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 and L 2 - L ∞ observability estimates on unbounded dom...

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Bibliographic Details
Published in:SN partial differential equations and applications 2023-04, Vol.4 (2), Article 12
Main Author: Waters, Alden
Format: Article
Language:English
Subjects:
1. Theory of PDEs
Analysis
Mathematical and Computational Biology
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Original Paper
Partial Differential Equations
Theoretical
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Summary: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 ∞ observability estimates on unbounded domains ω 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 ω c = Ω , a connected bounded domain, is observable, but data centered over 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.
ISSN:2662-2963
2662-2971
DOI:10.1007/s42985-023-00229-z