Geodesic
Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schr\"odinger Operator with Complex Potential
Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 229-248

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We describe the asymptotics of the spectrum of the operator $$ \widehat H\biggl(x,-\imath h\frac{\partial}{\partial x}\biggr)=-h^2\frac{\partial^2}{\partial x^2}+\imath(\cos x+\cos2x) $$ as $h\to0$ 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.
Keywords: Schrödinger operator, semiclassical spectrum of an operator, Riemannian surface, holomorphic form, Stokes line, turning point.
Mots-clés : quantization condition, monodromy matrix 
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     author = {A. I. Esina and A. I. Shafarevich},
     title = {Quantization {Conditions} on {Riemannian} {Surfaces} and the {Semiclassical} {Spectrum} of the {Schr\"odinger} {Operator} with {Complex} {Potential}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {229--248},
     publisher = {mathdoc},
     volume = {88},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_2_a6/}
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A. I. Esina; A. I. Shafarevich. Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schr\"odinger Operator with Complex Potential. Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 229-248. http://geodesic.mathdoc.fr/item/MZM_2010_88_2_a6/