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Journal of Spectral Theory

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Volume 9, Issue 2, 2019, pp. 513–546
DOI: 10.4171/JST/254

Published online: 2018-10-24

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

Horst Behncke[1] and Don B. Hinton[2]

(1) Universität Osnabrück, Germany
(2) University of Tennessee, Knoxville, USA

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.

Keywords: $m$-functions, singular operators, essential spectrum, non-selfadjoint operators, $\mathcal C$-symmetric Hamiltonian systems, Green's functions

Behncke Horst, Hinton Don: $\mathcal C$-symmetric Hamiltonian systems with almost constant coefficients. J. Spectr. Theory 9 (2019), 513-546. doi: 10.4171/JST/254