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

For a general subcritical second-order elliptic operator $P$ in a domain $Ω\subset \mathbb{R}^n$ (or noncompact manifold), we construct Hardy-weight $W$ which is optimal in the following sense. The operator $P - λW$ is subcritical in $Ω$ for all $λ< 1$, null-critical in $Ω$ for $λ= 1$, and supercritical near any neighborhood of infinity in $Ω$ for any $λ> 1$. 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 both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation $Pu=0$, the existence of which depends on the subcriticality of $P$ in $Ω$.