Absorbing Boundary Conditions for Variable Potential Schrödinger Equations via Titchmarsh-Weyl Theory (Communications in Computational Physics, 2025)
We propose a novel approach to simulate the solution of the time-dependentSchrödinger equation with a general variable potential. The key idea is to approximate the Titchmarsh-Weyl $m$-function (exact Dirichlet-to-Neumann operator) by a rationalfunction with respect to an appropriate spectral parameter. By using this method, weovercome the usual high-frequency restriction associated with absorbing boundaryconditions in general variable potential problems. The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequencyrange.
Keywords: Absorbing boundary conditions; variable potential; Schrödinger equation; Titchmarsh-Weyl $m$-function; unbounded domain.