The study of dispersive properties of Schr{\"o}dinger 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 Schr{\"o}dinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$-$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness 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 $q \geq 3$.

}, isbn = {978-3-319-58904-6}, doi = {10.1007/978-3-319-58904-6_11}, url = {https://doi.org/10.1007/978-3-319-58904-6_11}, author = {Felice Iandoli and Raffaele Scandone}, editor = {Alessandro Michelangeli and Gianfausto Dell{\textquoteright}Antonio} } @article {2014, title = {Dynamics on a graph as the limit of the dynamics on a "fat graph"}, number = {SISSA;69/2014/MATE}, year = {2014}, institution = {SISSA}, abstract = {We discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (\fat graph") when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph. Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.}, url = {http://urania.sissa.it/xmlui/handle/1963/7485}, author = {Gianfausto Dell{\textquoteright}Antonio and Alessandro Michelangeli} } @article {2005, title = {Decay of a bound state under a time-periodic perturbation: a toy case}, journal = {J. Phys. A 38 (2005) 4769-4781}, number = {SISSA;54/2004/FM}, year = {2005}, abstract = {We study the time evolution of a three dimensional quantum particle, initially in a bound state, under the action of a time-periodic zero range interaction with {\textquoteleft}{\textquoteleft}strength\\\'\\\' (\\\\alpha(t)). Under very weak generic conditions on the Fourier coefficients of (\\\\alpha(t)), we prove complete ionization as (t \\\\to \\\\infty). We prove also that, under the same conditions, all the states of the system are scattering states.}, doi = {10.1088/0305-4470/38/22/002}, url = {http://hdl.handle.net/1963/2298}, author = {Michele Correggi and Gianfausto Dell{\textquoteright}Antonio} } @article {1998, title = {Diffusion of a particle in presence of N moving point sources}, journal = {Annales Poincare Phys.Theor.69:413-424,1998}, number = {SISSA;9/96/ILAS/FM}, year = {1998}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/134}, author = {Gianfausto Dell{\textquoteright}Antonio and Rodolfo Figari and Alessandro Teta} }