Abstract
Bloch solutions of the difference Schrödinger equation with periodic complex potential on the real line are discussed. The case where the spectral parameter is outside the spectrum of the corresponding Schrödinger operator is considered.
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Notes
For \(\omega\in \mathbb{Q}\), these observations are obvious, and for \(\omega\notin\mathbb{Q}\), they follow from Weyl’s version of the ergodic theorem [6]. Moreover, in the latter case, the limits do not depend on \(x\) and equal \(\frac1\omega\int_{0}^\omega\ln|\lambda(x)|\,dx\).
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Funding
The research of A. A. Fedotov was supported by the Russian Science Foundation, project no. 22-11-00092, https://rscf.ru/project/22-11-00092/.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 3–16 https://doi.org/10.4213/faa4018.
Translated by O. V. Sipacheva
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Borisov, D.I., Fedotov, A.A. On Bloch Solutions of Difference Schrödinger Equations. Funct Anal Its Appl 56, 239–250 (2022). https://doi.org/10.1134/S0016266322040013
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DOI: https://doi.org/10.1134/S0016266322040013