Carleman estimates for the Schrödinger operator and applications to unique continuation
Ihyeok Seo
Communications on Pure & Applied Analysis 2012, 11(3): 1013-1036 doi: 10.3934/cpaa.2012.11.1013
We extend previously known Carleman estimates [18, 16, 11] for the (time-dependent) Schrödinger operator $i\partial_t+\Delta$ to a wider range for which inhomogeneous Strichartz estimates ([9, 27]) are known to hold. Then we apply them to obtain new results on unique continuation for the Schrödinger equation which include more general classes of potentials. Also, we obtain a unique continuation result for nonlinear Schrödinger equations.
keywords: unique continuation Carleman estimate Schrödinger equation. Strichartz estimate
Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications
Youngwoo Koh Ihyeok Seo
Discrete & Continuous Dynamical Systems - A 2017, 37(9): 4877-4906 doi: 10.3934/dcds.2017210

We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-\Delta)^{a/2}u=F(x, t)$, $u(x, 0)=f(x)$, of general orders $a>1$ with radial data $f, F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in [1] concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted $L^2$ estimates to the well-posedness theory for the Schrödinger equations with time-dependent potentials in the class.

keywords: Strichartz estimates well-posedness Schrödinger equations Morrey-Campanato class

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