On the parabolic kernel of the Schrödinger operator
type: Item
instance of: scholarly article
title: On the parabolic kernel of the Schrödinger operator
zbMATH Open document ID: 0611.58045
DOI: 10.1007/BF02399203
author: Peter Li
author: Shing-Tung Yau
published in: Acta Mathematica
publication date: 1986-01-01T00:00:00Z
review text: Parabolic equations of the type \[ (-\Delta -q(x,t)-\partial /\partial t)u(x,t)=0 \tag{\(*\)} \] on a Riemannian manifold are studied. Gradient estimates and a Harnack inequality for positive solutions of (\(*\)) are proved. This is applied to obtain upper bounds for the fundamental solution of (\(*\)) in the case \(q=q(x)\) and lower bounds in that case if the Ricci curvature is nonnegative. For a nonempty convex boundary Dirichlet and Neumann conditions are admitted. Estimates on the heat kernel and on Green's function are given, furthermore estimates on the eigenvalues of the Laplacian and on the Betti numbers. Finally the behaviour of the fundamental solution of the Schrödinger operator \((\Delta -\lambda^ 2q(x)-\partial /\partial t)\) is described as \(\lambda\to \infty\).
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 3988331
zbMATH Keywords: heat equation
zbMATH Keywords: gradient estimates
zbMATH Keywords: Harnack inequality
zbMATH Keywords: Betti numbers
zbMATH Keywords: Schrödinger operator
reviewed by: Reinhard Racke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3988331