High order schemes for solving the time-dependent Schr\"odinger equation
D. A. Volobueva1,2, A. A. Gusev 1,3,4, O. Chuluunbaatar,1,4,5, V. L. Derbov6, S. I. Vinitsky1,7, 1Joint Institute for Nuclear Research, Dubna, Russia; 2 Dubna branch of Lomonosov Moscow State University, Dubna, Russia; 3 Dubna State University, Dubna, Russia; 4Mongolian University of Science and Technology, Ulaanbaatar, Mongolia; 5 Institute of Mathematics and Digital Technology, Ulaanbaatar, Mongolia; 6Saratov State University, Saratov, Russia; 7 RUDN University, Moscow, Russia
Abstract
Implicit high-order operator-difference schemes for solving initial-boundary value problems (IBVPs) for the time-dependent Schrödinger equation (TSDE) are constructed. These multilayer schemes are based on the Magnus expansion of the evolution operator and Padé approximations. The solution is discretized with respect to the spatial variable using the finite element method with Hermite interpolation polynomials, ensuring the required smoothness of the solution and the symmetry property of the matrices of the corresponding algebraic problems. The convergence of the proposed schemes is demonstrated using benchmark IBVP calculations for the TSDE, which describes the evolution of several quantum systems.
Speaker
Vladimir L. Derbov
Saratov State University
Russia
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