Observability for Schrödinger equations with quadratic Hamiltonians

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 estima...

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Bibliographic Details
Published inarXiv.org
Main Author Waters, Alden
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.05.2022
Subjects
Harmonic oscillators
Mathematical analysis
Schrodinger equation
Time dependence
Wave packets
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Summary: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.
ISSN:2331-8422