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 $q \geq 3$.

JF - Advances in Quantum Mechanics: Contemporary Trends and Open Problems PB - Springer International Publishing CY - Cham SN - 978-3-319-58904-6 UR - https://doi.org/10.1007/978-3-319-58904-6_11 ER - TY - RPRT T1 - Dynamics on a graph as the limit of the dynamics on a "fat graph" Y1 - 2014 A1 - Gianfausto Dell'Antonio A1 - Alessandro Michelangeli AB - 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. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7485 U1 - 7598 ER - TY - JOUR T1 - Decay of a bound state under a time-periodic perturbation: a toy case JF - J. Phys. A 38 (2005) 4769-4781 Y1 - 2005 A1 - Michele Correggi A1 - Gianfausto Dell'Antonio AB - 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. UR - http://hdl.handle.net/1963/2298 U1 - 1718 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Diffusion of a particle in presence of N moving point sources JF - Annales Poincare Phys.Theor.69:413-424,1998 Y1 - 1998 A1 - Gianfausto Dell'Antonio A1 - Rodolfo Figari A1 - Alessandro Teta PB - SISSA Library UR - http://hdl.handle.net/1963/134 U1 - 75 U2 - LISNU U3 - Interdisciplinary Laboratory for Advanced Studies ER -