On fundamental solutions of generalized Schrödinger operators
type: Item
instance of: scholarly article
title: On fundamental solutions of generalized Schrödinger operators
zbMATH Open document ID: 0936.35051
author: Zhongwei Shen
published in: Journal of Functional Analysis
publication date: 1999-12-19T00:00:00Z
review text: Let \(\mu\) be a nonnegative Radon mesure on \(\mathbb{R}^n\), \(n\geq 3\), and consider the Schrödinger operator. If \(\mu\) satisfies certain scaling invariant Kato conditions and doubling conditions, the following estimates for the fundamental solution \(\Gamma_\mu\) of \(-\Delta+\mu\) are proved: \[ {ce^{-\varepsilon_2 d(x,y;\mu)} \over|x-y|^{u-2}} \leq\Gamma_\mu (x,y)\leq {Ce^{-\varepsilon_1 d(x,y;\mu)} \over|x-y|^{n-2}}, \] where \(d(x,y;\mu)\) is the distance function for the modified Agmon metric associated with \(\mu\). Further the boundedness of the corresponding Riesz transform \(\text{grad}(-\Delta +\mu)^{-1/2}\) is investigated on \(L^p\).
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 1367857
zbMATH Keywords: Schrödinger operator with measure as potential
zbMATH Keywords: estimates for fundamental solutions
zbMATH Keywords: Agmon metric
zbMATH Keywords: generalized Riesz transform
reviewed by: Niels Jacob
MaRDI profile type: Publication
OpenAlex ID: W2168844446
cites work: Q3969238
cites work: Brownian motion and harnack inequality for Schrödinger operators
cites work: Harnack's Inequality for Schrodinger Operators and the Continuity of Solutions
cites work: On the \(\bar {\partial{}}\) equation in weighted \(L^ 2\) norms in \(\mathbb{C} ^ 1\)
cites work: Etude de l'équation (1/2) \(\Delta u-u\mu =0\), où \(\mu\) est une mesure positive. (Study of the equation (1/2) \(\Delta u-u\mu =0\), where \(\mu\) is a positive measure)
cites work: Q4040640
cites work: The Green function for uniformly elliptic equations
cites work: Decay at infinity of eigenfunctions of Schrödinger operators with polynomial electro-magnetic field
cites work: The trace inequality and eigenvalue estimates for Schrödinger operators
cites work: An Estimate on the Heat Kernel of Magnetic Schrödinger Operators and Uniformly Elliptic Operators with Non-Negative Potentials
cites work: \(L^ p\) estimates for Schrödinger operators with certain potentials
cites work: Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields
cites work: Estimates in L^p for magnetic Schrodinger operators
cites work: Q5607484
DOI: 10.1006/JFAN.1999.3455
description: scientific article; zbMATH DE number 1367857