A. M. Chebotarev, “The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation”, Mat. Zametki, 61:4 (1997), 612–622; Math. Notes, 61:4 (1997), 510

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Matematicheskie Zametki, 1997, Volume 61, Issue 4, Pages 612–622
DOI: https://doi.org/10.4213/mzm1539 (Mi mzm1539)

This article is cited in 29 scientific papers (total in 29 papers)

The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation

A. M. Chebotarev

M. V. Lomonosov Moscow State University, Faculty of Physics

Full-text PDF (249 kB) Citations (29)

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DOI: https://doi.org/10.4213/mzm1539

Abstract: 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.

Received: 11.12.1996

English version:
Mathematical Notes, 1997, Volume 61, Issue 4, Pages 510–518
DOI: https://doi.org/10.1007/BF02354995

Bibliographic databases:

UDC: 519.217

Language: Russian

Citation: A. M. Chebotarev, “The quantum stochastic equation is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation”, Mat. Zametki, 61:4 (1997), 612–622; Math. Notes, 61:4 (1997), 510–518

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mzm1539

https://doi.org/10.4213/mzm1539

https://www.mathnet.ru/eng/mzm/v61/i4/p612

This publication is cited in the following 29 articles:

Fernando Iemini, Darrick Chang, Jamir Marino, “Dynamics of inhomogeneous spin ensembles with all-to-all interactions: Breaking permutational invariance”, Phys. Rev. A, 109:3 (2024)
Luigi Accardi, Yungang Lu, “Phenomenological models versus deductive models: The stochastic limit of quantum theory”, Int. J. Mod. Phys. A, 37:20n21 (2022)
dos Prazeres L.F., Souza Leonardo da Silva, Iemini F., “Boundary Time Crystals in Collective D-Level Systems”, Phys. Rev. B, 103:18 (2021), 184308
Iemini F., Russomanno A., Keeling J., Schiro M., Dalmonte M., Fazio R., “Boundary Time Crystals”, Phys. Rev. Lett., 121:3 (2018), 035301
Gough J.E., James M.R., “The series product for Gaussian quantum input processes”, Rep. Math. Phys., 79:1 (2017), 111–133
Gough J.E., Nurdin H.I., “Can Quantum Markov Evolutions Ever Be Dynamically Decoupled?”, 2017 IEEE 56Th Annual Conference on Decision and Control (Cdc), IEEE Conference on Decision and Control, IEEE, 2017
Gough J.E., “Non-Markovian Quantum Feedback Networks II: Controlled Flows”, J. Math. Phys., 58:6 (2017), 063517
Nurdin H.I., Yamamoto N., “Mathematical Modeling of Linear Dynamical Quantum Systems”: Nurdin, HI Yamamoto, N, Linear Dynamical Quantum Systems: Analysis, Synthesis, and Control, Communications and Control Engineering, Springer-Verlag Berlin, 2017, 35–71
Gough J.E., “The Stratonovich formulation of quantum feedback network rules”, J. Math. Phys., 57:12 (2016), 123505
Gough J.E., “Scattering Processes in Quantum Optics”, Phys. Rev. A, 91:1 (2015), 013802
Gough J.E., “Characteristic Operator Functions For Quantum Input-Plant-Output Models and Coherent Control”, J. Math. Phys., 56:1 (2015), 013506
Gregoratti M., “the Hamiltonian Generating Quantum Stochastic Evolutions in the Limit From Repeated To Continuous Interactions”, Open Syst. Inf. Dyn., 22:4 (2015), 1550022
Gough J., “the Global Versus Local Hamiltonian Description of Quantum Input-Output Theory”, Open Syst. Inf. Dyn., 22:2 (2015), 1550009
Luc Bouten, Rolf Gohm, John Gough, Hendra Nurdin, “A Trotter-Kato theorem for quantum Markov limits”, EPJ Quantum Technol., 2:1 (2015)
von Waldenfels W., “The Singular Coupling Limit for a Simple Pure Number Process”, Stochastics, 84:2-3, SI (2012), 417–423
Gregoratti, M, “Dilations a la Hudson-Parthasarathy of Markov semigroups in Classical Probability”, Stochastic Analysis and Applications, 26:5 (2008), 1025
Barchielli, A, “Continual measurements in quantum mechanics and quantum stochastic calculus”, Open Quantum Systems III: Recent Developments, 1882 (2006), 207
Von Waldenfels W., “The Hamiltonian of a Simple Pure Number Process”, Quantum Probability and Infinite Dimensional Analysis, Qp-Pq Quantum Probability and White Noise Analysis, 18, eds. Schurmann M., Franz U., World Scientific Publ Co Pte Ltd, 2005, 518–524
A. M. Chebotarev, G. V. Ryzhakov, “On the Strong Resolvent Convergence of the Schrödinger Evolution to Quantum Stochastics”, Math. Notes, 74:5 (2003), 717–733
A. M. Chebotarev, “What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?”, Math. Notes, 71:3 (2002), 408–427

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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