---
abstract: 'We compute the value distributions of the eigenfunctions and spectral determinant of the Schrödinger operator on families of star graphs. The values of the spectral determinant are shown to have a Cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related. This is in contrast to the spectral determinants of random matrices, for which the logarithm is known to satisfy a Gaussian limit distribution. The value distribution of the eigenfunctions also differs from the corresponding random matrix result. We argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of Šeba-type billiards.'
author:
- 'J. P. Keating, J. Marklof and B. Winn'
date: 'Received: 29 October 2002'
title: Value distribution of the eigenfunctions and spectral determinants of quantum star graphs
---

Introduction
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The study of quantum graphs as model systems for quantum chaos was initiated by Kottos and Smilansky [@kot:1], [@kot:pot], who observed that the spectral statistics of fully-connected graphs are typical of those associated with generic classically chaotic systems. The relative simplicity of quantum graphs, 