An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter
By Konstantin Pankrashkin, Nicolas Popoff. 2015
58 citations recorded by the source corpus.
Abstract We consider the Laplacian on a class of smooth domains Ω ⊂ R ν , ν ≥ 2 , with attractive Robin boundary conditions: Q α Ω u = − Δ u , ∂ u ∂ n = α u on ∂ Ω , α > 0 , where n is the outer unit normal, and study the asymptotics of its eigenvalues E j ( Q α Ω ) as well as some other spectral properties for α tending to +∞. We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact C 2 boundaries we show that, for each fixed j, E j ( Q α Ω ) = − α 2 + μ j ( α ) + O ( log α ) , where μ j ( α ) is the jth eigenvalue of the operator − Δ S − ( ν − 1 ) α H with ( − Δ S ) and H being respectively the positive Laplace–Beltrami operator and the mean curvature on ∂Ω. Analogous results are obtained for a class of domains with non-compact boundaries. In particular, we discuss the existence of eigenvalues for non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions. The effective Hamiltonian − Δ S − ( ν − 1 ) α H enters the framework of semi-classical Schrodinger operators on manifolds, and we provide the asymptotics of its eigenvalues for large α under various geometric assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of Q α Ω for large α.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8897efc0205c89221b11ba04eee02f994a010db3
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In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “Particular attention has been paid to the spectral properties of realizations with local and nonlocal Robin boundary conditions in [41, 106, 179, 180, 226, 325, 361, 413, 543, 628, 629, 665].”