Spectral properties of the monodromy matrix for Harper equation. (English) Zbl 0874.39006

Buslaev, V.; Fedotov, A.

Journ. Équ. Dériv. Partielles, St.-Jean-de-Monts 1996, Exp. No. 4, 11 p. (1996).

The authors describe properties of the entire analytic solutions of the Harper equation with a periodic coefficient of the form \((1/2)[\psi(x+h)+\psi(x-h)]+\cos x \psi(x)=E(x)\), where \(x\in{\mathbb{R}}\), \(\psi(x)\in{\mathbb{C}}\), \(h\) is fixed, \(0<h<2\pi\), and \(E\) is the spectral parameter, by means of the monodromization procedure. The treatment is independent of any semiclassical hypothesis on the number \(h\). The main object of the theory consists in the coefficients which characterize the asymptotic behaviour of minimal solutions in the vicinity of the singular point of the equation. A canonical basis consisting of two minimal solutions reflecting the symmetries of the equation is constructed and the monodromy matrix corresponding to this basis is introduced.

Reviewer: R.Vaillancourt (Ottawa)

MSC:

39A11	Stability of difference equations (MSC2000)
39A10	Additive difference equations

Keywords:

Harper equation; monodromy matrix; spectral properties; entire analytic solutions; asymptotic behaviour; minimal solutions