Details
| Originalsprache | Englisch |
|---|---|
| Aufsatznummer | 12 |
| Fachzeitschrift | Partial Differential Equations and Applications |
| Jahrgang | 4 |
| Ausgabenummer | 2 |
| Frühes Online-Datum | 8 März 2023 |
| Publikationsstatus | Veröffentlicht - Apr. 2023 |
Abstract
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Computational Mathematics
- Mathematik (insg.)
- Analysis
- Mathematik (insg.)
- Angewandte Mathematik
- Mathematik (insg.)
- Numerische Mathematik
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in: Partial Differential Equations and Applications, Jahrgang 4, Nr. 2, 12, 04.2023.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Observability for Schrödinger equations with quadratic Hamiltonians
AU - Waters, Alden Marie Seaburg
N1 - Funding Information: The author would like to thank the anonymous referees for careful reading of the manuscript and helpful suggestions. She would also like to thank Karel Pravda-Starov for writing [38] which is the basis for the construction here.
PY - 2023/4
Y1 - 2023/4
N2 - 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 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, 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.
AB - 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 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, 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.
KW - Control theory
KW - Observability
KW - Schrödinger equations
UR - http://www.scopus.com/inward/record.url?scp=85150172399&partnerID=8YFLogxK
U2 - 10.1007/s42985-023-00229-z
DO - 10.1007/s42985-023-00229-z
M3 - Article
VL - 4
JO - Partial Differential Equations and Applications
JF - Partial Differential Equations and Applications
SN - 2662-2971
IS - 2
M1 - 12
ER -