We construct a large class of superoscillating sequences, more generally of {mathscr {F}}-supershifts, where {mathscr {F}} is a family of smooth functions in (t, x) (resp. distributions in (t, x), or hyperfunctions in x depending on the parameter t) indexed by lambda in {mathbb {R}}. The frame in which we introduce such families is that of the evolution through Schrödinger equation (ipartial /partial t - {mathscr {H}}(x))(psi )=0 ({mathscr {H}}(x) = -(partial ^2/partial x^2)/2 + V(x)), V being a suitable potential). If {mathscr {F}}= {(t,x) mapsto varphi _lambda (t,x),;, lambda in {mathbb {R}}}, where varphi _lambda is evolved from the initial datum xmapsto e^{ilambda x}, {mathscr {F}}-supershifts will be of the form {sum _{j=0}^N C_j(N,a) varphi _{1-2j/N}}_{Nge 1} for ain {mathbb {R}}{setminus }[-1,1], taking C_j(N,a) =left( {begin{array}{c}N jend{array}}right) (1+a)^{N-j}(1-a)^j/2^N. Our results rely on the fact that integral operators of the Fresnel type govern, as in optical diffraction, the evolution through the Schrödinger equation, such operators acting continuously on the weighted algebra of entire functions mathrm{Exp}({mathbb {C}}). Analyzing in particular the quantum harmonic oscillator case forces us, in order to take into account singularities of the evolved datum that occur when the stationary phasis in the Fresnel operator vanishes, to enlarge the notion of {mathscr {F}}-supershift, {mathscr {F}} being a family of C^infty functions or distributions in (t, x), to that where {mathscr {F}} is a family of hyperfunctions in x, depending on t as a parameter.
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