4. The Weyl Functions

The main goal of this chapter is the analysis of the limiting qualitative behavior as the (real or complex) parameter \(\lambda\) tends to 0 of the flows determined by families of linear Hamiltonian systems with coefficient matrix of the form \(H +\lambda J^{-1}\varGamma\). Under a fundamental hypothesis, which is in particular satisfied if Γ is an Atkinson perturbation, the existence of the derivative at \(\lambda = 0\) of the rotation number \(\alpha _{\varGamma }(\lambda )\) is proved, and its value is worked out. If, in addition, Γ is positive definite (or if this is the case for the corresponding perturbation matrix in the Schrödinger case), then the existence of the limits in the L ¹ topology of the Weyl functions \(M^{\pm }(\lambda )\) for \(\lambda = i\varepsilon\) as \(\varepsilon \rightarrow 0^{+}\) is proved; and, as in the case of α′(0), it is shown that the values of the limits can be determined from Γ alone. All these fact taken together amount to an extension of the well-known Kotani theory. As a consequence it is possible to generalize to the n-dimensional Schrödinger equation a famous inequality (obtained in the scalar case by Moser (1980) and by Deift and Simon (1983)) involving the rotation number and its derivative. The chapter ends with a description of a scenario in which the convergence of the Weyl functions is uniform.