Superoscillating sequences and supershifts in families of generalized functions

Superoscillations arise naturally from the theory of weak values in quantum physics, and they were originally introduced by Aharonov and, independently an in a different context, by Berry. In the last several years there has been a significant interest in the mathematical properties of these objects, including a now well understood approach to the study of longevity of superoscillations when they evolve according to the Schrodinger equation. In this talk, however, I will focus on how to construct a large class of superoscillating sequences. IN order to do so, we will use the Schrodinger equation with suitable hamiltonians, to evolve the basic examples of superoscillations. Even in the simple case of the quantum harmonic oscillator, we are forced to deal with unavoidable singularities, that in turn encouraged us to study the notion of superoscillations for hyperfunctions. The talk is based on a paper jointly coauthored with F.Colombo, I.Sabadini (both from the Politecnico of Milano, Italy) and A.Yger (from the University of Bordeaux).