Observability for Schr\'odinger equations with quadratic Hamiltonians
| Autor: | Waters, Alden |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Theorem 4, Example 1, and Corollary 2, are new to this version, Lemma 3 has been replaced with a more general argument. Typos corrected and details in the computations expanded |
| Databáze: | arXiv |
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