Ricerc@Sapienza

We report here on some recent results obtained in collaboration with V. Maz'ya and G. Schmidt cite{LMS2017}.
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.

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.