CPAA
Carleman estimates for the Schrödinger operator and applications to unique continuation
Ihyeok Seo
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
DCDS
On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications
Chu-Hee Cho Youngwoo Koh Ihyeok Seo
In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrödinger equation in the radial case. Then we apply them to the well-posedness theory for the equation $i\partial_{t}u+|\nabla|^{\alpha}u=V(x,t)u$, $1<\alpha<2$, with radial $\dot{H}^\gamma$ initial data below $L^2$ and radial potentials $V\in L_t^rL_x^w$ under the scaling-critical range $\alpha/r+n/w=\alpha$.
keywords: Strichartz estimates well-posedness Schrödinger equations.
DCDS
Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications
Youngwoo Koh Ihyeok Seo

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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