we find in theorem \[thm:6\] completely determines the value distribution of the star graph eigenfunctions (see the appendix) and has a significantly different shape (c.f. equation [(\[eq:Q:asympt\])]{} and figure \[fig:9:5\] below). Other quantum systems for which the value distribution of the eigenfunctions has a non-random-matrix limit are the Cat Maps [@par:cat].

In [@berk:2] a correspondence was noted between the two-point spectral correlation functions for star graphs and a class of systems known as Šeba billiards. The original Šeba billiard [@seba:1] was a rectangular quantum billiard perturbed by a point singularity. More generally, we describe any integrable system perturbed in such a way as belonging to the same class [@seba:2]. We conjecture that the results derived in the present work will also apply to systems in the Šeba class.

The remainder of this paper is structured as follows. In section \[sec:2\] we prove theorems \[thm:1\] and \[thm:2\]. In section \[sec:3\] we treat the finite $v$ cases, theorem \[thm:3\] and \[thm:5\]. In sections \[sec:4\] and \[sec:5\] we prove, respectively, theorems \[thm:4\] and \[thm:6\], developing the necessary machinery in section \[sec:4\]. Section \[sec:6\] is devoted to numerical computations that illustrate our results. We develop more fully the connections between the present work and 