Eigenvalues of Robin Laplacians in infinite sectors

For $\alpha\in(0,\pi)$, let $U_\alpha$ denote the infinite planar sector of opening $2\alpha$, \[ U_\alpha=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|<\alpha \big\}, \] and $T^\gamma_\alpha$ be the Laplacian in $L^2(U_\alpha)$, $T^\gamma_\alpha u= -\Delta u$, with the Robin boundary condition $\partial_\nu u=\gamma u$, where $\partial_\nu$ 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 $\alpha<\frac\pi 2$. 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 $E_n(T^\gamma_\alpha)$ 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.