Compressed Modes for Variational Problems in Mathematics and Physics

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.