Abstract
We discuss the difference Schrödinger operator on a line with a periodic, possibly complex, potential. We show that this operator has no eigenvalues. The proof is based on the use of the notion of Bloch solutions introduced by Buslaev and Fedotov for difference equations on a line.
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Acknowledgments
The author discussed this work with V. S. Buslaev when the study of non-self-adjoint problems did not yet seem to be relevant. In the self-adjoint case, the “instantaneous” proof of Theorem 1 (given after its statement) was pointed out to V. S. Buslaev by B. Simon. When preparing the paper, the author discussed it with M. A. Lyalinov. The author is very grateful to V. S. Buslaev and M. A. Lyalinov.
Funding
This work is supported by the Russian Science Foundation (grant No. 22-11-00092), https://rscf.ru/project/22-11-00092/.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 213, pp. 450–458 https://doi.org/10.4213/tmf10346.
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Fedotov, A.A. On the absence of eigenvalues of the difference Schrödinger operator on a line with a periodic potential. Theor Math Phys 213, 1698–1705 (2022). https://doi.org/10.1134/S0040577922120042
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DOI: https://doi.org/10.1134/S0040577922120042