Bounded-potential weak UCP

Let $x_0\in \mathrm{\Omega }$ have some neighborhood on which $\psi =0$ almost everywhere. Choose any $R>0$ with $\overline{B_R(x_0)}\subset \mathrm{\Omega }$; no vanishing on this larger ball is assumed. Take a smooth cutoff $\chi$ supported in $B_R(x_0)$, equal to one on $B_{R\mathrm{/}2}(x_0)$, and put $u=\chi \psi$, $e=2\mathrm{\nabla }\chi \cdot \mathrm{\nabla }\psi +(\mathrm{\Delta }\chi )\psi$. By [F1], $u$ is compactly supported $H^2$, $\mathrm{\Delta }u=Bu+e$, and $e\in L^2$ has support in $R\mathrm{/}2\le r<R$, where $r=\mathrm{\mid }x-x_0\mathrm{\mid }$. The initial known zero neighborhood ensures that $u$ vanishes near $x_0$, exactly as required by [F2]. Local boundedness on the compact ball gives $M=\mathrm{\parallel }B{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(B_R(x_0))}<\mathrm{\infty }$.

Set $X_s=\mathrm{\parallel }{r}^{-\tau }u{\mathrm{\parallel }}_2$, $\tau =s+1$. All weighted norms are finite since $u$ vanishes near the center. The estimate [F2] and the triangle inequality give $X_s\le (2\mathrm{/}s)(\mathrm{\parallel }{r}^{-\tau +2}Bu{\mathrm{\parallel }}_2+\mathrm{\parallel }{r}^{-\tau +2}e{\mathrm{\parallel }}_2)$. On the support of $u$ one has $r^2\mathrm{\mid }B\mathrm{\mid }\le M{R}^2$, so the first norm is at most $M{R}^2{X}_s$. For every integer $s\ge \max \{1,4M{R}^2\}$ this term can be absorbed, giving $X_s\le (4\mathrm{/}s)\mathrm{\parallel }{r}^{-\tau +2}e{\mathrm{\parallel }}_2$. Since $-\tau +2=1-s\le 0$ and $e$ is supported where $r\ge R\mathrm{/}2$, its norm is bounded by $(R\mathrm{/}2{)}^{-\tau +2}\mathrm{\parallel }e{\mathrm{\parallel }}_2$. This absorption uses only the modulus of the complex function $B$.

Fix $0<\rho <R\mathrm{/}2$. On $B_{\rho }(x_0)$ the cutoff is one and $r^{\tau }\le {\rho }^{\tau }$, hence $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{L^2(B_{\rho }(x_0))}\le {\rho }^{\tau }{X}_s\le (4\mathrm{/}s)(2\rho \mathrm{/}R{)}^{s+1}(R\mathrm{/}2{)}^2\mathrm{\parallel }e{\mathrm{\parallel }}_2$ for all integers $s$ from step 2.1. The cutoff and $e$ are fixed independently of $s$. Because $2\rho \mathrm{/}R<1$, the right-hand side tends to zero as $s\to \mathrm{\infty }$, proving $\psi =0$ almost everywhere on $B_{\rho }(x_0)$. Exhausting $B_{R\mathrm{/}2}(x_0)$ by a countable sequence of such balls proves vanishing almost everywhere on $B_{R\mathrm{/}2}(x_0)$. Thus any initially known zero neighborhood at a center expands to this ball for every admissible $R$, irrespective of the size of the initial neighborhood.

Choose an initial point $x_0$ in the given nonempty zero open subset, and let $y\in \mathrm{\Omega }$ be arbitrary. There is a finite polygonal path $\gamma :[0,1]\to \mathrm{\Omega }$ from $x_0$ to $y$: the subset reachable from $x_0$ by such paths is relatively open, since a small ball around each reachable point permits an extra line segment; its complement is relatively open by the same ball argument, and connectedness forces the reachable subset to be all of $\mathrm{\Omega }$. The compact image $K$ of $\gamma$ has a positive distance from ${\mathbb{R}}^3\setminus \mathrm{\Omega }$ when that complement is nonempty; if $\mathrm{\Omega }={\mathbb{R}}^3$ choose any positive distance parameter. Choose a fixed $R>0$ smaller than this distance, so $\overline{B_R(z)}\subset \mathrm{\Omega }$ for every $z\in K$. Uniform continuity of the finite polygonal path supplies a finite partition $0=t_0<\cdots <t_n=1$ with $\mathrm{\mid }\gamma (t_{j+1})-\gamma (t_j)\mathrm{\mid }<R\mathrm{/}4$.

Apply step 3.1 at $\gamma (t_0)=x_0$ to establish vanishing on $B_{R\mathrm{/}2}(x_0)$. Inductively, if vanishing is established on $B_{R\mathrm{/}2}(\gamma (t_j))$, the next center $\gamma (t_{j+1})$ lies strictly inside this ball and therefore has a zero neighborhood. Step 4.1 with the same radius $R$ then establishes vanishing on $B_{R\mathrm{/}2}(\gamma (t_{j+1}))$. Finitely many applications prove vanishing in a neighborhood of $y$. Thus the open union of zero neighborhoods is all of $\mathrm{\Omega }$. To conclude a single almost-everywhere assertion on $\mathrm{\Omega }$, use the countable base of rational balls: all rational balls contained in some such neighborhood cover $\mathrm{\Omega }$, and $\psi$ vanishes almost everywhere on each of them. Their countable union has a null exceptional set. Only finite path constructions and countable approximation or null-set unions were used; no additional regularity, real-valuedness of $B$, or stronger choice hypothesis is required.