The rotation number for non-autonomous linear Hamiltonian systems II:

the Floquet coefficient

with Roberta Fabbri and Russell Johnson,

Z. Angew. Math. Phys.

This paper is the second of a two-part series in which we review the properties of

the rotation number for a random family of linear non-autonomous Hamiltonian

systems. In Part I, the rotation number for such a family was defined and its basic

properties discussed. Here, a complex quantity -the Floquet coefficient w- for such

a family is defined and studied. The rotation number is the imaginary part of w. A

basic trace formula satisfied by w is derived, and applications to Atkinson-type

spectral problems are given. In particular, w is used to discuss the convergence

properties of the Weyl M-functions, the Kotani theory, and the gap-labelling

phenomenon for these problems.