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Upon reading the link on the spectrum of the Laplacian on $\mathbb{R}^n$ I thought of considering a slight generalization:

Let $p$ be a positive function such that $\frac{1}{p} \in L^1_{loc}(\mathbb{R})$. Consider the Sturm-Liouville operator $L_pu = -(pu')'$ on $\mathbb{R}$. For simplicity, suppose $L_p$ is in the limit point case at $\pm \infty$ so that $L_p$ defined on its maximal domain $$\{u \in L^2(\mathbb{R}) : u,pu' \in AC_{loc}(\mathbb{R}), L_pu \in L^2(\mathbb{R})\}$$ is self-adjoint.

Question: Is $L_p$ a positive operator?

Erdberg
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Let $u \in \mathcal{D}(L)$. Then $$ \langle Lu,u\rangle = \int_{-\infty}^{\infty}-(pu')'udx=\int_{-\infty}^{\infty} pu'u'dx \ge 0. $$ The reason this works is that $p \ge 0$, and $L$ is in the limit-point case at $\pm\infty$, which guarantees that the evaluation terms vanish at $\infty$.

Disintegrating By Parts
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