Abstract:
The study of dispersive properties of Schrödinger operators with point
interactions is a fundamental tool for understanding the behavior of many body
quantum systems interacting with very short range potential, whose dynamics can
be approximated by non linear Schrödinger equations with singular interactions. In
this work we proved that, in the case of one point interaction in R3, the perturbed
Laplacian satisfies the same Lp -Lq estimates of the free Laplacian in the smaller
regime q ∈ 2 [2;3). These estimates are implied by a recent result concerning the
Lp boundedness of the wave operators for the perturbed Laplacian. Our approach,
however, is more direct and relatively simple, and could potentially be useful to
prove optimal weighted estimates also in the regime q ≥ 3.
