Gaussian decay of Harmonic Oscillators and related models

TL;DR

This paper investigates how the Gaussian decay properties of harmonic oscillator eigenfunctions are affected by complex perturbations, revealing instability and characterizing optimal decay rates linked to Hardy's Uncertainty Principle.

Contribution

It demonstrates the instability of Gaussian decay under complex perturbations and provides a quantitative characterization of the sharpest possible decay for related Schrödinger evolutions.

Findings

Decay is not stable under bounded complex perturbations.

Characterization of the optimal Gaussian decay as a function of system parameters.

Connection established with Hardy's Uncertainty Principle.

Abstract

We prove that the decay of the eigenfunctions of harmonic oscillators, uniform electric or magnetic fields is not stable under 0-order complex perturbations, even if bounded, of these Hamiltonians, in the sense that we can produce solutions to the evolutionary Schr\"odinger flows associated to the Hamiltonians, with a stronger Gaussian decay at two distinct times. We then characterize, in a quantitative way, the sharpest possible Gaussian decay of solutions as a function of the oscillation frequency or the strength of the field, depending on the Hamiltonian which is considered. This is connected to the Hardy's Uncertainty Principle for free Schr\"odinger evolutions.