Related papers: Schr\"odinger operators and unique continuation. T…
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…
We prove the unique continuation property for the differential inequality $|(-\Delta)^{\alpha/2}u|\leq|V(x)u|$, where $0<\alpha<n$ and $V\in L_{\textrm{loc}}^{n/\alpha,\infty}(\mathbb{R}^n)$, $n\geq3$.
We prove a quantitative unique continuation principle for Schr\"odinger operators $H=-\Delta+V$ on $\mathrm{L}^2(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^d$ and $V$ is a singular potential: $V \in \mathrm{L}^\infty(\Omega)…
The purpose of this paper is to study the unique continuation property for a Schr\"odinger-type equation $ \bar\partial u = Vu$ on a domain in $\mathbb C^n$, where the solution $u$ may be a scalar function, or a vector-valued function.…
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 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…
The aim of this paper is to review and compare the spectral properties of (the closed extension of) --$\Delta$ + U (V $\ge$ 0) and --$\Delta$ + iV in L 2 (R^d) for C $\infty$ real potentials U or V with polynomial behavior. The case with…
We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schr\"odinger operators. Let $\Lambda_L = (-L/2,L/2)^d$ and $H_L = -\Delta_L + V_L$ be a Schr\"odinger operator on $L^2 (\Lambda_L)$ with a…
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…
We show that formal Schr\"odinger operators with singular potentials from the space W^{-1}_{2,unif}(R) can be naturally defined to give selfadjoint and bounded below operators, which depend continuously in the uniform resolvent sense on the…
We obtain a global unique continuation result for the differential inequality $|(i\partial_t+\Delta)u|\leq|V(x)u|$ in $\mathbb{R}^{n+1}$. This is the first result on global unique continuation for the Schr\"odinger equation with…
We prove quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + Vu = \lambda u$, where $\lambda \in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim \langle…
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…
We investigate the quantitative unique continuation property for solutions to $$\Delta^2_{X} u = V u,$$ where $\Delta_{X} = \Delta_{x} + |x|^{2\beta} \Delta_{y}$ ($0 < \beta \leq 1$), with $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^{n}$,…
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 revisit \cite[Theorem 6.3]{JK}. Following the main ideas used to prove this theorem, we establish a quantitative version of the strong unique continuation property for the Sch\"odinger operator with unbounded potential. We also show that…
We study the one-dimensional Schr\"odinger operators $$ S(q)u:=-u"+q(x)u,\quad u\in \mathrm{Dom}\left(S(q)\right), $$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the…
We prove strong unique continuation property for the differential inequality $|(\partial_t +\Delta)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in…
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 $u$…
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.