Extension technique for complete Bernstein functions of the Laplace operator

Abstract

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 Schrödinger 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.

30 pages