Value distribution of the eigenfunctions and spectral determinants of quantum star graphs

TL;DR

This paper analyzes the value distributions of eigenfunctions and spectral determinants of quantum star graphs, revealing a Cauchy distribution for spectral determinants and contrasting with random matrix theory, with implications for Seba-type billiards.

Contribution

It provides the first detailed analysis of the value distributions of eigenfunctions and spectral determinants on quantum star graphs, showing they differ from random matrix predictions.

Findings

Spectral determinants follow a Cauchy distribution in the limit.

Eigenfunction values differ from random matrix results.

Distributions are similar to those of Seba-type billiards.

Abstract

We compute the value distributions of the eigenfunctions and spectral determinant of the Schrodinger 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 Seba-type billiards.