Localization of zeros for Cauchy transforms

We study the localization of zeros of Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.