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Spectra of operator pencils with small \(\mathcal{PT}\)-symmetric periodic perturbation. (English) Zbl 1444.47023

The paper deals with spectral properties of an operator pencil with a small \(\mathcal{PT}\)-perturbation and a fixed localized potential. It is shown that the continuous spectrum has a bounded structure. The isolated eigenvalues converging to the continuous spectrum are studied. The asymptotic behaviour of these eigenvalues with respect to a small parameter is found.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
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