A fast solution method for time dependent multidimensional Schrödinger equations
This work provides an efficient numerical method for solving high-dimensional Schrödinger equations, which is a known bottleneck in computational physics and quantum mechanics.
The paper proposes fast solution methods for the multidimensional Schrödinger equation using basis functions from approximate approximations, achieving high-order approximations up to order 6 in up to 200 dimensions with negligible saturation error.
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.