×

Semiclassical analysis for Harper’s equation. (English) Zbl 0759.47022

Recent developments in hyperbolic equations, Proc. Conf., Pisa/Italy 1987, Pitman Res. Notes Math. Ser. 183, 312-321 (1988).
[For the entire collection see Zbl 0713.00007.]
The authors are interested in the spectrum of Harper’s operator in \(p^ 2(\mathbb{Z})\) given by: \[ H_{\theta,\lambda,h}u(n)=\left[{{u(n+1)+u(n- 1)}\over 2}\right] +\lambda \cos(hn+\theta)u(n) \] in the special case \(\lambda=1\) and \(h/2\pi\not\in\mathbb{Q}\). In that case, it has been conjectured that the spectrum is a Cantor set of measure 0. This appears in particular in the work of Hofstadter whose results are numerical and indicate clearly this structure. Another source is particularly important for the authors: Wilkinson’s papers, which are based on a WKB analysis with infinitely many “potential” wells in the space \(T^*\mathbb{R}\), which interact by tunnel effect; the method here is intuitive, and, using the techniques they have developed in previous works, extended to the case of infinitely many wells by V. Carlsson, the authors manage to give a rigorous presentation of Wilkinson’s ideas.

MSC:

47B39 Linear difference operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10 Spectrum, resolvent
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Citations:

Zbl 0713.00007
PDFBibTeX XMLCite