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Estimations asymptotiques des intervalles d’instabilité pour l’équation de Hill. (Asymptotic estimates for the intervals of unstability for the Hill equation). (French) Zbl 0644.34021
Relationships between the length decrease \(\gamma_ n\) of the intervals where the solutions of Hill equation are not bounded and the regularity properties of the potential V(x) have been studied previously [see for ex. M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, (1978; Zbl 0401.47001)]. In this paper an asymptotic majoration of \(\gamma_ n\) for \(n\to \infty\) is given for V(x) real analytic. For V(x) a trigonometric polynomial, an asymptotic development is obtained as modul of a finite sum of exponentially small terms. The use method consists on studying the equation in the complex domain and applying the WKB method. Turning points play a fundamental role.
Reviewer: A.de Castro

MSC:
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
30E15 Asymptotic representations in the complex plane
34D99 Stability theory for ordinary differential equations
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