Schrِdinger Semigroups on Manifolds

We derive uniform upper bounds for the transition density (or parabolic kernel) pV of the Schrödinger operator −12Δ + V on a Riemannian manifold (with Ricci curvature bounded from below) of the form pV(t, x, y) ≤ C · m−1/2(B[formula](x)) · m−1/2(B[formula](y))[formula] or, more suggestively, of the form pV(t, x, y) ≤ γ · p(βt, x, y) · e−αt (where p(·, ·, ·) denotes the heat kernel on the manifold), and also similar lower bounds. The assumptions on the potential V are very weak (i.e., close to be necessary). For instance, on Rd the assumptions for the upper and lower bound are satisfied by the highly singular, oscillating potentials V(x) = ||x||−k · sin(||x||−k) (with arbitraryk > 0) and V(x) = exp(1/||x||) · sin(exp(l/||x||)). We also admit to replace the functions V by signed measures μ in order to obtain estimates for the generalized Schrödinger operators −12Δ + μ. For manifolds with nonnegative Ricci curvature, the number α (in our upper bound) can be chosen arbitrarily close to the optimal value, namely the L2-spectral bound λμ. In this case, therefore, the Lp-growth bound of Schrödinger semigroups is independent of p ∈ [1, ∞]. Both assertions (as well as the above bounds) seem to be new even in the Euclidean case. In the Euclidean case, our estimates improve those of Blanchard and Ma derived under more restrictive assumptions on the potential (which are not satisfied by the above examples). For V ≡ 0, our bounds reduce to the heat kernel estimates of Li and Yau (or, more precisely, to the improvements of these estimates by Davies, resp. the author).