Abstract:
The spectral problem in a bounded domain $\Omega\subset R^n$ is considered for the equation $-\Delta u=\lambda u$ in $\Omega$, $-u=\lambda\,\partial u/\partial\nu$ on the boundary of $\Omega$ ($\nu$ the interior normal to the boundary, $\Delta$, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues $\{\lambda_j^0\}_{j=1}^\infty$ and $\{\lambda_j^\infty\}_{j=1}^\infty$, converging respectively to 0 and $+\infty$. It is also established that
\begin{gather*}
N^0(\lambda)=\sum_{\operatorname{Re}\lambda_j^0\ge1/\lambda}1\approx\mathrm{const}\,\lambda^{b-1},
\\
N^\infty(\lambda)\equiv\sum_{\operatorname{Re}\lambda_j^\infty\le\lambda}1\approx\mathrm{const}\,\lambda^{n/2},
\end{gather*}
The constants are explicitly calculated.
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