nonadapted quantum stochastic differential equation (QSDE) as an equation for the strong limit of the family of unitary groups satisfying the Schrödinger equation with singularly de

Banach Center Publications•Jan 1, 1998•5 citations

STOCHASTIC EVOLUTION AS A QUASICLASSICAL LIMIT OF A BOUNDARY VALUE PROBLEM FOR SCHRÖDINGER EQUATIONS

V. P. Belavkin, Vassili Nikitich Kolokoltsov

We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Mar 1, 2002•8 citations

Stochastic Evolutions As Boundary Value Problems (Infinite Dimensional Analysis and Quantum Probability Theory)

V. P. Belavkin, Vassili Nikitich Kolokoltsov

Solutions to linear stochastic and quantum stochastic equations are proved to provide the interaction representations for the solutions to certain Dirac type equations with boundary conditions in pseudo Fock spaces.The latter are presented as the semi-classical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for general Schr\"odinger equations.

数理解析研究所講究録•Aug 1, 2001•0 citations

On the Generator of the Solution of a Quantum Stochastic Differential Equation

Oswaldo Gonzalez-Gaxiola, Roberto B. Quezada

If the Hamiltonian of the system is a bounded selfadjoint operator, we give a simple alternative characterization of the infinitesimal generator iC of the unitary group associated with a Quantum Stochastic Differential Equation (QSDE). By using the defect indices of the symmetric operator C, we give a sufficient condition that assures that iC is the restriction of a generator of a strongly continuous semigroup of isometries in the case when the Hamiltonian of the system is an unbounded symmetric operator.

Stochastic Models•Sep 12, 2006•0 citations
We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of Schrödinger equations in Fock space: the strong resolvent limit is unitary equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitary equivalent to the symmetric (or Stratonovich) form of QSDE. We prove that QSDE is unitary equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps of the phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.}
We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.
We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schrödinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the ``master equation'') is derived from the boundary value problem for the Schrödinger equation.
We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.
If the Hamiltonian of the system is a bounded selfadjoint operator, we give a simple alternative characterization of the infinitesimal generator iC of the unitary group associated with a Quantum Stochastic Differential Equation (QSDE). By using the defect indices of the symmetric operator C, we give a sufficient condition that assures that iC is the restriction of a generator of a strongly continuous semigroup of isometries in the case when the Hamiltonian of the system is an unbounded symmetric operator.