Numerical solutions of the time-dependent Schrödinger equation with position-dependent effective mass. (English) Zbl 1535.65165
Summary: Numerical solution of the time-dependent Schrödinger equation with a position-dependent effective mass is challenging to compute due to the presence of the non-constant effective mass. To tackle the problem we present operator splitting-based numerical methods. The wavefunction will be propagated either by the Krylov subspace method-based exponential integration or by an asymptotic Green’s function-based time propagator. For the former, the wavefunction is given by a matrix exponential whose associated matrix-vector product can be approximated by the Krylov subspace method; and for the latter, the wavefunction is propagated by an integral with retarded Green’s function that is approximated asymptotically. The methods have complexity \(O(N\log N)\) per step with appropriate algebraic manipulations and fast Fourier transform, where \(N\) is the number of spatial points. Numerical experiments are presented to demonstrate the accuracy, efficiency, and stability of the methods.
{© 2023 Wiley Periodicals LLC.}
{© 2023 Wiley Periodicals LLC.}
MSC:
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 65F25 | Orthogonalization in numerical linear algebra |
| 65F55 | Numerical methods for low-rank matrix approximation; matrix compression |
| 65F60 | Numerical computation of matrix exponential and similar matrix functions |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
| 65M80 | Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
Keywords:
asymptotic Green’s function; effective mass; exponential integration; fast Fourier transform; Krylov subspace; Schrödinger equation; WKBJSoftware:
expmARPACKReferences:
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