$\mathcal{C}$-symmetric Hamiltonian systems with almost constant coefficients

Abstract

We consider a $\mathcal{C}$-Symmetric Hamiltonian System of differential equations on a half interval or the real line. We determine the spectrum and construct the resolvent for the system. The essential spectrum is found to be a subset of an algebraic curve $\Sigma$ defined by a characteristic polynomial for the system. The results are first proved for a constant coefficient system and then for an almost constant coefficient system. The results are applied to a number of examples including the complex hydrogen atomand the complex relativistic electron.