The trace formula for quantum graphs with general self adjoint boundary conditions

Abstract

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-$t$ asymptotics for the trace of the heat kernel.

36 pages; improved, final version to appear in Ann. H. Poincare

References in corpus (2)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• The inverse scattering problem for metric graphs and the traveling salesman problem

Cited by in corpus (38)

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• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Universality of the momentum band density of periodic networks
• Non-self-adjoint graphs
• Quantum graphs with singular two-particle interactions
• The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations
• Free fermions and the classical compact groups
• Quantum graphs with two-particle contact interactions
• The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
• Zeta functions of quantum graphs
• Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs
• Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons
• Exactly solvable interacting two-particle quantum graphs
• On a two-particle bound system on the half-line
• Many-particle quantum graphs and Bose-Einstein condensation
• Trace Formulae for quantum graphs with edge potentials
• A new proof of the Voronoï summation formula
• Two interacting particles on the half-line
• Spectral determinants and zeta functions of Schrödinger operators on metric graphs
• An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics
• Instability of Bose-Einstein condensation on quantum graphs under repulsive perturbations
• A trace formula for metric graphs with piecewise constant potentials and multi-mode graphs
• Differences between Robin and Neumann eigenvalues on metric graphs
• Inverse Steklov spectral problem for curvilinear polygons
• Zero modes of quantum graph Laplacians and an index theorem
• A Gelfand-Levitan trace formula for generic quantum graphs
• Bounds on the negative eigenvalues of Laplacians on finite metric graphs
• Hidden symmetries in non-self-adjoint graphs
• Many-particle quantum graphs: A review
• Concrete method for recovering the Euler characteristic of quantum graphs
• Quasi-isospectrality on quantum graphs
• Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge
• Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core
• Quantum Systems with jump-discontinuous mass. I
• On fully supported eigenfunctions of quantum graphs
• Finite deficiency indices and uniform remainder in Weyl's law
• Solvable models of interacting n-particle systems on quantum graphs
• Changing gears: Isospectrality via eigenderivative transplantation