Abstract
We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schrödinger 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.
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