Eigenvalues of Robin Laplacians in infinite sectors

Abstract

For $\alpha \in (0,\pi )$, let $U_{\alpha }$ denote the infinite planar sector of opening $2\alpha$, \[ U_α=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|<α\big\}, \] and $T_{\alpha }^{\gamma }$ be the Laplacian in $L^2(U_{\alpha })$, $T_{\alpha }^{\gamma }u=-\Delta u$, with the Robin boundary condition ${\partial }_{\nu }u=\gamma u$, where ${\partial }_{\nu }$ stands for the outer normal derivative and $\gamma >0$. The essential spectrum of $T_{\alpha }^{\gamma }$ does not depend on the angle $\alpha$ and equals $[-{\gamma }^2,+\infty )$, and the discrete spectrum is non-empty iff $\alpha <\frac{\pi }{2}$. 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 $E_n(T_{\alpha }^{\gamma })$ of $T_{\alpha }^{\gamma }$ behaves as \[ E_n(T^γ_α)=-\dfrac{γ^2}{(2n-1)^2 α^2}+O(1) \] 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.

arXiv comment34 pages

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Cited by in corpus (6)

• On radial Schroedinger operators with a Coulomb potential
• Absence of eigenvalues of non-self-adjoint Robin Laplacians on the half-space
• Robin eigenvalues on domains with peaks
• Hydrogenoid spectra with central perturbations
• A numerical study of the Dirichlet-to-Neumann operator in planar domains
• On Schrödinger operators with ${\delta }^{\prime }$-potentials supported on star graphs