ABSTRACT. Let H = -\L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and regularity, and in particular allow V which are unbounded below. We give a general survey of the properties of e~t H , t > 0, and related mappings given in terms of solutions of initial value problems for the differential equation du/dt + Hu = 0. Among the subjects treated are L ^-properties of these maps, existence of continuous integral kernels for them, and regularity properties of eigenfunctions, including Harnack's inequality.
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