Sharp decay estimates of Lq-norms for nonnegative Schrödinger heat semigroups

Let H=−Δ+V be a nonnegative Schrödinger operator on L2(RN), where N⩾3 and V is a radially symmetric nonpositive function in RN decaying quadratically at the space infinity. For any 1⩽p⩽q⩽∞, we denote by ‖e−tH‖q,p the operator norm of the Schrödinger heat semigroup e−tH from Lp(RN) to Lq(RN). In this paper, under suitable conditions on V, we give the exact and optimal decay rates of ‖e−tH‖q,p as t→∞ for all 1⩽p⩽q⩽∞. The decay rates of ‖e−tH‖q,p depend on whether the operator H is subcritical or critical and on the behavior of the positive harmonic function for the operator H.