Rotation number for non-autonomous linear Hamiltonian systems II: The Floquet coefficient
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, we defined the rotation number for such a family and discussed its basic properties. Here we define and study a complex quantity - the Floquet coefficient w - for such a family. The rotation number is the imaginary part of w. We derive a basic trace formula satisfied by w, and give applications to Atkinson-type spectral problems. In particular we use w to discuss the convergence properties of the Weyl M-functions, the Kotani theory, and the gap-labelling phenomenon for these problems.
Mathematics Subject Classification (2000).Primary: 34A30 37B55 Secondary: 34D09 47A10
Keywords.Hamiltonian system Rotation number Floquet coefficient Atkinson spectral problem Weyl M-functions
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