Observability for Schr\"odinger equations with quadratic Hamiltonians

TL;DR

This paper develops a Gaussian wavepacket-based parametrix for Schrödinger equations with quadratic Hamiltonians, establishing observability estimates on unbounded domains and analyzing conditions for initial data observability.

Contribution

It introduces a precise Gaussian wavepacket parametrix for time-dependent harmonic oscillators and derives new observability estimates for specific initial data classes.

Findings

Observable initial data must be away from the domain's complement

Data centered over the domain requires near-Gaussian form

Counterexamples challenge existing principles for harmonic oscillators

Abstract

We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schr\"odinger equation using Gaussian wavepackets. This parametrix of Gaussian wavepackets is precise and tractable. Using this parametrix we prove $L^2$ and $L^2-L^{\infty}$ observability estimates on unbounded domains $\omega$ for a restricted class of initial data. This data includes a class of compactly supported piecewise $C^1$ functions which have been extended from characteristic functions. Initial data of this form which has the bulk of its mass away from $\omega^c=\Omega$, a connected bounded domain, is observable, but data centered over $\Omega$ must be very nearly a single Gaussian to be observable. We also give counterexamples to established principles for the simple harmonic oscillator in the case of certain time dependent harmonic oscillators.