Perfect absorption in Schr\"odinger-like problems using non-equidistant complex grids

TL;DR

This paper introduces two non-equidistant grid methods for perfect absorption in the time-dependent Schrödinger equation, demonstrating efficient and robust absorption comparable to basis set discretizations, with exponential convergence and high accuracy.

Contribution

It presents novel non-equidistant grid implementations for exterior complex scaling that achieve perfect absorption, contrasting with previous literature, and introduces generalized Q-point schemes for finite differences.

Findings

Achieves exponential convergence of absorption with grid spacing.

Attains local relative errors around 10^{-9} in ionization problems.

Demonstrates efficiency comparable to basis set discretizations.

Abstract

Two non-equidistant grid implementations of infinite range exterior complex scaling are introduced that allow for perfect absorption in the time dependent Schr\"odinger equation. Finite element discrete variables grid discretizations provide as efficient absorption as the corresponding finite elements basis set discretizations. This finding is at variance with results reported in literature [L. Tao et al., Phys. Rev. A 48, 063419 (2009)]. For finite differences, a new class of generalized $Q$-point schemes for non-equidistant grids is derived. Convergence of absorption is exponential $\sim \Delta x^{Q-1}$ and numerically robust. Local relative errors $\sim10^{-9}$ are achieved in a standard problem of strong-field ionization.