On Evgrafov–Fedoryuk’s theory and quadratic differentials

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• 作者：Boris Shapiro
• 关键词：Spectral asymptotics ; Quadratic differentials ; Singular planar metric ; Geodesics ; Primary 34M40 ; 34E20 ; Secondary 34E10
• 刊名：Analysis and Mathematical Physics
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：5
• 期：2
• 页码：171-181
• 全文大小：457 KB
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  7.Evgrafov, M., Fedoryuk, M.: Asymptotic behavior of solutions of the equation \(w^{\prime \prime }(z)-p(z,\,\lambda )w(z)=0\) as \(\lambda \rightarrow \infty \) in the complex \(z\) -plane (Russian). Usp. Math. Nauk 21(1), 3-0 (1966)MATH
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  14.Jenkins, J.A.: Univalent functions and conformal mapping. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe: Moderne Funktionentheorie. Springer, Berlin (1958)
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  18.Shin, K. C.: Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT -Symmetry, Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 6, 015 (2010)
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• 作者单位：Boris Shapiro (1)
  
  1. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1664-235X

文摘

The purpose of this note is to recall the theory of the (homogenized) spectral problem for Schr?dinger 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 \(d \atopwithdelims ()2\).