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

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


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
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