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Quantization conditions on riemannian surfaces and the semiclassical spectrum of the Schrödinger operator with complex potential

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Abstract

We describe the asymptotics of the spectrum of the operator

$$ \hat H\left( {x, - \imath h\frac{\partial } {{\partial x}}} \right) = - h^2 \frac{{\partial ^2 }} {{\partial x^2 }} + \imath \left( {\cos x + \cos 2x} \right) $$

as h → 0 and show that the spectrum concentrates near some graph on the complex plane. We obtain equations for the eigenvalues, which are conditions on the periods of a holomorphic form on the corresponding Riemannian surface.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Moscow, Russia

    A. I. Esina

  2. Moscow State University, Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow, Russia

    A. I. Shafarevich

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  1. A. I. Esina
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  2. A. I. Shafarevich
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Correspondence to A. I. Esina.

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Original Russian Text © A. I. Esina, A. I. Shafarevich, 2010, published in Matematicheskie Zametki, 2010, Vol. 88, No. 2, pp. 229–248.

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Esina, A.I., Shafarevich, A.I. Quantization conditions on riemannian surfaces and the semiclassical spectrum of the Schrödinger operator with complex potential. Math Notes 88, 209–227 (2010). https://doi.org/10.1134/S0001434610070205

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  • DOI: https://doi.org/10.1134/S0001434610070205

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