Abstract
In the paper, the review of the papers [26,27,28,29,30, 32,33,34] devoted to the semiclassical asymptotic behavior of the eigenvalues of some nonself-adjoint operators important for applications is given. These operators are the Schrödinger operator with complex periodic potential and the operator of induction. It turns out that the asymptotics of the spectrum can be calculated using the quantization conditions, which can be represented as the condition that the integrals of a holomorphic form over the cycles on the corresponding complex Lagrangian manifold, which is a Riemann surface of constant energy, are integers. In contrast to the real case (the Bohr–Sommerfeld–Maslov formulas), to calculate a chosen spectral series, it is sufficient to assume that the integral over only one of the cycles takes integer values, and different cycles determine different parts of the spectrum.
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Acknowledgements
I thank the referee for very useful comments. The research was supported by the Russian Scientific Foundation (grant 16-11-10069).
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Shafarevich, A. (2018). Quantization Conditions on Riemannian Surfaces and Spectral Series of Non-selfadjoint Operators. In: Filipuk, G., Lastra, A., Michalik, S. (eds) Formal and Analytic Solutions of Diff. Equations . FASdiff 2017. Springer Proceedings in Mathematics & Statistics, vol 256. Springer, Cham. https://doi.org/10.1007/978-3-319-99148-1_9
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