On the eigenfunction expansion for the Hamilton operators

TL;DR

This paper develops a spectral representation for solutions to linear Hamilton equations with nonnegative energy, extending previous work and applying Krein's spectral theory to justify eigenfunction expansions in nonlinear relativistic equations.

Contribution

It introduces a spectral representation for Hamilton equations with nonnegative energy using Krein's theory, advancing the mathematical framework for such systems.

Findings

Spectral representation for Hamilton equations established

Eigenfunction expansion justified for nonlinear relativistic Ginzburg-Landau equations

Extension of previous positive definite energy results

Abstract

A spectral representation for solutions to linear Hamilton equations with nonnegative energy in Hilbert spaces is obtained. This paper continues our previous work on Hamilton equations with positive definite energy. Our approach is a special version of M. Krein's spectral theory of $J$-selfadjoint operators in Hilbert spaces with indefinite metric. As a principal application of these results, we justify the eigenfunction expansion for linearized nonlinear relativistic Ginzburg-Landau equations.