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Li, Peter; Yau, Shing Tung

On the parabolic kernel of the Schrödinger operator. (English) Zbl 0611.58045

Acta Math. 156, 154-201 (1986).
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 \(\lambda\to \infty\).
Reviewer: R.Racke

Cited in 30 Reviews
Cited in 484 Documents

MathOverflow Questions:

Gaussian bounds on Dirichlet heat kernel
Gaussian upper heat kernel bounds on closed Riemannian manifolds
Quick question on the constants involved in heat kernel upper bounds

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J32 Boundary value problems on manifolds
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

heat equation; gradient estimates; Harnack inequality; Betti numbers; Schrödinger operator
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\textit{P. Li} and \textit{S. T. Yau}, Acta Math. 156, 154--201 (1986; Zbl 0611.58045)
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