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Ihyeok Seo

Publications and source records attributed to Ihyeok Seo.

51 records · Page 3Linked to original sources
arXiv2017

Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications

We obtain weighted $L^2$ Strichartz estimates for Schrödinger equations $i\partial_tu+(-Δ)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, 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 \cite{BBCRV} 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.

math.AP↗
arXiv2016

Inhomogeneous Strichartz estimates for Schrödinger's equation

Foschi and Vilela in their independent works (\cite{F},\cite{V}) showed that the range of $(1/r,1/\widetilde{r})$ for which the inhomogeneous Strichartz estimate $ \big\|\int_{0}^{t}e^{i(t-s)Δ}F(\cdot,s)ds\big\|_{L^{q}_tL^{r}_x} \lesssim \|F\|_{L^{\widetilde{q}'}_tL^{\widetilde{r}'}_x} $ holds for some $q,\widetilde{q}$ is contained in the closed pentagon with vertices $A,B,B',P,P'$ except the points $P,P'$ (see Figure 1). We obtain the estimate for the corner points $P,P'$.

math.AP↗
arXiv2015

On weighted $L^2$ estimates for solutions of the wave equation

In this paper we consider weighted $L^2$ integrability for solutions of the wave equation. For this, we obtain some weighed $L^2$ estimates for the solutions with weights in Morrey-Campanato classes. Our method is based on a combination of bilinear interpolation and localization argument which makes use of Littlewood-Paley theorem and a property of Hardy-Littlewood maximal functions. We also apply the estimates to the problem of well-posedness for wave equations with potentials.

math.AP↗
arXiv2015

On absolute continuity of the spectrum of periodic Schrödinger operators

In this paper we find a new condition on a real periodic potential for which the self-adjoint Schrödinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted $L^2$ spaces on the torus, and an oscillatory integral theorem.

math.SP↗
arXiv2015

A remark on unique continuation for the Cauchy-Riemann operator

In this note we obtain a unique continuation result for the differential inequality $|\bar{\partial}u|\leq|Vu|$, where $\bar{\partial}=(i\partial_y+\partial_x)/2$ denotes the Cauchy-Riemann operator and $V(x,y)$ is a function in $L^2(\mathbb{R}^2)$.

math.AP↗
arXiv2014

From resolvent estimates to unique continuation for the Schrödinger equation

In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation $(i\partial_t+Δ)u=V(x)u$. In general the problem is to find a class of potentials $V$ which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted $L^2$ resolvent estimates $\|(-Δ-z)^{-1}f\|_{L^2(|V|)}\leq C\|f\|_{L^2(|V|^{-1})}$. We carry out it in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of limiting absorption principle and Kato $H$-smoothing theorem in spectral theory, and employ some tools from harmonic analysis. Once the resolvent estimate is set up for a potential class, from our abstract theory the unique continuation would follow from the same potential class. Also, it turns out that there can be no dented surface on the boundary of the maximal open zero set of the solution $u$. In this regard, another main issue for us is to know which class of potentials allows the resolvent estimate. We establish such a new class which contains previously known ones, and will also apply it to the problem of well-posedness for the equation.

math.AP↗
arXiv2013

On minimal support properties of solutions of Schrödinger equations

In this paper we obtain minimal support properties of solutions of Schrödinger equations. We improve previously known conditions on the potential for which the measure of the support of solutions cannot be too small. We also use these properties to obtain some new results on unique continuation for the Schrödinger operator.

math.AP↗
arXiv2013

Global unique continuation from a half space for the Schrödinger equation

We obtain a global unique continuation result for the differential inequality $|(i\partial_t+Δ)u|\leq|V(x)u|$ in $\mathbb{R}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials $V(x)$ in $\mathbb{R}^{n}$. Our method is based on a new type of Carleman estimates for the operator $i\partial_t+Δ$ on $\mathbb{R}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.

math.AP↗
arXiv2013

On unique continuation for Schrödinger operators of fractional and higher orders

In this note we study the property of unique continuation for solutions of $|(-Δ)^{α/2}u|\leq|Vu|$, where $V$ is in a function class of potentials including $\bigcup_{p>n/α}L^p(\mathbb{R}^n)$ for $n-1\leqα<n$. In particular, when $n=2$, our result gives a unique continuation theorem for the fractional ($1<α<2$) Schrödinger operator $(-Δ)^{α/2}+V(x)$ in the full range of $α$ values.

math.AP↗