Noncompact Lagrangian manifolds corresponding to the spectral series of the Schrödinger operator with delta-potential on a surface of revolution
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.
scientific article; zbMATH DE number 6124966
scientific article