A fast solution method for time dependent multidimensional Schrödinger equations

ABSTRACT

In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.

KEYWORDS:

• Schrödinger equation
• higher dimensions
• separated representations
• error estimates

AMS SUBJECT CLASSIFICATIONS:

• 65D32
• 35Q41
• 41A30
• 41A63

Acknowledgements

The publication was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.21.0008).

Notes

No potential conflict of interest was reported by the authors.

In memory of V. V. Zhikov.