Related papers: Zeta functions of quantum graphs
Contour integral representations for Riemann's Zeta function and Dirichelet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800's, but somehow they do not…
We introduce a new method for constructing isospectral quantum graphs that is based on transplanting derivatives of eigenfunctions. We also present simple digraphs with the same reversing zeta function, which generalizes the Bartholdi zeta…
In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order…
In this paper, we explore the properties of zeta functions associated with infinite graphs of groups that arise as quotients of cuspidal tree-lattices, including all non-uniform arithmetic quotients of the tree of rank one Lie groups over…
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…
The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.
Following the seminal works of Asorey-Ibort-Marmo and Mu\~{n}oz-Casta\~{n}eda-Asorey about selfadjoint extensions and quantum fields in bounded domains, we compute all the heat kernel coefficients for any strongly consistent selfadjoint…
The $L^2$-zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y, the normalized zeta functions of the finite graphs converge to the $L^2$-zeta…
This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties…
We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which…
In this article, we study the zeta function $\zeta_q$ associated to the Laplace operator $\Delta_q$ acting on the space of the smooth $(0,q)$-forms with $q=0,\ldots,n$ on the complex projective space $\mathbb{P}^n(\mathbb{C})$ endowed with…
We present a pedagogical exposition of some applications of functional methods in quantum field theory: we use heat-kernel and zeta-function techniques to study the Casimir effect, the pair production in strong electric fields, quantum…
We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then…
We construct models of exactly solvable two-particle quantum graphs with certain non-local two-particle interactions, establishing appropriate boundary conditions via suitable self-adjoint realisations of the two-particle Laplacian. Showing…
We provide a coherent overview of a number of recent results obtained by the authors in the theory of schemes defined over the field with one element. Essentially, this theory encompasses the study of a functor which maps certain geometries…
We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an…
Quantum graphs have recently emerged as models of nonlinear optical, quantum confined systems with exquisite topological sensitivity and the potential for predicting structures with an intrinsic, off-resonance response approaching the…
We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function…
Heat-kernel expansion and zeta function regularisation are discussed for Laplace type operators with discrete spectrum in non compact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several…
The quantum walk is a quantum counterpart of the classical random walk. On the other hand, the absolute zeta function can be considered as a zeta function over F_1. This paper presents a connection between the quantum walk and the absolute…