Schrödinger Flow’s Dispersive Estimates in a regime of Re-scaled Potentials

The problem of monitoring the (constants in the estimates that quantify the) dispersive behaviour of the flow generated by a Schrödinger operator is posed in terms of the scaling parameter that expresses the small size of the support of the potential, along the scaling limit towards a Hamiltonian of point interaction. At positive size, dispersive estimates are completely classical, but their dependence on the short range of the potential is not explicit, and the understanding of such a dependence would be crucial in connecting the dispersive behaviour of the short-range Schrödinger operator with the zero-range Hamiltonian. The general set-up of the problem is discussed, together with preliminary answers, open questions, and plausible conjectures, in a ‘propaganda’ spirit for this subject.