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

Author

Listed:
  • 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
    Institute for Information Transmission Problem (Kharkevich Institute) of RAS, Bolshoy Karetny per. 19, Build. 1, Moscow 127051, Russia)

  • E. A. Zhizhina

    (Faculty of Engineering Science and Technology, UiT The Arctic University of Norway, P.O. Box 385, 8505 Narvik, Norway
    Institute for Information Transmission Problem (Kharkevich Institute) of RAS, Bolshoy Karetny per. 19, Build. 1, Moscow 127051, Russia)

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.

Suggested Citation

  • D. I. Borisov & A. L. Piatnitski & E. A. Zhizhina, 2023. "Spectrum of One-Dimensional Potential Perturbed by a Small Convolution Operator: General Structure," Mathematics, MDPI, vol. 11(19), pages 1-26, September.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4042-:d:1246341
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