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- W4290543276 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 Schrodinger 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." @default.
- W4290543276 created "2022-08-08" @default.
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- W4290543276 date "2017-07-08" @default.
- W4290543276 modified "2023-10-16" @default.
- W4290543276 title "Extension technique for complete Bernstein functions of the Laplace operator" @default.
- W4290543276 doi "https://doi.org/10.48550/arxiv.1707.02475" @default.
- W4290543276 hasPublicationYear "2017" @default.
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