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Register a new userEigenvalues of Robin Laplacians in infinite sectors
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For $alphain(0,pi)$, let $U_alpha$ denote the infinite planar sector of opening $2alpha$, [ U_alpha=big{ (x_1,x_2)inmathbb R^2: big|arg(x_1+ix_2) big|<alpha big}, ] and $T^gamma_alpha$ be the Laplacian in $L^2(U_alpha)$, $T^gamma_alpha u= -Delta u$, with the Robin boundary condition $partial_ u u=gamma u$, where $partial_ u$ stands for the outer normal derivative and $gamma>0$. The essential spectrum of $T^gamma_alpha$ does not depend on the angle $alpha$ and equals $[-gamma^2,+infty)$, and the discrete spectrum is non-empty iff $alpha<fracpi 2$. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle $alpha$. In particular, there is just one discrete eigenvalue for $alpha ge frac{pi}{6}$. As $alpha$ approaches $0$, the number of discrete eigenvalues becomes arbitrary large and is minorated by $kappa/alpha$ with a suitable $kappa>0$, and the $n$th eigenvalue $E_n(T^gamma_alpha)$ of $T^gamma_alpha$ behaves as [ E_n(T^gamma_alpha)=-dfrac{gamma^2}{(2n-1)^2 alpha^2}+O(1) ] and admits a full asymptotic expansion in powers of $alpha^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $delta$-interactions on star graphs.
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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.
By introducing a weight function to the Laplace operator, Bakry and Emery defined the drift Laplacian to study diffusion processes. Our first main result is that, given a Bakry-Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $R^n$ and Neumann eigenvalues of domains in $R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize emph{all the results in Riemannian geometry based on gradient estimates to Bakry-Emery manifolds}.
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 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 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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