Spectrum of One-Dimensional Potential Perturbed by a Small Convolution Operator: General Structure

D. I. Borisov (), A. L. Piatnitski and E. A. Zhizhina
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D. I. Borisov: Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa 450008, Russia
A. L. Piatnitski: Faculty of Engineering Science and Technology, UiT The Arctic University of Norway, P.O. Box 385, 8505 Narvik, Norway
E. A. Zhizhina: Faculty of Engineering Science and Technology, UiT The Arctic University of Norway, P.O. Box 385, 8505 Narvik, Norway

Mathematics, 2023, vol. 11, issue 19, 1-26

Abstract: We consider an operator of multiplication by a complex-valued potential in L 2 ( R ) , to which we add a convolution operator multiplied by a small parameter. The convolution kernel is supposed to be an element of L 1 ( R ) , while the potential is a Fourier image of some function from the same space. The considered operator is not supposed to be self-adjoint. We find the essential spectrum of such an operator in an explicit form. We show that the entire spectrum is located in a thin neighbourhood of the spectrum of the multiplication operator. Our main result states that in some fixed neighbourhood of a typical part of the spectrum of the non-perturbed operator, there are no eigenvalues and no points of the residual spectrum of the perturbed one. As a consequence, we conclude that the point and residual spectrum can emerge only in vicinities of certain thresholds in the spectrum of the non-perturbed operator. We also provide simple sufficient conditions ensuring that the considered operator has no residual spectrum at all.

Keywords: convolution operator; potential; perturbation; spectrum; emerging eigenvalues (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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