# Localized computation of eigenstates of random Schr\"odinger operators

## TL;DR

This paper introduces a new numerical scheme for approximating localized low-energy eigenstates of random Schrödinger operators, leveraging a preconditioned inverse iteration with multigrid methods for efficient local computation.

## Contribution

The paper presents a novel localized numerical method for eigenstates of random Schrödinger operators, combining inverse iteration with multigrid solvers for improved efficiency.

## Key findings

- Method effectively approximates localized eigenstates.
- Numerical experiments demonstrate good performance in 2D and 3D.
- Applicable to nonlinear random Schrödinger operators.

## Abstract

This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schr\"odinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates localize in the sense of an exponential decay of their moduli. We propose a reliable numerical scheme which provides localized approximations of such localized states. The method is based on a preconditioned inverse iteration including an optimal multigrid solver which spreads information only locally. The practical performance of the approach is illustrated in various numerical experiments in two and three space dimensions and also for a non-linear random Schr\"odinger operator.