Spectrum of the weighted Laplace operator in unbounded domains. (Q2881213)

Let \(\Omega \subset \mathbb {R}^n\), \(n\geq 2\) be an unbounded domain with the boundary \(\Gamma \) such that the closure of \(\Omega \) does not contain \(0\). The author considers the self-adjoint Friedrichs extension \(L\) of \(lu=-r^s\Delta u\), \(r=| x| \), \(s\geq 0\) in the Hilbert space \(L_{2,s}(\Omega )\) with the norm \(\| u\| ^2=\int _{\Omega }r^{-s}| u| ^2\, dx\). Assume that the set \(\mathbb {R}^n\backslash \Omega \) satisfies the following star-shapeness condition with respect to the origin: \(\Sigma _{\eta _1}\Sigma _{\eta _2}\) if \(0<\eta _1<\eta _2\), where \(\Sigma _{\eta }=\{x\:| x| =1,\eta x\in \Omega \cap \{r=\eta \}\}.\) The author characterizes the spectrum of \(L\) when \(s>2\), \(s=2\), \(0\leq s\leq 2\), respectively. The paper generalizes the former results of \textit{R. T. Lewis} [Trans. Am. Math. Soc. 271, 653--666 (1982; Zbl 0507.35069)] and \textit{D. M. Eidus} [J. Funct. Anal. 100, No. 2, 400--410 (1991; Zbl 0762.35020)].