Related papers: The bundle Laplacian on discrete tori

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of…

We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and of bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of…

We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over…

Riemann surface carries a natural line bundle, the determinant bundle. The space of sections of this line bundle (or its multiples) constitutes a natural non-abelian generalization of the spaces of theta functions on the Jacobian. There has…

Extending functions from boundary values plays an important role in various applications. In this thesis we consider discrete and continuous formulations of the problem based on $p$-Laplacians, in particular for $p=\infty$ and tight…

We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation…

A cuspidal end is a type of metric singularity, described as a product $S^1 \times \left] a, +\infty \right[$ with the Poincar\'e metric. The underlying set can also be seen as $\mathbb{R} \times \left] a, +\infty \right[$ subject to the…

The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As…

Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $\mu_j \in {\mathbb C},\: {\rm Im}\: \mu_j > 0,$ be the resonances of the Laplacian in the…

We initiate the study of spectral zeta functions $\zeta_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions.…

In this paper we observe that isomorphism classes of certain metrized vector bundles over P^1-{0,infinity} can be parameterized by arithmetic quotients of loop groups. We construct an asymptotic version of theta functions, which are defined…

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely…

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\chi$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $\Delta$ denote the hyperbolic…

We present an explicit formula for the determinant on the Metzler matrix of a digraph $D$. Furthermore, we introduce a walk-type zeta function with respect to this Metzler matrix of the symmetric digraph of a finite torus, and express its…

We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8, and in…

We prove that the zeta-function $\zeta_\Delta$ of the Laplacian $\Delta$ on a self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues,…

We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence…

We express the zeta function associated to the Laplacian operator on $S^1_r\times M$ in terms of the zeta function associated to the Laplacian on $M$, where $M$ is a compact connected Riemannian manifold. This gives formulas for the…

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one…

We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang…