Abstract
We discuss the 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.
Highlights
A classical result identifies the Dirichlet-to-Neumann operator in half-space with the square root of the Laplace operator; namely, if u(t, x) is harmonic in H = (0, ∞)×Rd with boundary value f (x) = u(0, x), given some boundedness condition on u, we have−(− )1/2 f (x) = ∂t u(0, x).The above observation was extended to general fractional powers of the Laplace operator by Caffarelli and Silvestre in [5]: for α ∈ (0, 2), if u satisfies the elliptic equation∇t,x (t 1−α∇t,x u(t, x)) = 0 (1.1)in H with boundary value f (x) = u(0, x), under appropriate boundedness assumption on u, we have −(− )α/2 f (x) = |
Any complete Bernstein function of − can be represented in this way, if one allows for certain singularities of a(t)
This is complemented by two applications for non-local Schrödinger operators ψ(− ) + V (x): a Courant–Hilbert theorem on nodal domains, and an upper bound for the eigenvalues
Summary
Any complete Bernstein function of − can be represented in this way, if one allows for certain singularities of a(t) Even though this method has already been mentioned in the literature (see, for example, [21,37,51]), finding a reference is problematic. The main purpose of this article is to fill in this gap and discuss rigorously the abovementioned general extension technique This is complemented by two applications for non-local Schrödinger operators ψ(− ) + V (x): a Courant–Hilbert theorem on nodal domains (for the extension problem), and an upper bound for the eigenvalues. We prove uniqueness of harmonic extensions, given appropriate boundedness condition; the first result of this kind was proved in [49] for fractional powers of certain elliptic operators. In [18] a closely related, but essentially different extension technique is developed in a non-commutative setting, for the sub-Laplacian on the Heisenberg group
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