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Deviation Estimates for Random Walks and Stochastic Methods for Solving the Schrödinger Equation

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Abstract

The stochastic representation of solutions of the Cauchy problem for the Schrödinger equation is used in order to construct unitary matrix approximations of the resolving operator. We show that the probability distribution of deviations of random walks allows one to estimate the increase rate of derivatives and the support of solutions.

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REFERENCES

  1. J. Manz,"Molecular wavepacket dynamics:Theory for experiments 1926-1996,"in:Femtochemistry and Femtobiology:Ultrafast Reaction Dynamics at Atomic-Scale Resolution (V. Sundstr ¨om,editor), Imperial College Press, London,1997,pp.80–318.

    Google Scholar 

  2. H. Tal-Ezer and R. Kosloff, "An accurate and efficient scheme for propagating the time dependent Schr ¨odinger equation," J. Chem. Phys., 81 (1984), 3967–3971.

    Google Scholar 

  3. C. Leforestier et al., "A comparison of different propagation schemes for the time dependent Schr ¨odinger equation," J.Comp.Phys., 94 (1991),59–80.

    Google Scholar 

  4. Numerical Grid Methods and Their Applications to Schr¨odinger Equation (C. Cerjan,editor),Kluwer, Netherlands,1993.

    Google Scholar 

  5. Y. Huang, D.J. Kouri,and D.K. Hoffman,"A general,energy-separable polynomial representation of the time-independent full Green operator with application to time-independent wavepacket forms of Schr ¨odinger and Lippmann-Schwinger equations,"Chem.Phys.Lett.,225 (1994),37–45.

    Google Scholar 

  6. S.M. Ermakov and G.A. Mikhailov,Statistical Modeling [in Russian ],Nauka, Moscow,1982.

    Google Scholar 

  7. A.M. Chebotarev,"Representation of solutions of the Schr ¨odinger equation as mathematical expectation of functionals of jump process,"Mat.Zametki [Math.Notes ],24 (1978),o.5,699–706.

    Google Scholar 

  8. A.M. Chebotarev and V.P. Maslov,"Processus de sauts et leurs applications dans la m ´ecanique quantique,"Lecture Notes in Phys.,106 (1979),58–72.

    Google Scholar 

  9. A.A. Konstantinov, V.P. Maslov,and A.M. Chebotarev,"Stochastic representations of solutions of the Cauchy problems for equations of quantum mechanics,"Uspekhi Mat.Nauk [R ssian Math. Surveys ],45 (1990),o.6,3–24.

    Google Scholar 

  10. A.M. Chebotarev and R. Quezada-Batalla,"Stochastic approach to time-dependent quantum tunnel-ing,"R ssian J.Math.Phys.,4 (1996),o.3,275–286.

    Google Scholar 

  11. Yu.V. Prokhorov and Yu.A. Rozanov,Probability Theory [in Russian ],Nauka, Moscow,1973.

    Google Scholar 

  12. K. Ito and H.P. McKean, Diffusion Processes and Their Sample Paths,Springer-Verlag, Berlin, 1965.

    Google Scholar 

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Authors and Affiliations

  1. M. V. Lomonosov Moscow State University, russia

    A. M. Chebotarev

  2. russia

    A. V. Polyakov

Authors
  1. A. M. Chebotarev
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  2. A. V. Polyakov
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Chebotarev, A.M., Polyakov, A.V. Deviation Estimates for Random Walks and Stochastic Methods for Solving the Schrödinger Equation. Mathematical Notes 76, 564–577 (2004). https://doi.org/10.1023/B:MATN.0000043486.02521.da

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