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Symbolic algorithm for factorization of the evolution operator of the time-dependent Schrödinger equation

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Abstract

A symbolic algorithm for the decomposition of the unitary evolution operator is developed. This algorithm allows one to generate multilayer implicit schemes for solution of the time-dependent Schrödinger equation. Some additional gauge transformations are also implemented in the algorithm. This allows one to distinguish symmetric operators, which are required for constructing efficient evolutionary schemes. The efficiency of the generated schemes is demonstrated by integrable models.

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References

  1. Derbov, V.L., Melnikov, L.A., Umansky, I.M., and Vinitsky, S.I., Multipulse Laser Spectroscopy of p-bartHe+: Measurement and Control of the Metastable State Populations, Phys. Rev. A, 1997, vol. 55, pp. 3394–3400.

    Article  Google Scholar 

  2. Butkovskiy, A.G. and Samoilenko, Yu.I., Control of Quantum-Mechanical Processes and Systems, Dordrecht Hardbound: Kluwer, 1990.

    Google Scholar 

  3. Bankov, N.G., Kaschiev, M.S., and Vinitsky, S.I., Adaptive Method for Solving the Time-Dependent Schrödinger Equation, Comptes rendus de l’Akademie bulgare des Sciences, 2002, vol. 55, pp. 25–30.

    MathSciNet  Google Scholar 

  4. Derbov, V.L., Kaschiev, M.S., Serov, V.V., Gusev, A.A., and Vinitsky, S.I., Adaptive Numerical Methods for Time-Dependent Schroedinger Equation in Atomic and Laser Physics, Proc. SPIE, 2003, vol. 5069, pp. 52–60.

    Google Scholar 

  5. Marchuk, G.I., Partial Differential Equations: II SYNSPADE-1970, New York: Academic, 1971.

    Google Scholar 

  6. Samarskii, A.A., Teoriya raznostnykh skhem (The Theory of Difference Schemes), Moscow: Nauka, 1977.

    Google Scholar 

  7. Strang, G. and Fix, G., An Analysis of the Finite Element Method, Englewood Cliffs: Prentice-Hall, 1973.

    Google Scholar 

  8. Bathe, K.J., Finite Element Procedures in Engineering Analysis, New York: Englewood Cliffs, Prentice Hall, 1982.

    Google Scholar 

  9. Abrashkevich, A.G., Abrashkevich, D.G., Kaschiev, M.S., and Puzynin, I.V., Finite-Element Solution of the Coupled-Channel Schrödinger Equation Using High-Order Accuracy Approximations, Comput. Phys. Commun., 1995, vol. 85, pp. 40–65.

    Google Scholar 

  10. Abrashkevich, A.G., Kaschiev, M.S., and Vinitsky, S.I., A New Method for Solving an Eigenvalue Problem for a System of Three Coulomb Particles within the Hyperspherical Adiabatic Representation, J. Comput. Phys., 2000, vol. 163, pp. 328–348.

    Article  MathSciNet  Google Scholar 

  11. Gusev, A.A., Chekanov, N.A., Rostovtsev, V.A., Vinitsky, S.I., and Uwano, Y., A Comparison of Algorithms for the Normalization and Quantization of Polynomial Hamiltonians, Programmirovanie, 2004, vol. 30, no. 2, pp. 27–36 [Programming and Comput. Software. (Engl. Transl.), 2004, vol. 30, no. 2, pp. 75–82].

    MathSciNet  Google Scholar 

  12. Ukolov, Yu.A., Chekanov, N.A., Gusev, A.A., Rostovtsev, V.A., Vinitsky, S.I., and Uwano, Y., LINA01: A REDUCE Program for the Normalization of Polynomial Hamiltonians, Comput. Phys. Commun., 2005, vol. 166, pp. 66–80.

    MathSciNet  Google Scholar 

  13. Puzynin, I.V., Selin, A.V., and Vinitsky, S.I., A High-Order Accuracy Method for Numerical Solving of the Time-Dependent Schrödinger Equation, Comput. Phys. Commun., 1999, vol. 123, pp. 1–6.

    MathSciNet  Google Scholar 

  14. Puzynin, I.V., Selin, A.V., and Vinitsky, S.I., Magnus-Factorized Method for Numerical Solving the Time-Dependent Schrödinger Equation, Comput. Phys. Commun., 2000, vol. 126, pp. 158–161.

    Google Scholar 

  15. Selin, A.V., A Method for Approximate Analysis of a Linear Evolutionary Equation in a Hilbert Space, Zh. Vychisl. Mat. Mat. Fiz., 2002, vol. 42, pp. 937–949 [Comp. Math. Math. Phys. (Engl. Transl.), 2002, vol. 42, pp. 901–914].

    MATH  MathSciNet  Google Scholar 

  16. Wilcox, R.M., Exponential Operators and Parameter Differentiation in Quantum Physics, J. Math. Phys., 1967, vol. 8, pp. 962–982.

    Article  MATH  MathSciNet  Google Scholar 

  17. Magnus, W., On the Exponential Solution of Differential Equations for a Linear Operator, Commun. Pure Appl. Math., 1954, vol. 7, pp. 649–673.

    MATH  MathSciNet  Google Scholar 

  18. Crank, J. and Nicholson, P., A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type, Proc. Cambridge Philos. Soc., 1947, vol. 43, pp. 50–67.

    MathSciNet  Google Scholar 

  19. Berezin, I.S. and Zhidkov, N.P., Metody vychislenii (Numerical Analysis), vol. 1, Moscow: Fizmatlit, 1959.

    Google Scholar 

  20. http://www.nag.co.uk/numeric/numerical_libraries.asp.

  21. Flügge, S., Practical Quantum Mechanics, Heidelberg: Springer, 1971.

    Google Scholar 

  22. Kalitkin, N.N., Chislennye metody (Numerical Analysis), Moscow: Nauka, 1978.

    Google Scholar 

  23. Jannussis, A., Quantum Equations of Motion in the Finite-Difference Approach, Lett. al Nuovo Cim., 1984, vol. 40, pp. 250–256.

    MathSciNet  Google Scholar 

  24. Jannussis, A., Difference Equations in the Lie-Admissible Formulation, Lett. al Nuovo Cim., 1985, vol. 42, pp. 129–133.

    MathSciNet  Google Scholar 

  25. Fizicheskaya entsiklopediya (Encyclopedia of Physics), vol. 5, Moscow: Bol’shaya Rossiiskaya entsiklopediya, 1998.

  26. Olver, P.J., Application of Lie Groups to Differential Equations, New York: Springer, 1986.

    Google Scholar 

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Original Russian Text © S.I. Vinitsky, V.P. Gerdt, A.A. Gusev, M.S. Kaschiev, V.A. Rostovtsev, V.N. Samoylov, T.V. Tupikova, Y. Uwano, 2006, published in Programmirovanie, 2006, Vol. 32, No. 2.

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Vinitsky, S.I., Gerdt, V.P., Gusev, A.A. et al. Symbolic algorithm for factorization of the evolution operator of the time-dependent Schrödinger equation. Program Comput Soft 32, 103–113 (2006). https://doi.org/10.1134/S0361768806020083

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  • DOI: https://doi.org/10.1134/S0361768806020083

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