# Elliptic Harnack inequality for 1D Schrodinger operator?

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:

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 $r$.

I see results such as the "strong Harnack inequality" in Aizenman and Simon's 1982 paper "Brownian Motion and Harnack Inequality for Schrödinger Operators". But their equivalent of the constant $C_{H}$ appears to depend on BOTH $x$ and $r$.

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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 $x \rightarrow \infty$.