L2-exponential lower bounds to solutions of the Schroedinger equation

We study decay properties of solutions psi of the Schroedinger equation (-Δ+V)psi=Epsi. Typical of our results is one which shows that if V=o( vertical stroke x vertical stroke -1sup(/)2) at infinity of if V is a homogeneous N-body potential (for example atomic or molecular), then if E<0 and α>√-E, esup(a)sup(|)sup(x)sup(|)psi. We also construct examples to show that previous essential spectrum-dependent upper bounds can be far from optimal if psi is not the ground state. (orig.WL)