  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.
