On a one-dimensional quadratic operator pencil with a small periodic perturbation

We consider a quadratic operator pencil with a small periodic perturbation multiplied by the spectral parameter. It is motivated, in particular, by a one-dimensional Klein-Gordon equation with a time-parity-symmetric perturbation. We study in details the structure of the considered operator pencil. We show that its essential spectrum has a band structure and at certain thresholds, the bands bifurcate into small parabolas. We then study how the isolated limiting eigenvalues behave under the perturbation. We show that if zero is a limiting isolated eigenvalue, under the perturbation it remains an eigenvalue but an additional isolated eigenvalue can emerge from zero. The most part of the paper is devoted to studying the isolated eigenvalues converging to the essential spectrum. We establish sufficient conditions for the existence and absence of such eigenvalues and in the case of the existence, we calculate the leading terms of their asymptotic expansions.