Extension technique for complete Bernstein functions of the Laplace operator

We discuss representation of certain functions of the Laplace operator $\mathrm{\Delta }$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-\mathrm{\Delta }{)}^{1\mathrm{/}2}$, the square root of the $d$-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $(d+1)$-dimensional Laplace operator ${\mathrm{\Delta }}_{t,x}$ in $(0, \infty) \times $\mathbf{R}$^d$\mathrm{.}CaffarelliandSilvestreextendedthistofractionalpowers$(-\Delta)^{\alpha/2}$,whichcorrespondtooperators$\nabla_{t,x} (t^{1 - \alpha} \nabla_{t,x})$. We provide an analogous result for all complete Bernstein functions of $-\mathrm{\Delta }$ 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 $\psi (-\mathrm{\Delta })+V(x)$, 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.