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Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators acting on spaces of entire hyperholomorphic functions. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different from each other, in both cases, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite-order differential operators acting on these two classes of functions. This is particularly remarkable since the exponential function is not in the kernel of the Dirac operator, but it plays an important role in the theory of entire monogenic functions with growth conditions. [ABSTRACT FROM AUTHOR]

We consider the evolution, for a time-dependent Schrödinger equation, of the so-called Dirac comb. We show how this evolution allows us to recover explicitly (indeed optically) the values of the quadratic generalized Gauss sums. Moreover we use the phenomenon of superoscillatory sequences to prove that such Gauss sums can be asymptotically recovered from the values of the spectrum of any sufficiently regular function compactly supported on R. The fundamental tool we use is the so called Galilean transform that was introduced and studied in the context on non-linear time dependent Schrödinger equations. Furthermore, we utilize this tool to understand in detail the evolution of an exponential e i ω x in the case of a Schrödinger equation with time-independent periodic potential. [ABSTRACT FROM AUTHOR]

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The study of the evolution of superoscillations as initial datum of field equations requires the notion of supershift, which generalizes the concept of superoscillations. The present paper has a dual purpose. The first one is to give an updated and self-contained explanation of the strategy to study the evolution of superoscillations by referring to the quantum-mechanical Schrödinger equation and its variations. The second purpose is to treat the Dirac equation in relativistic quantum theory. The treatment of the evolution of superoscillations for the Dirac equation can be deduced by recent results on the Klein–Gordon equation, but further additional considerations are in order, which are fully described in this paper. [ABSTRACT FROM AUTHOR]

In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist in time. In this paper, we investigate the centrifugal potential, which is a singular potential, and we show that the techniques developed to study the evolution of superoscillations in the case of the Schrödinger equation with a quadratic Hamiltonian apply to this setting. We also specify, in the case of the centrifugal potential, the notion of super-shift of the limit solution, a fact explained in the last section of this paper. It then becomes apparent that superoscillations are just a particular case of super-shift. [ABSTRACT FROM AUTHOR]

Weak measurement and weak values have a very deep meaning in quantum mechanics, and new phenomena associated to them have recently been observed experimentally. These measurements give rise to the notion of superoscillating sequences of functions. In the recent years the authors have started an intensive study of this topic from the mathematical point of view. In this paper we use a generalization of the Schrödinger equation in which the spatial derivative is replaced by a suitable convolution operator to prove the existence of a large class of superoscillating sequences. The method we use also allows us to construct such sequences explicitly. [ABSTRACT FROM AUTHOR]

Abstract: Superoscillatory functions were introduced in Aharonov and Vaidman (1990) [5], and recently studied in detail in Aharonov et al. (2011) [2], Berry (1994) [7] and Berry and Popescu (2006) [9]. In this paper we study the time evolution of a superoscillating function, by taking it as initial value for the Cauchy problem for the Schrödinger equation. By using convolution operators on spaces of entire functions with suitable growth conditions, we prove the surprising fact that the superoscillatory phenomenon persists for all values of t. [Copyright &y& Elsevier]