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Register a new userOn the discrete spectrum of Robin Laplacians in conical domains
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Added by
Konstantin Pankrashkin
Publication date
2015
fields
Physics
and research's language is
English
Authors
Konstantin Pankrashkin
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We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
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We study the discrete spectrum of the Robin Laplacian $Q^{Omega}_alpha$ in $L^2(Omega)$, [ umapsto -Delta u, quad dfrac{partial u}{partial n}=alpha u text{ on }partialOmega, ] where $Omegasubset mathbb{R}^{3}$ is a conical domain with a regular cross-section $Thetasubset mathbb{S}^2$, $n$ is the outer unit normal, and $alpha>0$ is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of $Q^{Omega}_alpha$ is $-alpha^2$ and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of $Q^Omega_alpha$ is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of $Q^{Omega}_alpha$ in $(-infty,-alpha^2-lambda)$, with $lambda>0$, behaves for $lambdato0$ as [ dfrac{alpha^2}{8pi lambda} int_{partialTheta} kappa_+(s)^2d s +oleft(frac{1}{lambda}right), ] where $kappa_+$ is the positive part of the geodesic curvature of the cross-section boundary.
We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrodinger operators with attractive $delta$-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.
We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_circ ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.
Let $Omega$ be a curvilinear polygon and $Q^gamma_{Omega}$ be the Laplacian in $L^2(Omega)$, $Q^gamma_{Omega}psi=-Delta psi$, with the Robin boundary condition $partial_ u psi=gamma psi$, where $partial_ u$ is the outer normal derivative and $gamma>0$. We are interested in the behavior of the eigenvalues of $Q^gamma_Omega$ as $gamma$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^gamma_Omega$ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with $partial Omega$. In the particular case of a polygon with straight edges the first eigenpairs are exponentially close to those of the model operators. Finally, we prove a Weyl asymptotics for the eigenvalue counting function of $Q^gamma_Omega$ for a threshold depending on $gamma$, and show that the leading term is the same as for smooth domains.
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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