Small-tau expansion for the form factor of glued quantum star graphs

TL;DR

This paper derives the small-tau expansion of the form factor for glued quantum star graphs, revealing how the glueing affects spectral properties depending on the relative number of glueing edges.

Contribution

It provides the first explicit third-order small-tau expansion for the form factor of glued quantum star graphs, considering backscattering orbits and boundary conditions.

Findings

Glueing has no effect when glueing edges are negligible.

Glueing influences the $ au^2$ term when glueing and non-glueing edges are comparable.

The expansion accounts for dominant backscattering orbits.

Abstract

We compute the small-tau expansion up to the third order for the form factor of two glued quantum star graphs with Neumann boundary conditions, by taking into account only the most backscattering orbits. We thus show that the glueing has no effect if the number of glueing edges is negligible compared to the number of edges of the graph, whereas it has an effect on the $\tau^2$ term when the numbers of glueing and non glueing edges are of the same order.