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.
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This work was supported by the Polish National Science Centre (NCN) Grant No. 2015/19/B/ST1/01457.
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Kwaśnicki, M., Mucha, J. Extension technique for complete Bernstein functions of the Laplace operator. J. Evol. Equ. 18, 1341–1379 (2018). https://doi.org/10.1007/s00028-018-0444-4
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DOI: https://doi.org/10.1007/s00028-018-0444-4