My research topics include:
  • Linear and nonlinear dispersive equations. Asymptotic behavior of solutions and a priori estimates for dispersive flows (time decay, Strichartz, local smoothing, Kato-smoothing, well-posedness+scattering vs finite-time blow-up). Decompositions in dispersive profiles (concentration-compactness methods) and rigidity properties of dispersive groups (Morawetz and virial-type methods). Fourier Restriction Theorems and existence of maximizers. Electromagnetic potentials and the role they play for dispersion. Scaling-critical models and the role of angular momenta. Relativistic Hardy inequalities, diamagnetism and angular momenta. Spectral properties of spherical Laplace-Beltrami operators and their linear perturbations. Mapping properties of wave operators.
  • Helmholtz Equation. Agmon-Hörmander Theory. First-order perturbations of magnetic type. Exterior domains, a priori estimates in Morrey-Campanato spaces. Cross sections and far-field patterns, inverse scattering. Relation with Schrodinger and wave equations. Lamé operators and their relations with Helmholtz.
  • Uncertainty Principle and Unique Continuation. Sharpest possible vanishing and infinity decay-rate of solutions PDE’s. Laplace equation and existence of waveguides. Carleman estimates. Sharp versions of uncertainty principles and their connections with unique continuation properties of evolution flows.
  • Spectral Theory. Spectral stability for non self-adjoint Schrödinger Hamiltonians. Absence of eigenvalues. Multiplier methods for Spectral Theory.