相关论文: Unique continuation of Schr\"odinger-type equation…
In this paper, we extend our earlier unique continuation results \cite{PZ2} for the Schr\"odinger-type inequality $ |\bar\partial u| \le V|u|$ on a domain in $\mathbb C^n$ by removing the smoothness assumption on solutions $u = (u_1,…
In this paper, we investigate the unique continuation property for the inequality $|\bar\partial u| \le V|u|$, where $u$ is a vector-valued function from a domain in $\mathbb C^n$ to $\mathbb C^N$, and the potential $V\in L^2$. We show that…
We study two types of unique continuation properties for the higher order Schr\"{o}dinger equation with potential $$ i\partial_tu=(-\Delta_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$…
In this note we study the property of unique continuation for solutions of $|(-\Delta)^{\alpha/2}u|\leq|Vu|$, where $V$ is in a function class of potentials including $\bigcup_{p>n/\alpha}L^p(\mathbb{R}^n)$ for $n-1\leq\alpha<n$. In…
In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schr\"odinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
We prove the unique continuation property for the differential inequality $|(-\Delta)^{\alpha/2}u|\leq|V(x)u|$, where $0<\alpha<n$ and $V\in L_{\textrm{loc}}^{n/\alpha,\infty}(\mathbb{R}^n)$, $n\geq3$.
In this paper we develop an abstract method to handle the problem of unique continuation for the Schr\"odinger equation $(i\partial_t+\Delta)u=V(x)u$. In general the problem is to find a class of potentials $V$ which allows the unique…
We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the…
In this article we prove the property of unique continuation (also known for C^\infty functions as quasianalyticity) for solutions of the differential inequality |\Delta u| \leq |Vu| for V from a wide class of potentials (including…
We study uniqueness properties of solutions of Schr\"odinger equations. The aim is to obtain sufficient conditions on the decay behavior of the difference of two solution $u_1-u_2$ of the equation at two different times $t_0=0$ and $t_1=1$…
We obtain unique continuation results for Schrodinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear…
Let $u: \Omega \subset \mathbb C^n \to \mathbb C^m$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)^2 -1 \leq q < \infty$ such that $\displaystyle \frac{1}{p} + \frac{1}{p'} = 1$ and $\displaystyle \frac{1}{p} -…
We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for…
The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schr\"odinger equation. This property guarantees that if a solution of the Schr\"odinger equation vanishes on a set of positive…
We obtain a unique continuation result for the differential inequality $| (i\partial_t +\Delta)u | \leq |Vu| + | W\cdot\nabla u |$ by establishing $L^2$ Carleman estimates. Here, $V$ is a scalar function and $W$ is a vector function, which…
We prove unique continuation properties for solutions of the evolution Schr\"odinger equation with time dependent potentials. As an application of our method we also obtain results concerning the possible concentration profiles of blow up…
This article deals with the weak and strong unique continuation principle for fractional Schr\"odinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable…
We consider the steady fractional Schr\"odinger equation $L u + V u = f$ posed on a bounded domain $\Omega$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ is a potential…
Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.
In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation \begin{equation*} \Delta^2_{X}u=Vu, \end{equation*} where $\Delta_{X}=\Delta_{x}+|x|^{2\alpha}\Delta_{y}$…