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 $\mathbb{R}^3$, 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 $q \geq 3$.

%B Advances in Quantum Mechanics: Contemporary Trends and Open Problems %I Springer International Publishing %C Cham %P 187–199 %@ 978-3-319-58904-6 %G eng %U https://doi.org/10.1007/978-3-319-58904-6_11 %R 10.1007/978-3-319-58904-6_11 %0 Report %D 2014 %T Dynamics on a graph as the limit of the dynamics on a "fat graph" %A Gianfausto Dell'Antonio %A Alessandro Michelangeli %X 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. %I SISSA %G en_US %U http://urania.sissa.it/xmlui/handle/1963/7485 %1 7598 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-12-12T13:16:15Z No. of bitstreams: 1 sissa-preprint-69-2014-mate.pdf: 137454 bytes, checksum: 4d4571bd97df46ee020c81057ce78620 (MD5) %0 Journal Article %J J. Phys. A 38 (2005) 4769-4781 %D 2005 %T Decay of a bound state under a time-periodic perturbation: a toy case %A Michele Correggi %A Gianfausto Dell'Antonio %X 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 ``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. %B J. Phys. A 38 (2005) 4769-4781 %G en_US %U http://hdl.handle.net/1963/2298 %1 1718 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-25T08:22:58Z\\nNo. of bitstreams: 1\\n0408003v2.pdf: 207599 bytes, checksum: 256b29c96ea071184e9455fd1dd92717 (MD5) %R 10.1088/0305-4470/38/22/002 %0 Journal Article %J Annales Poincare Phys.Theor.69:413-424,1998 %D 1998 %T Diffusion of a particle in presence of N moving point sources %A Gianfausto Dell'Antonio %A Rodolfo Figari %A Alessandro Teta %B Annales Poincare Phys.Theor.69:413-424,1998 %I SISSA Library %G en %U http://hdl.handle.net/1963/134 %1 75 %2 LISNU %3 Interdisciplinary Laboratory for Advanced Studies %$ Made available in DSpace on 2004-09-01T12:22:22Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1996