# Rapidly-oscillating scatteringless non-Hermitian potentials and the absence of Kapitza stabilization

## TL;DR

This paper investigates non-Hermitian oscillating potentials in quantum and classical wave systems, revealing that certain imaginary amplitude modulations can effectively cancel the potential, preventing dynamical stabilization phenomena like Kapitza stabilization.

## Contribution

It introduces a non-Hermitian extension of the Schrödinger equation showing that specific imaginary oscillations eliminate the potential's effects, unlike in traditional Hermitian systems.

## Key findings

- Imaginary amplitude modulations with one-sided Fourier spectra cancel the potential effects.
- No Kapitza stabilization occurs under these non-Hermitian oscillations.
- The potential's influence is effectively nullified for a wide class of modulations.

## Abstract

In the framework of the ordinary non-relativistic quantum mechanics, it is known that a quantum particle in a rapidly-oscillating bound potential with vanishing time average can be scattered off or even trapped owing to the phenomenon of dynamical (Kapitza) stabilization. A similar phenomenon occurs for scattering and trapping of optical waves. Such a remarkable result stems from the fact that, even though the particle is not able to follow the rapid external oscillations of the potential, these are still able to affect the average dynamics by means of an effective -albeit small- nonvanishing potential contribution. Here we consider the scattering and dynamical stabilization problem for matter or classical waves by a bound potential with oscillating ac amplitude $f(t)$ in the framework of a non-Hermitian extension of the Schr\"odinger equation, and predict that for a wide class of imaginary amplitude modulations $f(t)$ possessing a one-sided Fourier spectrum the oscillating potential is effectively canceled, i.e. it does not have any effect to the particle dynamics, contrary to what happens in the Hermitian case