Spectra of operator pencils with small 𝒫𝒯-symmetric periodic perturbation

¹ Institute of Mathematics with Computer Center, Ufa Scientific Center, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa 450008, Russia.
² University of Hradec Králové 62, Rokitanského, Hradec Králové 50003, Czech Republic.
³ Bashkir State Pedagogical University, October Rev. Str. 3a, Ufa 450000, Russia.
⁴ Department of Engineering, University of Sannio, Corso Garibaldi, 107, 82100 Benevento, Italy.

^* Corresponding author: giuseppe.cardone@unisannio.it

Received: 28 May 2019
Accepted: 19 November 2019

Abstract

We study the spectrum of a quadratic operator pencil with a small 𝒫𝒯-symmetric periodic potential and a fixed localized potential. We show that the continuous spectrum has a band structure with bands on the imaginary axis separated by usual gaps, while on the real axis, there are no gaps but at certain points, the bands bifurcate into small parabolas in the complex plane. We study the isolated eigenvalues converging to the continuous spectrum. We show that they can emerge only in the aforementioned gaps or in the vicinities of the small parabolas, at most two isolated eigenvalues in each case. We establish sufficient conditions for the existence and absence of such eigenvalues. In the case of the existence, we prove that these eigenvalues depend analytically on a small parameter and we find the leading terms of their Taylor expansions. It is shown that the mechanism of the eigenvalue emergence is different from that for small localized perturbations studied in many previous works.