Compressed Modes for Variational Problems in Mathematics and Physics

TL;DR

This paper introduces a formalism for deriving localized solutions to variational problems in physics and mathematics, enabling better analysis and basis construction for differential operators like the Laplacian.

Contribution

It presents a novel approach to obtain localized solutions and construct localized bases for differential operators, extending traditional methods like plane waves.

Findings

Provides a general formalism for localized solutions in variational problems

Develops a localized basis spanning eigenspaces of differential operators

Applicable to Schrödinger's equation and electromagnetic problems

Abstract

This paper describes a general formalism for obtaining localized solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems. This class includes the important cases of Schr\"odinger's equation in quantum mechanics and electromagnetic equations for light propagation in photonic crystals. These ideas can also be applied to develop a spatially localized basis that spans the eigenspace of a differential operator, for instance, the Laplace operator, generalizing the concept of plane waves to an orthogonal real-space basis with multi-resolution capabilities.