Extension technique for complete Bernstein functions of the Laplace operator

We discuss representation of certain functions of the Laplace operator $\Delta$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-\Delta)^{1/2}$, the square root of the $d$-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $(d + 1)$-dimensional Laplace operator $\Delta_{t,x}$ in $(0, \infty) \times \mathbf{R}^d$. Caffarelli and Silvestre extended this to fractional powers $(-\Delta)^{\alpha/2}$, which correspond to operators $\nabla_{t,x} (t^{1 - \alpha} \nabla_{t,x})$. We provide an analogous result for all complete Bernstein functions of $-\Delta$ using Krein's spectral theory of strings. Two sample applications are provided: a Courant--Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schr\"odinger operators $\psi(-\Delta) + V(x)$, as well as an upper bound for the eigenvalues of these operators. Here $\psi$ is a complete Bernstein function and $V$ is a confining potential.