Related papers: Green function and self-adjoint Laplacians on poly…
Let $\Delta_k$ be the Dunkl Laplacian relative to a fixed root system $\mathcal{R}$ in $\mathbb{R}^d$, $d\geq2$, and to a nonnegative multiplicity function $k$ on $\mathcal{R}$. Our first purpose in this paper is to solve the…
At coupling strengths lambda = 1/2, 1, or 2, the Calogero-Sutherland model (CSM) is related to Brownian motion in a Wigner-Dyson random matrix ensemble with orthogonal, unitary, or symplectic symmetry. Using this relation in conjunction…
We study the Laplacian on Stenzel spaces (generalized deformed conifolds), which are tangent bundles of spheres endowed with Ricci flat metrics. The (2d-2)-dimensional Stenzel space has SO(d) symmetry and can be embedded in C^d through the…
An analytical Green's function is developed to study the acoustic scattering by a flat plate with a serrated edge. The scattered pressure is solved using the Wiener-Hopf technique in conjunction with the adjoint technique. It is shown that…
In this article we derive the lattice Green Functions (GFs) of graphene using a Tight Binding Hamiltonian incorporating both first and second nearest neighbour hoppings and allowing for a non-orthogonal electron wavefunction overlap. It is…
Precise asymptotics known for the Green function of the Laplacian have found their analogs for bounded below periodic elliptic operators of the second-order below and at the bottom of the spectrum. Due to the band-gap structure of the…
We elaborate and make rigorous various speculations about the implications of spectral properties of self-adjoint operators on spaces of automorphic forms for location of zeros of $L$-functions. Some of these ideas arose in work of Colin de…
On the unit square, we introduce a method for accurately computing source-neutral Green's functions of the fractional Laplacian operator with either periodic or homogeneous Neumann boundary conditions. This method involves analytically…
We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian $\Delta_k$ in $\mathbb{R}^d$. As applications we derive the Poisson-Jensen formula for $\Delta_k$-subharmonic functions and…
We investigate the spectral properties of rooted trees with the intention of improving the currently existing results that deal with this matter. The concept of an assigned rational function is recursively defined for each vertex of a…
The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian…
We study the low-energy behavior of the Green function for one-dimensional Fokker-Planck and Schr\"odinger equations with periodic potentials. We derive a formula for the power series expansion of reflection coefficients in terms of the…
We establish a local Harnack inequality in a neighborhood of an indecomposable singular point of a stationary integral varifold. Extending the method of Gr\"uter and Widman \cite{gruter1982green}, we construct the Green function on a…
Precise asymptotics known for the Green's function of the Laplace operator have found their analogs for periodic elliptic operators of the second order at and below the bottom of the spectrum. Due to the band-gap structure of the spectra of…
Homogeneous and inhomogeneous biharmonic equation are considered on the $n$-dimensional unit sphere. The Green function is given as a series of Gegenbauer polynomials. In the paper, explicit representations of the Green function are found…
It is proved that the Green's function of the simple random walk on a surface group of large genus decays exponentially at the spectral radius. It is also shown that Ancona's inequalities extend to the spectral radius R, and therefore that…
In this paper, we generalize results of Bruinier on automorphic Green functions on Hilbert modular surfaces to arbitrary ideals. For instance, we compute the Fourier expansion of the unregularized Green functions, use it to regularize them,…
In this article we study the role of the Green function for the Laplacian in a compact Riemannian manifold as a tool for obtaining well-distributed points. In particular, we prove that a sequence of minimizers for the Green energy is…
We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace's equation in three dimensions. An expansion in curvilinear coordinates leads us to…
We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schr\"odinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation,…