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Analysis, Probability, Applications, and Computation pp 27–36Cite as

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Cubature of Multidimensional Schrödinger Potential Based on Approximate Approximations

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Abstract

We report here on some recent results obtained in collaboration with Maz’ya and Schmidt (Appl Anal 98:408–429, 2019). We derive semi-analytic cubature formulas for the solution of the Cauchy problem for the Schrödinger equation which are fast and accurate also if the space dimension is greater than or equal to 3. We follow ideas of the method of approximate approximations, which provides high-order semi-analytic cubature formulas for many important integral operators of mathematical physics. The proposed method is very efficient in high dimensions if the data allow separated representations.

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References

  1. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968)

    MATH  Google Scholar 

  2. G. Beylkin, M. Mohlenkamp, Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26, 2133–2159 (2005)

    Article  MathSciNet  Google Scholar 

  3. W. Hackbusch, B.N. Khoromskij, E.E. Tyrtyshnikov, Hierarchical Kronecker tensor-product approximations. J. Numer. Math. 13, 119–156 (2005)

    Article  MathSciNet  Google Scholar 

  4. M.R. Hestenes, Extension of the range of differentiable functions. Duke Math. J. 8, 183–192 (1941)

    Article  MathSciNet  Google Scholar 

  5. B.N. Khoromskij, Tensors-structured numerical methods in scientific computing: survey on recent advances. Chemom. Intell. Lab. Syst. 110, 1–19 (2012)

    Article  Google Scholar 

  6. F. Lanzara, G. Schmidt, On the computation of high-dimensional potentials of advection–diffusion operators. Mathematika 61, 309–327 (2015)

    Article  MathSciNet  Google Scholar 

  7. F. Lanzara, V. Maz’ya, G. Schmidt, On the fast computation of high dimensional volume potentials. Math. Comp. 80(274), 887–904 (2011)

    Article  MathSciNet  Google Scholar 

  8. F. Lanzara, V. Maz’ya, G. Schmidt, Accurate cubature of volume potentials over high-dimensional half-spaces. J. Math. Sci. (New York) 173(6), 683–700 (2011)

    Article  MathSciNet  Google Scholar 

  9. F. Lanzara, V.G. Maz’ya, G. Schmidt, Computation of volume potentials over bounded domains via approximate approximations. J. Math. Sci. (New York) 189, 508–524 (2013)

    Article  MathSciNet  Google Scholar 

  10. F. Lanzara, V. Maz’ya, G. Schmidt, Fast cubature of volume potentials over rectangular domains by approximate approximations. Appl. Comput. Harmon. Anal. 36(1), 167–182 (2014)

    Article  MathSciNet  Google Scholar 

  11. F. Lanzara, V. Maz’ya, G. Schmidt, Approximation of solutions to multidimensional parabolic equations by approximate approximations. Appl. Comput. Harmon. Anal. 41, 749–767 (2016)

    Article  MathSciNet  Google Scholar 

  12. F. Lanzara, V. Maz’ya, G. Schmidt, Accurate computation of the high dimensional diffraction potential over hyper-rectangles. Bull. TICMI 22, 91–102 (2018)

    Google Scholar 

  13. F. Lanzara, V. Maz’ya, G. Schmidt, A fast solution method for time dependent multidimensional Schrödinger equations. Appl. Anal. 98, 408–429 (2019)

    Article  MathSciNet  Google Scholar 

  14. V. Maz’ya, G. Schmidt, Approximate approximations and the cubature of potentials. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6, 161–184 (1995)

    Google Scholar 

  15. V. Maz’ya, G. Schmidt, Approximate Approximations. Mathematical Surveys and Monographs, vol. 141 (American Mathematical Society, Providence, 2007)

    Chapter  Google Scholar 

  16. H. Takahasi, M. Mori, Doubly exponential formulas for numerical integration. Publ. RIMS, Kyoto Univ. 9, 721–741 (1974)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Sapienza University, Rome, Italy

    Flavia Lanzara

Authors
  1. Flavia Lanzara
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Corresponding author

Correspondence to Flavia Lanzara .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Karl‐Olof Lindahl

  2. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Torsten Lindström

  3. Department of Mathematics, University of Torino, Torino, Italy

    Luigi G. Rodino

  4. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Joachim Toft

  5. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Patrik Wahlberg

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Lanzara, F. (2019). Cubature of Multidimensional Schrödinger Potential Based on Approximate Approximations. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_3

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