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Register a new userEffective operators for Robin eigenvalues in domains with corners
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Added by
Konstantin Pankrashkin
Publication date
2018
fields
Physics
and research's language is
English
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We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schrodinger-type operator on the boundary of the domain with boundary conditions at the corners.
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We study Schroedinger operators with Robin boundary conditions on exterior domains in $R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.
We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
Let $Omegasubsetmathbb{R}^N$, $Nge 2,$ be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian $umapsto -Delta u$ in $Omega$ with the Robin boundary condition $partial_n u=alpha u$ on $partialOmega$ with $partial_n$ being the outward normal derivative and $alpha>0$ being a parameter. We show that for large $alpha$ the associated eigenvalues $E_j(alpha)$ behave as $E_j(alpha)sim -epsilon_j alpha^ u$, where $ u>2$ and $epsilon_j>0$ depend on the dimension and the peak geometry. This is in contrast with the well-known estimate $E_j(alpha)=O(alpha^2)$ for the Lipschitz domains.
This note aims to give prominence to some new results on the absence and localization of eigenvalues for the Dirac and Klein-Gordon operators, starting from known resolvent estimates already established in the literature combined with the renowned Birman-Schwinger principle.
In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Omega; mathbb{C}^4)$, where $Omega subset mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that $Omega$ is unbounded, and corresponding trace formulas.
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