ETD

• dispersive
• functional analysis
• Kato smoothing
• limiting absorption principle
• pdes
• point interactions
• resolvent estimates
• Schrödinger equation
• Strichartz

We construct point-interactions as self-adjoint extensions of the free Laplacian on R^n minus the origin for n=1, 2, 3, and characterize their domain, action, resolvent and spectrum. In particular, we show that the resolvent is a rank-one perturbation of the free Laplacian one. We then prove weighted L^2 resolvent estimates for the point-interaction for n=1, 2, 3, by bounding the free resolvent and the perturbation separately, employing the explicit expression of the free resolvent integral kernel with Bessel functions. In particular we prove an estimate, independent of the spectral parameter, for a resolvent difference for n=2. We then prove the classical Strichartz estimates for the Schrödinger equation, observe that for n=2 we don't have the endpoint (time exponent=2), and, thanks to the resolvent estimate for n=2, prove a similar dispersive estimate (with decaying weights) for time exponent=2, using the abstract Kato smoothing technique. We finally sketch a proof of point-interactions Strichartz estimates and mix them up with smoothing ones.