Related papers: On Green's Functions and Positive, Self-Adjoint, E…
We give two-sided, global (in all variables) estimates of the heat kernel and the Green function of the fractional Schr\"odinger operator with a non-negative and locally bounded potential $V$ such that $V(x) \to \infty$ as $|x| \to \infty$.…
In this paper, we give the explicit expressions of high-order Green operators on the disk and the polydisc, and hence the kernel functions of high-order Green operators are also presented. As applications, we present the explicit integral…
We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…
We study the generalized eigenvalue problem on the whole space for a class of integro-differential elliptic operators. The nonlocal operator is over a finite measure, but this has no particular structure. Some of our results even hold for…
The heat operator with a pure soliton potential is considered and its Green's function, depending on a complex spectral parameter k, is derived. Its boundedness properties in all variables and its singularities in the spectral parameter k…
It is well known that positive Green's operators are not necessarily positivity preserving. In this paper we investigate the matter of just how far from being positivity preserving a positive Green's operator can be. We will also identify a…
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's…
We present some new results concerning perturbation theory for positive solutions of second-order linear elliptic operators, including further study of the equivalence of positive minimal Green functions and the validity of a Liouville…
In this paper, we study an asymptotic expansion of the heat kernel for a Laplace operator on a smooth Riemannian manifold without a boundary at enough small values of the proper time. The Seeley-DeWitt coefficients of this decomposition…
We obtain an off-diagonal upper bound for Green and heat kernel of Laplace type operator on symmetric spaces.
If P is a differential operator with constant coefficients, an identity is derived to calculate the action of exp(P) on the product of two functions. In many-body theory, P describes the interaction Hamiltonian and the identity yields a…
In this paper, we study the existence of positive solutions for nonlinear fractional differential equations with a singular weight. We derive Green's function and corresponding integral operator and then examine the compactness of the…
This paper introduces a probability density estimator based on Green's function identities. A density model is constructed under the sole assumption that the probability density is differentiable. The method is implemented as a binary…
In this paper we develop a way of obtaining Green's functions for partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat…
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional…
In this paper we will deduce several properties of the Green's functions related to the Hill's equation coupled to various boundary value conditions. In particular, the idea is to study the Green's functions of the second order differential…
The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and…
For a finite reflection group on $\b R^N,$ the associated Dunkl operators are parametrized first-order differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is - under…
In this paper I discuss a formulation of relativistic few-particle scattering theory where the dynamical input is a collection of reflection-positive Euclidean covariant Green functions. This formulation of relativistic quantum mechanics…
This article gives a fundamental discussion on variable coefficients, self-adjoint, formally partially hypoelliptic differential operators. A generalization of the results to pseudo differential operators, is given in a following article in…