  We construct in this paper a large class of superoscillating sequences, more
generally of $\mathscr F$-supershifts, where $\mathscr F$ is a family of smooth
functions (resp. distributions, hyperfunctions) indexed by a real parameter
$\lambda\in \R$. The key model we introduce in order to generate such families
is the evolution through a Schr\"odinger equation $(i\partial/\partial t -
\mathscr H(x))(\psi)=0$ with a suitable hamiltonian $\mathscr H$, in particular
a suitable potential $V$ when $\mathscr H(x) = -(\partial^2/\partial x^2)/2 +
V(x)$. The family $\mathscr F$ is in this case $\mathscr F= \{(t,x) \mapsto
\varphi_\lambda(t,x)\,;\, \lambda \in \R\}$, where $\varphi_\lambda$ is evolved
from the initial datum $x\mapsto e^{i\lambda x}$. Then $\mathscr F$-supershifts
will be of the form $\{\sum_{j=0}^N C_j(N,a) \varphi_{1-2j/N}\}_{N\geq 1}$ for
$a\in \R\setminus [-1,1]$, taking $C_j(N,a)
=\binom{N}{j}(1+a)^{N-j}(1-a)^j/2^N$. We prove the locally uniform convergence
of derivatives of the supershift towards corresponding derivatives of its
limit. We analyse in particular the case of the quantum harmonic oscillator,
which forces us, in order to take into account singularities of the evolved
datum, to enlarge the notion of supershifts for families of functions to a
similar notion for families of hyperfunctions, thus beyond the frame of
distributions.
