On completeness of random exponentials in the Bargmann–Fock space

Abstract

We study the completeness/incompleteness properties of a system of exponentials EΛ={eπλz; λ∈Λ}, viewed as elements of the Bargmann–Fock space of entire functions. We assume that the index set Λ is a realization of a random point field in ℂ (the support of a random measure). We prove that the properties are determined by the density of the field, i.e., by the mean number of the field points per unit area. We also discuss certain implications and motivations of our results, in particular, the jumps of the integrated density of states of the Landau Hamiltonian with the random potential, equal to the sum of point scatters.