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ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS

In this lecture we continue our discussion of connections with spectral theory for differential operators, started in Lecture 3, in more detail. We will only discuss the case of Schrödinger operators although similar theories exist for general canonical systems. See [4] and [2] for the basics of the spectral theory. Present discussion is a shortened version of selected sections of our paper with N. Makarov [3]. Recall that the H-model space of an inner function Θ, KΘ = H 2 ΘH = H ∩ΘH̄, is a Hilbert space with the Hilbert structure inherited from H. A function kλ from a Hilbert space H of analytic functions in a complex domain is called a reproducing kernel corresponding to the point λ from the domain if for any f ∈ H, < f, kλ >H= f(λ). In the case of KΘ the reproducing kernels are given by the formula