I got my Bachelor degree and my Master degree in Mathematics at "University of Bari".

From October 2014 to December 2017 I have been a PhD student under the supervision of Prof. Luis Vega. In my thesis, I considered the delta-shell interaction on bounded and smooth domains and its approximation by the coupling of the free Dirac operator with shrinking short-range potentials, getting surprising results related to Klein's Paradox. Moreover, I investigated the Dirac operator perturbed by a particular class of Coulomb-type spherically symmetric potentials, describing the self-adjoint realisations of this operator in terms of the behaviour of the functions of the domain in the origin.

From December 2017 to October 2018 I have been a postdoc under the supervision of Prof. Luis Vega. During this period I worked on Hardy-type inequalities for the Dirac operator. We exploited these inequalities to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix valued potentials of Coulomb type: we characterised its eigenvalues in terms of the Birman-Schwinger principle, and we bounded its discrete spectrum from below, showing that the ground state energy is reached if and only if the potential verifies some rigidity conditions.

I am currently working as a Postdoctoral Fellow at CEREMADE - Université of Paris-Dauphine under the supervision of Prof. Mathieu Lewin.

• F. Pizzichillo, H. Van Den Bosch: ''Self-Adjointness of two dimensional Dirac operators on corner domains''.
  arXiv preprint arXiv:1902.05010, (2019).
• T. Ourmières-Bonafos, F. Pizzichillo: ''Dirac operators and shell interactions: a survey''.
  arXiv preprint arXiv:1902.03901, (2019).

• B. Cassano, F. Pizzichillo, L. Vega: ''A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator''.
  Revista Matemática Complutense (2019); doi: 10.1007/s13163-019-00311-4.
• B. Cassano, F. Pizzichillo: ''Boundary triples for the Dirac operator with Coulomb-type spherically symmetric perturbations''.
  Journal of Mathematical Physics 60, 041502 (2019); doi: 10.1063/1.5063986.
• B. Cassano, F. Pizzichillo: ''Self-adjoint extensions for the Dirac operator with Coulomb-type spherically symmetric potentials''.
  Letters in Mathematical Physics, 1-33 (2018); doi: 10.1007/s11005-018-1093-9.
• A. Mas, F. Pizzichillo: ''Klein's paradox and the relativistic $\delta$-shell interaction in $\mathbb R^3$'',
  Analysis & PDE, 11(3), 705-744 (2017); doi: 10.2140/apde.2018.11.705
• T. Ourmières-Bonafos, K. Pankrashkin, F. Pizzichillo: ''Spectral asymptotics for $\delta$-interactions on sharp cones''.
  Journal of Mathematical Analysis and Applications, 458(1), 566-589 (2017); doi: 10.1063/1.5000381.
• A. Mas, F. Pizzichillo: ''The relativistic spherical $\delta$-shell interaction in $\mathbb R^3$: spectrum and approximations''.
  Journal of Mathematical Physics, 58(8), 082102 (2017); doi: 10.1063/1.5000381.

• PhD Thesis: Singular perturbations for the Dirac Hamiltonian.
  At BCAM - Basque Center for Applied Mathematics, under the supervision of Prof. Luis Vega.
• Master Thesis: Linear and non-linear damped wave equations.
  At University of Bari, under the supervision of Prof. Michael Reissig and Prof. Sandra Lucente.
• Bachelor Thesis: The spectral theorem for compact operators and applications to PDEs. (in italian).
  At University of Bari, under the supervision of Prof. Lorenzo D'Ambrosio .