For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is:




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There exist $C\_{H} > 0$ and $\delta \in (0,1)$ such that for any $B(x,r)$ in $\mathbb{R}$, if $u \geq 0$ satisfies $Hu \equiv 0$ in $B(x,r)$, then


$$\sup\_{B(x,\delta r)} u \leq C\_{H} \inf\_{B(x,\delta r)} u$$


where $C\_{H}$ and $\delta$ are indepedent of $x$ and $r$.




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I see results such as the "strong Harnack inequality" in Aizenman and Simon's 1982 paper ["Brownian Motion and Harnack Inequality for Schrödinger Operators"](http://sloan2.caltech.edu/SimonPapers/135.pdf). But their equivalent of the constant $C\_{H}$ appears to depend on BOTH $x$ and $r$. 



Elliptic Harnack inequality for 1D Schrodinger operator?
The constant $C\_H$ must depend on $x$ and $r$. Consider for example $u = e^{\frac{1}{2}|x|^2}$, which satisfies
$$\Delta u = (1+|x|^2)u.$$
Then $u$ has a minimum of $1$ at $0$, which clearly doesn't control the maximum of $u$ on $B\_{\delta r}$ times any constant independent of $r$.


Furthermore, $C\_H$ must depend on $x$ because if we fix $r$ then the ratio 
$$\frac{u(x+r)}{u(x-r)}$$
blows up as $x \rightarrow \infty$.


