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On the complex WKB analysis for a second order linear O.D.E. with a many-segment characteristic polygon. (English) Zbl 1082.34081
This paper concerns the one-dimensional Schrödinger equation $\varepsilon^{2h}{d^2y\over dx^2}= Q(x,\varepsilon) y,\quad Q(x,\varepsilon):= \sum^h_{j=0} a_j\varepsilon^j x^{m_j};$
$m_j:= {(h- j+1)(h-j)\over 2},\quad 0\neq a_j\in\mathbb C\;\forall j,$
$h= 2,3,4,\dots;\quad x,y\in\mathbb{C};\quad 0< \varepsilon\leq \varepsilon_0;\quad D: 0\leq|x|\leq x_0.$ The asymptotic behavior at the turning point $$x= 0$$ is studied when the characteristic polygon has many segments.
##### 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
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