On the negative spectrum of the Robin Laplacian in corner domains

Abstract

For a bounded corner domain $\Omega$, we consider the Robin Laplacian in $\Omega$ with large Robin parameter. Exploiting multiscale analysis and a recursive procedure, we have a precise description of the mechanism giving the ground state of the spectrum. It allows also the study of the bottom of the essential spectrum on the associated tangent structures given by cones. Then we obtain the asymptotic behavior of the principal eigenvalue for this singular limit in any dimension, with remainder estimates. The same method works for the Schrödinger operator in ${\mathbb{R}}^n$ with a strong attractive delta-interaction supported on $\partial \Omega$. Applications to some Erhling's type estimates and the analysis of the critical temperature of some superconductors are also provided.

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Cited by in corpus (12)

• Eigenvalues of Robin Laplacians in infinite sectors
• Dirichlet spectrum of the Fichera layer
• On the $p$-Laplacian with Robin boundary conditions and boundary trace theorems
• Discrete spectrum of interactions concentrated near conical surfaces
• A spectral isoperimetric inequality for cones
• Robin eigenvalues on domains with peaks
• Eigenvalue counting function for Robin Laplacians on conical domains
• Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian
• Strong coupling asymptotics for $\delta$-interactions supported by curves with cusps
• A remark on the effect of random singular two-particle interactions
• Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters
• Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks