Observability for Schrödinger equations with quadratic Hamiltonians

Alden Waters ()
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Alden Waters: University of Groningen

Partial Differential Equations and Applications, 2023, vol. 4, issue 2, 1-33

Abstract: 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 $$L^2$$ 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.

Keywords: Control theory; Schrödinger equations; Observability; 35R01; 35R30; 35L20; 58J45; 35A22 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00229-z

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