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We identify a class of non-local integro-differential operators $K$ in$\mathbb{R}$ with Dirichlet-to-Neumann maps in the half-plane $\mathbb{R}\times (0, \infty)$ for appropriate elliptic operators $L$. More precisely, weprove a bijective correspondence between L\'evy operators $K$ with non-localkernels of the form $\nu(y - x)$, where $\nu(x)$ and $\nu(-x)$ are completelymonotone functions on $(0, \infty)$, and elliptic operators $L = a(y)\partial_{xx} + 2 b(y) \partial_{x y} + \partial_{yy}$. This extends a numberof previous results in the area, where symmetric operators have been studied:the classical identification of the Dirichlet-to-Neumann operator for theLaplace operator in $\mathbb{R} \times (0, \infty)$ with$-\sqrt{-\partial_{xx}}$, the square root of one-dimensional Laplace operator;the Caffarelli--Silvestre identification of the Dirichlet-to-Neumann operatorfor $\nabla \cdot (y^{1 - \alpha} \nabla)$ with $(-\partial_{xx})^{\alpha/2}$for $\alpha \in (0, 2)$; and the identification of Dirichlet-to-Neumann mapsfor operators $a(y) \partial_{xx} + \partial_{yy}$ with complete Bernsteinfunctions of $-\partial_{xx}$ due to Mucha and the author. Our results rely onrecent extension of Krein's spectral theory of strings by Eckhardt andKostenko.

Harmonic extension technique for non-symmetric operators with completely monotone kernels