Observability for Schrödinger equations with quadratic Hamiltonians

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 $$L^2$$ L 2 and $$L^2-L^{\infty }$$ L 2 - L ∞ observability estimates on unbounded domains $$\omega $$ ω for a restricted class of initial data. This data includes a class of compactly supported piecewise $$C^1$$ C 1 functions which have been extended from characteristic functions. Initial data of this form which has the bulk of its mass away from $$\omega ^c=\Omega $$ ω c = Ω , a connected bounded domain, is observable, but data centered over $$\Omega $$ Ω 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.