Ricerc@Sapienza

We prove a sharp resolvent estimate in scale invariant norms of Amgon–Hörmander type for a magnetic Schrödinger operator on Rn, n ≥ 3 L = −(∂ + iA)2 + V with large potentials A, V of almost critical decay and regularity. The estimate is applied to prove sharp smoothing and Strichartz estimates for the Schrödinger, wave and Klein–Gordon flows associated to L.

We investigate the controllability of the Jaynes-Cummings dynamics in the resonant and nearly resonant regime. We analyze two different types of control operators acting on the bosonic part, corresponding—in the application to cavity Quantum Electro Dynamics—to an external electric and magnetic field, respectively. For these models, we prove approximate controllability for all values of the coupling constant g ∈ R, except those in a countable set S∗ which is explicitly characterized in the statement.

We report here on some recent results obtained in collaboration with V. Maz'ya and G. Schmidt cite{LMS2017}.
We derive semi-analytic cubature formulas for the solution of the Cauchy problem for the Schrödinger equation which are fast and accurate also if the space dimension is greater than or equal to 3.
We follow ideas of the method of approximate approximations, which provides high-order semi-analytic cubature formulas for many important integral operators of mathematical physics.

We show that the presence of negative eigenvalues in the spectrum of the angular component of an electromagnetic Schr\"odinger hamiltonian $H$ generically produces a lack of the classical time-decay for the associated Schr\"odinger flow $e^{-itH}$. This is in contrast with the fact that dispersive estimates (Strichartz) still hold, in general, also in this case.

We prove local smoothing, local energy decay and weighted Strichartz inequalities for fractional Schrodinger equations with a Aharonov-Bohm magnetic field in 2D. Explicit representations of the flows in terms of spherical expansions of the Hamiltonians are involved in the study. An improvement of the free estimate is proved, when the total flux of the magnetic field through the unit sphere is not an integer.