Description
For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $\Omega_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: \, |x' | < \varepsilon x_1\big\}$ and the operator $Q_{\varepsilon}^{\alpha}$ acting as the Laplacian $u\mapsto-\Delta u$ on $\Omega_{\varepsilon}$ with the Robin boundary condition $\partial_\nu u=\alpha u$ at $\partial\Omega_\varepsilon$, where $\partial_\nu$ is the outward normal derivative and $\alpha>0$. It is known from numerous earlier works that the essential spectrum of $Q^\alpha_\varepsilon$ is $[-\alpha^2,+\infty)$ and that the discrete spectrum is finite for $n=1$ and infinite for $n\ge 2$, but the behavior of individual eigenvalues with respect to the geometric parameter $\varepsilon$ was only addressed for $n=1$ so far. In the present work we consider arbitrary $n\ge2$ and look at the spectral asymptotics as $\varepsilon$ becomes small, i.e. as the cone becomes "sharp" and collapses to its central axis. Our main result is as follows: if $n\ge 2$, $\alpha>0$ and $j\in \mathbb{N}$ are fixed, then the $j$th eigenvalue $E_j(Q^\alpha_\varepsilon)$ of $Q^\alpha_\varepsilon$ behaves as $E_j(Q^\alpha_\varepsilon)=-\frac{n^2\alpha^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\frac{1}{\varepsilon}\right)$ as $\varepsilon\to 0^+$.
