## 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

### Keywords:

Hill equation; WKB method; Turning points

Zbl 0401.47001
Full Text:

### References:

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