On persistence of superoscillations for Schr\"{o}dinger equation with time-dependent quadratic Hamiltonians

TL;DR

This paper investigates how superoscillations in solutions to the Schrödinger equation with time-dependent quadratic Hamiltonians persist over time, providing explicit solutions and an operator framework for analysis.

Contribution

It introduces an explicit operator based on Riccati system solutions to prove superoscillation persistence in time-dependent quadratic Hamiltonian Schrödinger equations.

Findings

Explicit solutions for different oscillatory initial data.

Operator framework for superoscillation analysis.

Numerical illustrations for specific Hamiltonians.

Abstract

In this work we study the persistence in time of superoscillations for the Schr\"{o}dinger equation with quadratic time-dependent Hamiltonians. We have solved explicitly the Cauchy initial value problem with three different kind of oscillatory initial data. In order to prove the persistence of superoscillations we have defined explicitly an operator in terms of solutions of a Riccati system associated with the variable coefficients of the Hamiltonian. The operator is defined on a space of entire functions. Particular examples include Caldirola-Kanai and degenerate parametric harmonic oscillator Hamiltonians and more. For these examples we have illustrated numerically the convergence on real and imaginary parts.