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

Alexander Chebotarev; Dmitry Victorov

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 119-133
  • ISSN: 0137-6934

Abstract

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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 degenerating Hamiltonians in Fock space. Stochastic differentials of QSDE generate a nonadapted associative Ito multiplication table, and the coefficients of these differentials satisfy the formal unitarity conditions of the Hudson-Parthasarathy type [10].

How to cite

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Chebotarev, Alexander, and Victorov, Dmitry. "Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space." Banach Center Publications 43.1 (1998): 119-133. <http://eudml.org/doc/208831>.

@article{Chebotarev1998,
abstract = {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 degenerating Hamiltonians in Fock space. Stochastic differentials of QSDE generate a nonadapted associative Ito multiplication table, and the coefficients of these differentials satisfy the formal unitarity conditions of the Hudson-Parthasarathy type [10].},
author = {Chebotarev, Alexander, Victorov, Dmitry},
journal = {Banach Center Publications},
keywords = {stochastic processes; Fock space; Schrödinger’s equation; non-adapted quantum stochastic differential equation; strongly degenerating Hamiltonians; differentials},
language = {eng},
number = {1},
pages = {119-133},
title = {Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space},
url = {http://eudml.org/doc/208831},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Chebotarev, Alexander
AU - Victorov, Dmitry
TI - Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 119
EP - 133
AB - 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 degenerating Hamiltonians in Fock space. Stochastic differentials of QSDE generate a nonadapted associative Ito multiplication table, and the coefficients of these differentials satisfy the formal unitarity conditions of the Hudson-Parthasarathy type [10].
LA - eng
KW - stochastic processes; Fock space; Schrödinger’s equation; non-adapted quantum stochastic differential equation; strongly degenerating Hamiltonians; differentials
UR - http://eudml.org/doc/208831
ER -

References

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  2. [2] S. Albeverio, F. Gesztesy and R. Hoegh-Krohn, Solvable models in Quantum mechanics, Springer-Verlag, New-York, Berlin, London, 1988. Zbl0679.46057
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  5. [5] F. A. Berezin, The Method of Second Quantization, Academic Press, New-York, 1996. Zbl0956.83531
  6. [6] A. M. Chebotarev, Quantum stochastic differential equation as a strong resolvent limit of the Schrödinger evolution, in: IV Simposio de Probabilidad y Procesos Estochasticos, Gunajuato, Mexico, 1996, Societad Mat. Mexicana, 1996, 71-89. Zbl0924.60089
  7. [7] A. M. Chebotarev, Symmetric form of the Hudson-Parthasarathy equation, Mathematical Notes, Vol. 60, N5, 1996, 544-561. Zbl0904.60050
  8. [8] A. M. Chebotarev, Quantum stochastic differential equation is unitarily equivalent to a boundary value problem for the Schrödinger equation, Mathematical Notes, Vol. 61, N4, 1997, 510-519. Zbl0919.60093
  9. [9] C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: quantum statistical differential equations and the master equation, Phys. Rev. A, Vl. 31, 1985, 3761-3774. 
  10. [10] R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys., Vol. 93, N3, 1984, 301-323. Zbl0546.60058
  11. [11] Ka T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1980. 
  12. [12], Naukova dumka, Kiev, 1993. 
  13. [13] P. A. Meyer, Quantum probability for probabilists, Lecture Notes in Math., vol. 1338, 1993. Zbl0773.60098
  14. [14] K. R. Parthasarathy, An introduction to quantum stochastic calculus, Birkhäuser, Basel, 1992. Zbl0751.60046
  15. [15] P. Zoller and C. W. Gardiner, Quantum Noise in Quantum Optics: the Stochastic Schödinger Equation, To appear in: Lecture Notes for the Les Houches Summer School LXIII on Quantum Fluctuations in July 1995, Edited by E. Giacobino and S. Reynaud, Elsevier Science Publishers B.V., 1997. 

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