Fast localization of eigenfunctions via smoothed potentials

TL;DR

This paper introduces a fast computational method to predict the localization of low-lying eigenfunctions in domains with rapidly varying potentials, improving efficiency while maintaining accuracy.

Contribution

A new fast algorithm based on the function 1/u that efficiently predicts eigenfunction localization, matching the accuracy of previous methods with significantly reduced computation time.

Findings

Computation time is approximately O(n^2 log n) for an n x n grid.

The method produces a landscape similar to existing approaches.

It accurately predicts the localization of eigenfunctions.

Abstract

We study the problem of predicting highly localized low-lying eigenfunctions $(-\Delta +V) \phi = \lambda \phi$ in bounded domains $\Omega \subset \mathbb{R}^d$ for rapidly varying potentials $V$. Filoche & Mayboroda introduced the function $1/u$, where $(-\Delta + V)u=1$, as a suitable regularization of $V$ from whose minima one can predict the location of eigenfunctions with high accuracy. We proposed a fast method that produces a landscapes that is exceedingly similar, can be used for the same purposes and can be computed very efficiently: the computation time on an $n \times n$ grid, for example, is merely $\mathcal{O}(n^2 \log{n})$, the cost of two FFTs.