Research
I completed my three years Ph.D program in Mathematics at Sapienza University of Rome in December 2017. My dissertation focused on unique continuation properties for the Zakharov-Kuznetsov equation and on stability of spectral properties for the Lamé operator once it is perturbed by (possibly) complex-valued potentials.
In order to tackle the first problem, the main tool has been to establish Carleman type estimates of a linear differential operator acting on suitably localized functions in space-time.
As regards with the second topic, the main role has been played by providing uniform (w.r.t the spectral parameter) resolvent estimates, which follow from the corresponding estimates for the usual Laplace operator, as soon as the action of the Lamé is better and more deeply understood.
My most up-to-date research interests include:
Spectral Theory. Spectral stability for self-adjoint/ non self-adjoint Schrödinger operators and applications to Maxwell, Lamé, Pauli and Dirac Hamiltonians. Multipliers methods for Spectral Theory ( both on domains and in the whole Euclidean space). Absence of eigenvalues (discrete and embedded). Limiting Absorption Principles (LAP). Eigenvalues distribution and Lieb-Thirring-type inequalities.
Uncertainty Principle and Unique Continuation. Sharpest possible vanishing and infinity decay-rate of non trivial solutions to partial differential equations, such as Korteweg-de Vries, Schrödinger and Zakharov-Kuznetsov models. Persistence properties of the evolution flow. Carleman estimates.
Dispersive and Strichartz estimates for wave-type problems and linear Maxwell systems. Global-in-time Strichartz estimates for linear isotropic/anysotropic Maxwell equations. Dispersive estimate with loss in the exterior/interior of a strictly convex domain. Dispersive estimates in domains with points of zero curvature.
Hardy, Rellich, Hardy-Rellich inequalities. Weighted inequalities. Improved inequalities. Diamagnetic phenomenon.