Quantum harmonic oscillator with superoscillating initial datum

TL;DR

This paper investigates how superoscillating initial data evolve in a quantum harmonic oscillator, revealing amplification of superoscillations and finite-time singularities, with implications for persistent superoscillating behavior over time.

Contribution

It demonstrates that superoscillations are amplified by the harmonic potential and that such behavior persists over time, leading to finite-time singularities in the solution.

Findings

Superoscillations are amplified by the harmonic potential.

The solution develops a finite-time singularity.

Superoscillating behavior persists over time.

Abstract

In this paper we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schr\"odinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.