On eigenfunction expansion of solutions to the Hamilton equations

TL;DR

This paper develops a spectral representation for solutions to linear Hamilton equations using Krein's spectral theory, and applies it to eigenfunction expansion of the linearized relativistic Ginzburg-Landau equation.

Contribution

It introduces a spectral approach for Hamilton equations based on Krein's theory and applies it to a specific relativistic PDE.

Findings

Established spectral representation for Hamilton equations

Applied eigenfunction expansion to relativistic Ginzburg-Landau equation

Extended Krein's spectral theory to indefinite metric spaces

Abstract

We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein's spectral theory of J-selfadjoint operators is the Hilbert spaces with an indefinite metric. Our main result is an application to the eigenfunction expansion for the linearized relativistic Ginzburg-Landau equation.