Finite pseudo orbit expansions for spectral quantities of quantum graphs

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.

arXiv comment23 pages, 4 figures, typos corrected, references added, vacuum energy calculation expanded

References in corpus (22)

• Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
• Kirchhoff's Rule for Quantum Wires
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Functional determinants for general Sturm-Liouville problems
• Spectral functions in mathematics and physics
• Vacuum Energy and Repulsive Casimir Forces in Quantum Star Graphs
• Spectral determinant on quantum graphs
• Repulsive Casimir Pistons
• Secular determinants of random unitary matrices
• Unitary stochastic matrix ensembles and spectral statistics
• The trace formula for quantum graphs with general self adjoint boundary conditions
• Functionals of the Brownian motion, localization and metric graphs
• Value distribution of the eigenfunctions and spectral determinants of quantum star graphs
• Quantum graphs where back-scattering is prohibited
• Relationship between scattering matrix and spectrum of quantum graphs
• Spectral determinant on graphs with generalized boundary conditions
• Zeta functions of quantum graphs
• The autocorrelation function for spectral determinants of quantum graphs
• Trace Formulae for quantum graphs with edge potentials
• The Riemannium
• Mathematical Aspects of Vacuum Energy on Quantum Graphs
• Vacuum energy, spectral determinant and heat kernel asymptotics of graph Laplacians with general vertex matching conditions

Cited by in corpus (19)

• Non-Weyl Microwave Graphs
• An elementary introduction to quantum graphs
• On the Effective Size of a Non-Weyl Graph
• Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral fluctuations
• Pseudo orbit expansion for the resonance condition on quantum graphs and the resonance asymptotics
• Asymptotics of resonances induced by point interactions
• Quantum graphs and microwave networks as narrow band filters for quantum and microwave devices
• Pseudo-orbit approach to trajectories of resonances in quantum graphs with general vertex coupling: Fermi rule and high-energy asymptotics
• Spectral Properties of Quantum Circulant Graphs
• A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs
• Application of topological resonances in experimental investigation of a Fermi golden rule in microwave networks
• Coupled unidirectional chaotic microwave graphs
• Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit
• Trace formulas for general Hermitian matrices: Unitary scattering approach and periodic orbits on an associated graph
• New approach to periodic orbit theory of spectral correlations
• Towards a semiclassical understanding of chaotic single- and many-particle quantum dynamics at post-Heisenberg time scales
• Lyndon word decompositions and pseudo orbits on q-nary graphs
• Geometric spectral theory of quantum graphs
• A combinatorial approach to counting primitive periodic and primitive pseudo orbits on circulant graphs