Distribution of zeros of holomorphic functions and resonances in logarithmic regions for Schroedinger operators on Euclidean space

Abstract

Motivated by scattering theory, this paper proves general results about the distribution of zeros in logarithmic neighborhoods of the real axis for a certain class of functions holomorphic in the closed lower half-plane. We define a new \emph{indicator function} that measures the growth of the holomorphic function along logarithmic curves, and connect this to the distribution of zeros of the function. This is related to classical results on the distribution of zeros in sectors for entire functions of completely regular growth. These results can be applied to the determinant of the scattering matrix of a Schrodinger operator on odd-dimensional Euclidean space, yielding bounds on resonance counting functions for logarithmic neighborhoods of the real axis. As a further application, we study a certain family of potentials in one dimension having jump-like singularities. Using our complex-analytic results we show that the singularities can lead to many -- and even infinitely many -- strings of resonances along logarithmic curves. We connect the properties of these strings of resonances, including their location and linear density, both to the parameters describing the singularities of the potential and to the asymptotic behavior of the scattering determinant.