Related papers: The Hodge Laplacian operator on 1-forms on $\mathb…

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{\Gamma,\Sigma}^{\rm o}(\lambda)$ associated to the spinor bundle $\Sigma$ on odd dimensional convex co-compact hyperbolic manifolds…

We examine a specific category of eigenfunctions of the lattice Laplacian on $\{p,q\}$-tessellations of the Poincar\'e disk that bear resemblance to plane waves in the continuum case. Our investigation reveals that the lattice eigenmodes…

Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami…

We define a set of holomorphic functions in terms of the Hauptmodul of a quotient Riemann surface and prove that these functions are holomorphic on the upper half-plane. It is also shown that these functions are automorphic forms of weight…

Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In this note, we initiate the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds,…

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${\lambda_{2g-2}}(S)$: ${\lambda_{2g-2}}(S) > 1/4 +…

This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define…

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles. The geodesic cycles we study are themselves hyperbolic manifolds of lower dimension. The restriction of an automorphic form to…

We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol…

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami…

This paper deals with the $\Gamma$-lattice points problem associated to a discrete subgroup of motions $\Gamma$ in the complex hyperbolic space $\mathbb{C} H^n$. We give two integral formulas for the local average of the number $N(T, z,…

The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator…

Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k)…

Let $X$ be a smooth, compact, projective K\"ahler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $\omega$ be a K\"ahler form on $X$ and $K_{X}$ the corresponding heat kernel…

If M is a smooth compact oriented Riemannian manifold of dimension n=4k+2, with or without boundary, and F is a vector bundle on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the…

The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space.…

Let O^1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indefinite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the…

In this paper, we examine eigenfunctions of a generalized Landau Magnetic Laplacian that models the physics of an electron confined to a plane in a magnetic field orthogonal to the plane. This operator has an infinite dimensional null space…

We study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic ball. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution.In doing so, we put…

We consider generalized Hodge-Laplace operators $\alpha d \delta + \beta \delta d$ for $\alpha, \beta > 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate…