Extension technique for complete Bernstein functions of the Laplace operator

We discuss the representation of certain functions of the Laplace operator $$\Delta $$Δ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $$(-\Delta )^{1/2}$$(-Δ)1/2, the square root of the d-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $$(d + 1)$$(d+1)-dimensional Laplace operator $$\Delta _{t,x}$$Δt,x in $$(0, \infty ) \times \mathbf {R}^d$$(0,∞)×Rd. Caffarelli and Silvestre extended this to fractional powers $$(-\Delta )^{\alpha /2}$$(-Δ)α/2, which correspond to operators $$\nabla _{t,x} (t^{1 - \alpha } \nabla _{t,x})$$∇t,x(t1-α∇t,x). We provide an analogous result for all complete Bernstein functions of $$-\Delta $$-Δ 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 $$\psi (-\Delta ) + V(x)$$ψ(-Δ)+V(x), as well as an upper bound for the eigenvalues of these operators. Here $$\psi $$ψ is a complete Bernstein function and V is a confining potential.