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On the WKB-theoretic structure of a Schrödinger operator with a merging pair of a simple pole and a simple turning point. (English) Zbl 1201.34141

Semiclassical analysis establishes the link between classical dynamics and the behavior as \(h\to 0\) of solutions of the Schrödinger equation
\[ ih\partial_t \psi = - h^2 \Delta \psi + Q\psi, \]
where \(h\) plays the role of Planck’s constant. In the present article, the authors perform an exact Wentzel-Kramers-Brillouin (WKB) analysis of the equation \[ \frac{d^2 \psi}{d x^2} - \eta^2Q(x, \eta) \psi = 0, \] where \(\eta\) is a large parameter (the reciprocal of \(h\)) and \(Q\) has a simple turning point and a simple pole, merging as \(\eta\to \infty\). This is done by converting the equation to a canonical (\(\infty\)-Whittaker) equation with \[ Q = \frac{1}{4} +\frac{\alpha}{x} +\eta^{-2}\frac{\gamma}{x^2}, \] the study of which has been announced in [T. Koike and Y. Takei On the Voros coefficient for the Whittaker equation with a large parameter: Some progress around Sato’s conjecture in exact WKB analysis, preprint (2010)].

MSC:

34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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