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

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.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations
Full Text: