Pontryagin's Theorem and Spectral Stability Analysis of Solitons

The main result of the present paper is the use of Pontryagin's theorem for proving a criterion, based on the difference in the number of negative eigenvalues between two self-adjoint operators $L_-$ and $L_+$, for the linear part of a Hamiltonian system to have eigenvalues with strictly positive real part (unstable eigenvalues).