Schrödinger Semigroups

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 > 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. CONTENTS A. Introduction Al. Overview A2. The class Kv A3. Literature on larger classes B. L ̂ -properties BI. L^-smoothing of semigroups B2. Sobolev estimates B3. Continuity and derivative estimates B4. Localization B5. Growth of L ̂ -semigroup norms as t -> oo B6. Weighted L-spaces B7. Integral kernels: General potentials B8. Integral kernels: Some special operators for some special potentials B9. Trace ideal properties BIO. Continuity in V Bl 1. Hypercontractive semigroups and all that B12. Some remarks on the case when H is unbounded below B13. The magnetic case C. Eigenfunctions CI. Harnack's inequality and subsolution estimates C2. Local estimates on v<p C3. Decay of eigenfunctions C4. Eigenfunctions and spectrum C5. Eigenfunction expansions Received by the editors March 4, 1982. 1980 Mathematics Subject Classification. Primary 81-02, 35-02; Secondary 47F05, 35P05. © 1982 American Mathematical Society 0273-0979/82/0000-0350/121.00