On the spectrum of convolution operator with a potential

This paper focuses on the spectral properties of a bounded self-adjoint operator in L2(Rd) being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the dis...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2023-01, Vol.517 (1), p.126568, Article 126568
Main Authors: Borisov, D.I., Piatnitski, A.L., Zhizhina, E.A.
Format: Article
Language:English
Subjects:
Convolution operator
Discrete spectrum
Essential spectrum
Minimax principle
Potential
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Summary:This paper focuses on the spectral properties of a bounded self-adjoint operator in L2(Rd) being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower and upper bounds for the number of discrete eigenvalues. Special attention is paid to the case of operators possessing countably many points of the discrete spectrum. We also compare the spectral properties of the operators considered in this work with those of classical Schrödinger operators.
ISSN:0022-247X
1096-0813
1096-0813
DOI:10.1016/j.jmaa.2022.126568