I will discuss recent results on several resolvent estimates, including uniform Sobolev inequalities of Kenig-Ruiz-Sogge type, for the Schrödinger operator with singular potentials. The inverse square potential, which is strictly bounded from below by the Hardy potential, is a typical example. Key ingredients are a simple perturbation method based on an iterated resolvent identity and a weighted resolvent estimate with a homogeneous weight. I will also discuss some applications to Keller-Lieb-Thirring type inequalities for individual eigenvalues of the Schrödinger operator with complex valued potentials, local smoothing and Strichartz estimates for the time-dependent Schrödinger equation.
This is joint work with Jean-Marc Bouclet (Toulouse).