On the instability intervals of the Mathieu-Hill operator

Abstract

Consider the Mathieu-Hill operator Ly = -y + (2h cos 2x)y, < x < + } in L2(R), where h ? (R)\{0} . We obtain the precise asymptotic formulas for the widths ? k of the instability intervals of L. The formula states the isolated terms of arbitrary number in the asymptotics of the sequence ? k for large k and verifies the results of Harrell (Am J Math suppl:139-150, 1981) and Avron and Simon (Ann Phys 134:76-84, 1981). © 2007 Springer.