Related papers: Higher Green's functions for modular forms

We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta…

Natural modes of helical structures are treated by using the periodic dyadic Green's functions in cylindrical coordinates. The formulation leads to an infinite system of one-dimensional integral equations in reciprocal (Fourier) space. Due…

In this paper, we explore the modular differential equation $\displaystyle y'' + F(z)y = 0$ on the upper half-plane $\mathbb{H}$, where $F$ is a weight 4 modular form for $\Gamma_0(2)$. Our approach centers on solving the associated…

We introduce a new type of pluricomplex Green function which has a logarithmic pole along a complex subspace A of a complex manifold X. It is the largest negative plurisubharmonic function on X whose Lelong number is at least the Lelong…

We verify the identity which relates the two-point Green functions of ${\cal N}=1$ SQED with $N_f$ flavors, regularized by higher derivatives, by explicit calculations in the three-loop approximation. This identity explains why in the limit…

The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible…

We generalize the definition of CM cycles beyond the small and big CM ones studied by various authors and give a uniform formula for the CM values of Green functions associated to these special cycles in general using the idea of…

We provide explicit representations of Green's functions for general linear fractional differential operators with {\it variable coefficients} and Riemann-Liouvilles derivatives. We assume that all their coefficients are continuous in $[0,…

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…

We study Green's matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of the Green's matrices.

Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a constitutive equation ${\bf J}({\bf x})={\bf…

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there…

A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved…

Let $\Gamma_{F,n}$ be the Hermitian modular group of degree $n>1$ in sense of Hel Braun with respect to an imaginary quadratic field $F$. Let $r$ be a natural number. There exists a multiplier system of weight $1/r$ (equivalently a…

We compile a list of 2-D and axisymmetric Green's functions for isotropic full and half spaces, to complement our letter "Linear elasticity of incompressible solids". We also extend the isotropic exactly incompressible linear theory from…

Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs,…

Let $X$ be a locally compact abelian group with countable base and let $\mathcal W$ be a convex cone of positive numerical functions on $X$ which is invariant under the group action and such that $(X,\mathcal W)$ is a balayage space or…

Lattice Green's Functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of…

High-order derivatives of Green's functions are a key ingredient in Taylor-based fast multipole methods, Barnes-Hut $n$-body algorithms, and quadrature by expansion (QBX). In these settings, derivatives underpin either the formation,…

The properties of the gauge invariant two-point quark Green's function are studied in the large-Nc limit of two-dimensional QCD. The analysis is done by means of an exact integrodifferential equation. The Green's function is found infrared…