On persistence of superoscillations for the Schrödinger equation with time‐dependent quadratic Hamiltonians

Abstract

In this work, we prove the persistence in time of superoscillations for the Schrödinger equation with time‐dependent coefficients. In order to prove the persistence of superoscillations, we have conditioned the coefficients to satisfy a Riccati system, and we have expressed the solution as a convolution operator in terms of solutions of this Riccati system. Further, we have solved explicitly the Cauchy initial value problem with three different kinds of superoscillatory initial data. The operator is defined on a space of entire functions. Particular examples include Caldirola‐Kanai and degenerate parametric harmonic oscillator Hamiltonians, and other examples could include Hamiltonians not self‐adjoint. For these examples, we have illustrated numerically the convergence on real and imaginary parts.