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We investigate the spectral properties of the differential operator $-r^s\Delta$, $s\ge 0$ with the Dirichlet boundary condition in unbounded domains whose boundaries satisfy some geometrical condition. Considering this operator as a self-adjoint operator in the space with the norm ${\parallel u\parallel }_{L_{2,s}\left(\Omega \right)}^2={\int }_{\Omega }{r}^{-s}{\left|u\right|}^2\mathrm{d}x$, we study the structure of the spectrum with respect to the parameter $s$. Further we give an estimate of the rate of condensation of discrete spectra when it changes to continuous.

We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega$, $\partial u/\partial \nu +\alpha u=0$ on $\partial \Omega$ where $\Omega \subset {ℝ}^n$, $n\ge 2$ is a bounded domain and $\alpha$ is a real parameter. We investigate the behavior of the eigenvalues ${\lambda }_k\left(\alpha \right)$ of this problem as functions of the parameter $\alpha$. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative ${\lambda }_1^{\text{'}}\left(\alpha \right)$. Assuming that the boundary $\partial \Omega$ is of class $C^2$ we obtain estimates to the difference ${\lambda }_k^D-{\lambda }_k\left(\alpha \right)$ between the $k$-th eigenvalue of the Laplace operator with Dirichlet...