Noncompact Lagrangian manifolds corresponding to the spectral series of the Schrödinger operator with delta-potential on a surface of revolution
type: Item
instance of: scholarly article
title: Noncompact Lagrangian manifolds corresponding to the spectral series of the Schrödinger operator with delta-potential on a surface of revolution
zbMATH Open document ID: 1266.58015
DOI: 10.1134/S1064562412050365
author: T. S. Ratiu
author: T. A. Filatova
author: A. I. Shafarevich
published in: Doklady Mathematics
publication date: 2013-01-14T00:00:00Z
review text: Let \(H:=-{h^2\over2}\Delta+\alpha\delta(x-x_0)\) for \(\alpha\in R\) be the Schrödinger operator on a compact surface of revolution diffeomorphic to the sphere. The authors show: { Theorem}. Suppose that there exists a number \(E=O(1)\) satisfying the quantization condition \[ \tan\left({1\over2h}\int(p,dx)+O(h)\right)={2\over\pi}\left(\ln\left({\sqrt{2E}\over h}\right)+{\pi h^2\over\alpha}+c\right) \] where \(\gamma\) is any orbit of the geodesic flow on the Lagrangian manifold \(\Lambda\) and \(c\) is the Euler constant. Then there exists an eigenvalue \(E_0\) of the operator \(H\) for which \(|E-E_0|=o(h)\) as \(\eta\rightarrow0\). The corresponding eigenfunction has the form \(\Psi=K_\Lambda(1)+O(h)\). The proof of the theorem is based on examining the monodromy of the equation to which the spectral problem for the operator \(H\) reduces after the separation of variables. There are two regular singular points corresponding to the poles of the surface, and both monodromy operators related to these points are equivalent to Jordan cells. The WKB method of [\textit{M. V. Fedoryuk}, Asymptotic methods for linear ordinary differential equations. Moskva: ``Nauka'' (1983; Zbl 0538.34001)] and the Langer-Wasow method [\textit{W. Wasow}, Asymptotic expansions for ordinary differential equations. Moskau: `Mir' (1968; Zbl 0169.10903)] play a central role.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 6124966
zbMATH Keywords: monodromy
zbMATH Keywords: regular singular point
zbMATH Keywords: Lagrangian manifold
zbMATH Keywords: WKB method
zbMATH Keywords: Langer-Wasow method
MaRDI profile type: Publication
Wikidata QID: Q125862045
cites work: Q3160268
cites work: Q4938042
cites work: Q5343650
cites work: Q4040558
cites work: Q3896271
full work available at URL: s1064562412050365
OpenAlex ID: W2031478569
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description: scientific article; zbMATH DE number 6124966