Zeta functions of quantum graphs

Abstract

In this article we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle. We start by considering the special case of the star graph with Neumann matching conditions at the center of the star. We then extend the technique to allow any matching conditions at the center for which the Laplace operator is self-adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions. In the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the Neumann star graph. In the second half of the article we apply the zeta function to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graphs. These have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature.

32 pages, typos corrected, references added

References in corpus (17)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Casimir force on a piston
• Casimir Forces in a Piston Geometry at Zero and Finite Temperatures
• Quantum graphs: an introduction and a brief survey
• Vacuum Energy and Repulsive Casimir Forces in Quantum Star Graphs
• Repulsive Casimir Pistons
• Kaluza-Klein models as pistons
• Approximation of a general singular vertex coupling in quantum graphs
• The trace formula for quantum graphs with general self adjoint boundary conditions
• Finite Temperature Casimir Effect in Kaluza-Klein Spacetime
• Computation of determinants using contour integrals
• Vacuum Stress and Closed Paths in Rectangles, Pistons, and Pistols
• The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations
• Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone
• An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics
• Mathematical Aspects of Vacuum Energy on Quantum Graphs
• $\zeta$-regularised spectral determinants on metric graphs

Cited by in corpus (14)

• Thermodynamics of photons on fractals
• Non-equilibrium Steady States of Quantum Systems on Star Graphs
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• The influence of geometry and topology of quantum graphs on their nonlinear-optical properties
• Spectral determinants and zeta functions of Schrödinger operators on metric graphs
• Zeta Functions of the Dirac Operator on Quantum Graphs
• An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics
• Spectral Properties of Quantum Circulant Graphs
• Entanglement entropy on finitely ramified graphs
• Vacuum energy, spectral determinant and heat kernel asymptotics of graph Laplacians with general vertex matching conditions
• Relating Zeta Functions of Discrete and Quantum Graphs
• Can One Hear the Spanning Trees of a Quantum Graph?
• Quantum graph walks II: Quantum walks on graph coverings
• Vacuum energy of Schrödinger operators on metric graphs