The spectrum of weighted Laplace operator in unbounded domains (Q656239)

The author studies the weighted Laplacian \[ Lu=-|x|^s\Delta u,\quad s\geq0 \] over the Hilbert space \(L_{2,s}(\Omega)\) endowed with the norm \(\|u\|^2_{L_{2,s}}=\int_\Omega |x|^{-s}|u|^2\,dx,\) where \(\Omega\subset \mathbb R^n,\) \(n\geq2,\) is an unbounded domain with closure not containing the origin. The dependence of the spectral properties of \(L\) (such as location of the spectrum, its density and structure) on the exponent \(s\) is studied.