Absorbing Boundary Conditions for Variable Potential Schr\"odinger Equations via Titchmarsh-Weyl Theory

TL;DR

This paper introduces a new method for simulating the Schrödinger equation with variable potentials by approximating the Titchmarsh-Weyl m-function, enabling accurate and efficient absorbing boundary conditions across all frequencies.

Contribution

It presents a novel rational approximation of the Titchmarsh-Weyl m-function to improve absorbing boundary conditions for variable potential Schrödinger equations.

Findings

Overcomes high-frequency restrictions in boundary conditions.

Provides a fast and accurate computational algorithm.

Ensures stability and precision over the entire frequency spectrum.

Abstract

We propose a novel approach to simulate the solution of the time-dependent Schr\"odinger equation with a general variable potential. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to an appropriate spectral parameter. By using this method, we overcome the usual high-frequency restriction associated with absorbing boundary conditions in general variable potential problems. The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequency range.