Superoscillating Sequences and Supershifts for Families of Generalized Functions

We construct a large class of superoscillating sequences, more generally of F-supershifts, where 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 is an element of R. The frame in which we introduce such families is that of the evolution through Schrodinger equation (i partial derivative/partial derivative t - H (x))(psi) = 0 (H(x) = - (partial derivative(2)/partial derivative x(2))/2+V (x)), V being a suitable potential). If F = {(t, x) bar right arrow phi(lambda)(t, x); lambda is an element of R}, where phi(lambda) is evolved from the initial datum x bar right arrow e(i lambda x), F-supershifis will be of the form {Sigma(N)(j=0) C-j (N, a)phi(1-2j/N)}(N >= 1) for a is an element of R\[-1, 1], taking C-j (N, a) = ((N)j)(1 + a)( N-j) (1 - a)(j) /2(N). Our results rely on the fact that integral operators of the Presnel type govern, as in optical diffraction, the evolution through the Schrodinger equation, such operators acting continuously on the weighted algebra of entire functions Exp(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 F-supershift, F being a family of C-infinity functions or distributions in (t, x), to that where F is a family of hyperfunctions in x, depending on t as a parameter.