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For a general subcritical second-order elliptic operator $P$ in a domain$\Omega \subset \mathbb{R}^n$ (or noncompact manifold), we constructHardy-weight $W$ which is optimal in the following sense. The operator $P -\lambda W$ is subcritical in $\Omega$ for all $\lambda < 1$, null-critical in$\Omega$ for $\lambda = 1$, and supercritical near any neighborhood of infinityin $\Omega$ for any $\lambda > 1$. Moreover, if $P$ is symmetric and $W>0$,then the spectrum and the essential spectrum of $W^{-1}P$ are equal to$[1,\infty)$, and the corresponding Agmon metric is complete.  Our method is based on the theory of positive solutions and applies to bothsymmetric and nonsymmetric operators. The constructed Hardy-weight is given byan explicit simple formula involving two distinct positive solutions of theequation $Pu=0$, the existence of which depends on the subcriticality of $P$ in$\Omega$.

Optimal Hardy Weight for Second-Order Elliptic Operator: An Answer to a Problem of Agmon