Sum of the negative eigenvalues for the semi-classical Robin Laplacian

Semi‐classical edge states for the Robin Laplacian

“…\end{equation}So, we get the following correspondence between the negative Robin eigenvalues $$\lbrace \lambda _n(\mathcal {T}_h)<0\rbrace$$ and the Steklov eigenvalues $$\lbrace \mu _n< h^{-1/2}\rbrace$$: $$\begin{equation*} {\mu}_{n}\sim {h}^{-1}\sqrt{h+{\lambda}_{n}({\mathcal{T}}_{h})}\hspace*{0.20202em}\hspace*{0.20202em}\mathrm{for}\hspace*{0.20202em}{h}^{-1/4}\ll \text{}{n}\leqslant \frac{L}{\pi}{h}^{-1/2}+\mathcal{O}({h}^{1/2})\, . \end{equation*}$$As a direct consequence of Theorem 1.3, we obtain a Weyl law extending earlier results [17, 19, 20]. Theorem Assume that Ω is simply connected.…”