An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics

TL;DR

This paper analyzes an infinite quantum graph with a non-standard Weyl asymptotics, deriving exact trace formulas, spectral zeta functions, and a Selberg-like zeta function, linking spectral properties to number theory and quantum chaos.

Contribution

It introduces an exact trace formula and zeta functions for an infinite quantum graph with non-standard Weyl asymptotics, connecting spectral theory to divisor problems and quantum chaos.

Findings

Eigenvalues relate to the divisor function d(n).

Derived explicit formulas for wave group trace, heat kernel, and spectral zeta functions.

Established a Selberg-like zeta function with a functional equation and Riemann hypothesis analogue.

Abstract

We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $\delta$-like potentials with strength $\kappa>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval $I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\frac{\pi}{n}$. We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each $\kappa>0$ and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by $\lambda_n=n^2$, $n\in\nz$, with multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics $\mathcal{N}(\lambda)=\frac{\sqrt{\lambda}}{2}\ln\lambda +\Or(\sqrt{\lambda})$ for the eigenvalue counting function. Based on an exact trace formula, the Vorono\"i summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph, for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true.