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Interacting Stochastic Schrödinger Equation

Author

Listed:
  • Lu Zhang

    (School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
    School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China)

  • Caishi Wang

    (School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China)

  • Jinshu Chen

    (School of Science, Lanzhou University of Technology, Lanzhou 730050, China)

Abstract

Being the annihilation and creation operators on the space h of square integrable Bernoulli functionals, quantum Bernoulli noises (QBN) satisfy the canonical anti-commutation relation (CAR) in equal time. Let K be the Hilbert space of an open quantum system interacting with QBN (the environment). Then K ⊗ h just describes the coupled quantum system. In this paper, we introduce and investigate an interacting stochastic Schrödinger equation (SSE) in the framework K ⊗ h , which might play a role in describing the evolution of the open quantum system interacting with QBN (the environment). We first prove some technical propositions about operators in K ⊗ h . In particular, we obtain the spectral decomposition of the tensor operator I K ⊗ N , where I K means the identity operator on K and N is the number operator in h , and give a representation of I K ⊗ N in terms of operators I K ⊗ ∂ k ∗ ∂ k , k ≥ 0 , where ∂ k and ∂ k ∗ are the annihilation and creation operators on h , respectively. Based on these technical propositions as well as Mora and Rebolledo’s results on a general SSE, we show that under some mild conditions, our interacting SSE has a unique solution admitting some regularity properties. Some other results are also proven.

Suggested Citation

  • Lu Zhang & Caishi Wang & Jinshu Chen, 2023. "Interacting Stochastic Schrödinger Equation," Mathematics, MDPI, vol. 11(6), pages 1-16, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1388-:d:1096010
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    References listed on IDEAS

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    1. Singh, Ajeet & Shukla, Anurag & Vijayakumar, V. & Udhayakumar, R., 2021. "Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    2. Barchielli, A. & Holevo, A. S., 1995. "Constructing quantum measurement processes via classical stochastic calculus," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 293-317, August.
    3. Barchielli, A. & Paganoni, A. M. & Zucca, F., 1998. "On stochastic differential equations and semigroups of probability operators in quantum probability," Stochastic Processes and their Applications, Elsevier, vol. 73(1), pages 69-86, January.
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