Eigenvalues of Robin Laplacians in infinite sectors

Abstract

For $α\in(0,π)$, let $U_α$ denote the infinite planar sector of opening $2α$, \[ U_α=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|<α\big\}, \] and $T^γ_α$ be the Laplacian in $L^2(U_α)$, $T^γ_αu= -Δu$, with the Robin boundary condition $\partial_νu=γu$, where $\partial_ν$ stands for the outer normal derivative and $γ>0$. The essential spectrum of $T^γ_α$ does not depend on the angle $α$ and equals $[-γ^2,+\infty)$, and the discrete spectrum is non-empty iff $α<\fracπ2$. 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 $α$. In particular, there is just one discrete eigenvalue for $α\ge \fracπ{6}$. As $α$ approaches $0$, the number of discrete eigenvalues becomes arbitrary large and is minorated by $κ/α$ with a suitable $κ>0$, and the $n$th eigenvalue $E_n(T^γ_α)$ of $T^γ_α$ 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 $α^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $δ$-interactions on star graphs.