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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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