Singular continuous spectrum for a class of almost periodic Jacobi matrices

Abstract

We consider operators (parametrized by a, Ô, \) on / 2 with matrix ôjy+i + ô /;-i + fl/ô/y with an = \ COS(2TTCW + 6).If ce is a Liouville number and \ > 2, we prove that for a.e.0, the operator's spectral measures are all singular continuous.We consider the operator// on l 2 (Z) depending upon three parameters, X, a, 0,In this note we will sketch the proof of the following result whose detailed proof will appear elsewhere [3].THEOREM 1. Fix a, an irrational number obeying(2) \a-p k /q k \<k~q k for a sequence q k -> «>.Fix X > 2. Then for a.e.0, //(X, a, 6) has purely singular continuous spectrum.

Locations

• CiteSeer X (The Pennsylvania State University) - View - PDF
• Bulletin of the American Mathematical Society - View - PDF