서이혁 교수

We prove the unique continuation property for the differential inequality $|(-\Delta )^{\alpha /2}u|\leq |V(x)u|$, where $0<\alpha <n$ and $V\in L_{\textrm {loc}}^{n/\alpha ,\infty }(\mathbb {R}^n)$, $n\geq 3$.

In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schrödinger equations $ iu_t + \Delta u = |u|^{p-1}u $, given initial data which is analytic with fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $ t $ cannot decay faster than $ 1/|t| $ as $ |t|\rightarrow\infty $. This extends the previous work of Tesfahun [19] for the cubic case $ p = 3 $ to the cases where $ p $ is any odd integer greater than $ 3 $.

In this note we study the property of unique continuation for solutions of , where V is in a function class of potentials including for . In particular, when , our result gives a unique continuation theorem for the fractional Schrödinger operator in the full range of α values.

In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i partial-differential Subscript t Baseline plus normal upper Delta right-parenthesis u equals upper V left-parenthesis x right-parenthesis u"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi mathvarian

We obtain a unique continuation result for fractional Schrödinger operators with\npotential in Morrey spaces. This is based on Carleman inequalities for fractional\nLaplacians.

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 th