Solving Time-Dependent Equations of Schrödinger-Type Using Mapped Infinite Elements

Abstract

We present a general technique to solve one-dimensional time-dependent Schrödinger-type equations. A mapped infinite elements approach is used to eliminate spurious reflections of outgoing wave packets from the boundaries of the interval of interest. This procedure leads to more precise solutions because the space coordinates are discretized to approximate the solution in the entire physical domain. We show its utility giving numerical results on three typical examples: a bounded short-range potential; the unbounded potential of a particle in a dc field; and a Hamiltonian with an intrinsic time dependence like the Hamiltonian of a charged particle in an ac electromagnetic field.

Multidomain spectral method for Schrödinger equations

“…which means we use a Möbius transformation to map the infinite interval I to the compact interval [−1, 1]. This approach is similar to infinite elements in finite elements methods, see [14] for Schrödinger equations. As before, l is sampled on N I + 1, N I ∈ N collocation points (3), and the corresponding differentiation matrix D I is introduced.…”

“…Note that we essentially use the coordinate s = 1/x in the compactified domain. With this coordinate, equation (1) takes the form (14) i∂…”

“…Because of this singular behavior, it is not necessary to impose a boundary condition at s = 0. Regular solutions at infinity satisfy with (14) i∂ t u(0, t) + V u(0, t) = 0. However, it is possible to impose boundary values compatible with the solution there via a τ -method, for instance the vanishing of the solution at infinity for asymptotically vanishing solutions.…”

“…But since we are also interested in solutions which do not vanish for x → ∞ as the Peregrine solution (2), we do not impose conditions at infinity. Note that it is straight forward to implement (14) with the differentiation matrices discussed above. All one has to do is to approach the operator s 4 ∂ ss + 2s 3 ∂ s which can be done with the same amount of work as on a finite domain.…”

“…A very efficient variant of the latter mainly used in astrophysics, see for instance [34], are multidomain methods with compactified exterior domains (CED). As in so called infinite element approaches, see for instance [14] for Schrödinger equations, the idea is to map the exterior domain with some Möbius transformation to a finite domain. In this paper, we apply a multidomain spectral approach with CED to Schrödinger equations and compare this to TBC and PML approaches for concrete examples.…”