Finite pseudo orbit expansions for spectral quantities of quantum graphs

TL;DR

This paper introduces a finite pseudo orbit expansion method for analyzing spectral properties of quantum graphs, simplifying previous approaches and enabling direct computation of spectral quantities from short irreducible pseudo orbits.

Contribution

It develops a finite expansion framework for spectral quantities of quantum graphs using irreducible pseudo orbits, simplifying calculations and deriving key spectral functions.

Findings

Finite pseudo orbit expansions accurately compute spectral quantities.

The secular equation is expressed in terms of irreducible pseudo orbits.

Spectral zeta function and related quantities are obtained as finite sums.

Abstract

We investigate spectral quantities of quantum graphs by expanding them as sums over pseudo orbits, sets of periodic orbits. Only a finite collection of pseudo orbits which are irreducible and where the total number of bonds is less than or equal to the number of bonds of the graph appear, analogous to a cut off at half the Heisenberg time. The calculation simplifies previous approaches to pseudo orbit expansions on graphs. We formulate coefficients of the characteristic polynomial and derive a secular equation in terms of the irreducible pseudo orbits. From the secular equation, whose roots provide the graph spectrum, the zeta function is derived using the argument principle. The spectral zeta function enables quantities, such as the spectral determinant and vacuum energy, to be obtained directly as finite expansions over the set of short irreducible pseudo orbits.