## Sur l’équation de Hill analytique.(French)Zbl 0567.34023

Sémin. Équations Dériv. Partielles 1984-1985, Exp. No. 16, 12 p. (1985).
The Hill (Schrödinger) equation with periodic potential is investigated. This equation has the form: $$-u''(x)+(V(x)-E)u(x)=0$$, $$V(x)$$ is a periodic function with periodic $$\pi$$. The periodic eigenvalues are denoted by $$E_ n^{\pm}$$ if n is even and if n is odd then $$E_ n^{\pm}$$ denotes the antiperiodic eigenvalues, more precisely $$[E^+_{n-1},E_ n^-] = \{\lambda_ n(k);(- 1)^{n+1})k\in [0,1],\;n\geq 1\}$$, where $$\lambda_ n(k)$$ is the nth eigenvalue of the Floquet problem: $-u''+(V(x)-\lambda)u(x) = 0; \quad u(\pi) = u(0)\exp (ik\pi); \quad u'(\pi)=u'(0)\exp (ik\pi).$ The relation $$E^+_{n-1}\leq E^-_ n\leq E^+_ n\leq E^-_{n+1}$$ holds. The interval $$(E^-_ n,E^+_ n)$$ is called the interval of instability. Let $$\gamma_ n=E^+_ n-E^-_ n$$. The main result is an estimate for $$\gamma_ n$$. This estimate is applied to the Mathieu differential equation: $$-u''+(\mu \cos 2x-E)u(x)=0$$ and it is stated: $n = (\mu^ n/8^{n-1})(1/[(n-1)!]^ 2)(1+O(1/n^ 2)).$
Reviewer: St.Fenyö

### MSC:

 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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