A preconditioned iterative solver for the scattering solutions of the Schr\"odinger equation

TL;DR

This paper introduces a preconditioned iterative solver for the Schrödinger equation's scattering solutions, transforming it into a coupled system with Helmholtz problems, and demonstrates its effectiveness through numerical experiments.

Contribution

It develops a novel multigrid preconditioner with a spectrum restricted to a quadrant, suitable for indefinite Helmholtz problems derived from the Schrödinger equation.

Findings

Preconditioner effectively restricts spectrum to a quadrant.

Compatible with Krylov subspace methods for indefinite Helmholtz problems.

Numerical results show the preconditioner's potential for automatic solution settings.

Abstract

The Schr\"odinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schr\"odinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjuction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasbility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.