In this paper we consider the simplest functional Schroedinger equation of a quantum field theory (in particular QCD) and study its solutions. We observe that the solutions to this equation must possess a number of properties. Its Taylor coefficients are multivalued functions with rational and logarithmic branchings and essential singularities of exponential type. These singularities occur along a locus defined by polynomial equations. The conditions we find define a class of functions that generalizes to multiple dimensions meromorphic functions with finite Nevanlinna type. We note that in perturbation theory these functions have local asymptotics that is given by multidimensional confluent hypergeometric functions in the sense of Gelfand-Kapranov-Zelevinsky.