Unique continuation for Schrödinger operators in dimension three or less

Sawyer, Eric T.. "Unique continuation for Schrödinger operators in dimension three or less." Annales de l'institut Fourier 34.3 (1984): 189-200. <http://eudml.org/doc/74642>.

@article{Sawyer1984,
abstract = {We show that the differential inequality $\vert \Delta u\vert \le v\vert u\vert $ has the unique continuation property relative to the Sobolev space $H^\{2,1\}_\{loc\}(\Omega )$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition\begin\{\} (K\_ n^\{\rm loc\})\ \lim \_\{r\rightarrow 0\}\sup \_\{x\in K\}\int \_\{\vert x-y\vert &lt; r\}\vert x-y\vert ^\{2-n\}v(y)dy=0 \end\{\}for all compact $K\subset \Omega $, where if $n=2$, we replace $\vert x- y\vert ^\{2-n\}$ by $-\log \vert x-y\vert $. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,\ldots $ and any $u\in H_c^\{2,1\}(\{\bf R\}^3-\lbrace 0\rbrace ),$\begin\{\} \{\vert u(x)\vert \over \vert x\vert ^N\}\le C\int \_\{\{\bf R\}^3\}\vert x-y\vert ^\{-1\}\{\vert \Delta u(y)\vert \over \vert y\vert ^N\}dy \text\{for\} \text\{a.e.\} x \text\{in\} \{\bf R\}^3.\end\{\}},
author = {Sawyer, Eric T.},
journal = {Annales de l'institut Fourier},
keywords = {unique continuation; Schrödinger operators; Sobolev space},
language = {eng},
number = {3},
pages = {189-200},
publisher = {Association des Annales de l'Institut Fourier},
title = {Unique continuation for Schrödinger operators in dimension three or less},
url = {http://eudml.org/doc/74642},
volume = {34},
year = {1984},
}

TY - JOUR
AU - Sawyer, Eric T.
TI - Unique continuation for Schrödinger operators in dimension three or less
JO - Annales de l'institut Fourier
PY - 1984
PB - Association des Annales de l'Institut Fourier
VL - 34
IS - 3
SP - 189
EP - 200
AB - We show that the differential inequality $\vert \Delta u\vert \le v\vert u\vert $ has the unique continuation property relative to the Sobolev space $H^{2,1}_{loc}(\Omega )$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition\begin{} (K_ n^{\rm loc})\ \lim _{r\rightarrow 0}\sup _{x\in K}\int _{\vert x-y\vert &lt; r}\vert x-y\vert ^{2-n}v(y)dy=0 \end{}for all compact $K\subset \Omega $, where if $n=2$, we replace $\vert x- y\vert ^{2-n}$ by $-\log \vert x-y\vert $. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,\ldots $ and any $u\in H_c^{2,1}({\bf R}^3-\lbrace 0\rbrace ),$\begin{} {\vert u(x)\vert \over \vert x\vert ^N}\le C\int _{{\bf R}^3}\vert x-y\vert ^{-1}{\vert \Delta u(y)\vert \over \vert y\vert ^N}dy \text{for} \text{a.e.} x \text{in} {\bf R}^3.\end{}
LA - eng
KW - unique continuation; Schrödinger operators; Sobolev space
UR - http://eudml.org/doc/74642
ER -