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On Evgrafov–Fedoryuk’s theory and quadratic differentials

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Abstract

The purpose of this note is to recall the theory of the (homogenized) spectral problem for Schrödinger equation with a polynomial potential and its relation with quadratic differentials. We derive from results of this theory that the accumulation rays of the eigenvalues of the latter problem are in \(1-1\)-correspondence with the short geodesics of the singular planar metrics induced by the corresponding quadratic differential. We prove that for a polynomial potential of degree \(d,\) the number of such accumulation rays can be any positive integer between \((d-1)\) and \(d \atopwithdelims ()2\).

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Acknowledgments

I am grateful to Professors Y. Baryshnikov, A. Zorich and my former Ph.D. student T. Holst for discussions around this topic. I want to thank the anonymous referee for constructive criticism which helped to improve the quality of the exposition.

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  1. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Boris Shapiro

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  1. Boris Shapiro
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Shapiro, B. On Evgrafov–Fedoryuk’s theory and quadratic differentials. Anal.Math.Phys. 5, 171–181 (2015). https://doi.org/10.1007/s13324-014-0092-y

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