2017/07/08 by Mateusz Kwaśnicki, Kwaśnicki, Mateusz, Jacek Mucha +1
Computer Science · Mathematics · #35J70 #47G20. Secondary: 60J60 #60J75 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Eigenfunction #Eigenvalues and eigenvectors #Elliptic operator #FOS: Mathematics #Functional Analysis (math.FA) #Harmonic function #Laplace operator #Mathematical analysis #Mathematics #Nabla symbol #Numerical methods in inverse problems #Operator (biology) #Physics #Primary: 35J25 #Pure mathematics #Quantum mechanics #Spectral Theory in Mathematical Physics #math.AP #math.FA #msc:35J25 #msc:35J70 #msc:47G20. #msc:60J60 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1707.02475
published in arXiv (Cornell University) (Cornell University) · 30 pages
arxiv created 2017/07/08 · openalex publication_date 2017/07/08 · arxiv updated 2017/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We discuss representation of certain functions of the Laplace operator\n\Δ as Dirichlet-to-Neumann maps for appropriate elliptic operators in\nhalf-space. A classical result identifies (-\Δ)1/2, the square root of\nthe d-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the\n(d + 1)-dimensional Laplace operator \Δt,x in (0, \∞) \×\n\Rd. Caffarelli and Silvestre extended this to fractional powers\n(-\Δ)\α/2, which correspond to operators \∇t,x (t1 -\n\α \∇t,x). We provide an analogous result for all complete\nBernstein functions of -\Δ using Krein's spectral theory of strings.\n Two sample applications are provided: a Courant--Hilbert nodal line theorem\nfor harmonic extensions of the eigenfunctions of non-local Schr "odinger\noperators \ψ(-\Δ) + V(x), as well as an upper bound for the eigenvalues\nof these operators. Here \ψ is a complete Bernstein function and V is a\nconfining potential.\n