Eigenvalues of Robin Laplacians in infinite sectors

For $\alpha \in (0,\pi )$, let $U_{\alpha }$ denote the infinite planar sector of opening $2\alpha$, $$U_{\alpha }=\{(x_1,x_2)\in {\mathbb{R}}^2:\mid \arg (x_1+i{x}_2)\mid <\alpha \},$$ and $T_{\alpha }^{\gamma }$ be the Laplacian in $L^2(U_{\alpha })$, $T_{\alpha }^{\gamma }u=-\mathrm{\Delta }u$, with the Robin boundary condition ${\mathrm{\partial }}_{\nu }u=\gamma u$, where ${\mathrm{\partial }}_{\nu }$ 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 $[-{\gamma }^2,+\mathrm{\infty })$, 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 $\kappa \mathrm{/}\alpha$ with a suitable $\kappa >0$, and the $n$th eigenvalue $E_n(T_{\alpha }^{\gamma })$ of $T_{\alpha }^{\gamma }$ behaves as $$E_n(T_{\alpha }^{\gamma })=-{\displaystyle \frac{{\gamma }^2}{(2n-1{)}^2{\alpha }^2}}+O(1)$$ 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.