MATHEMATICA BOHEMICA, Vol. 136, No. 4, pp. 415-427, 2011

Spectrum of the weighted Laplace operator in unbounded domains

Alexey Filinovskiy

Filinovskiy Alexey, Department of High Mathematics, Faculty of Fundamental Sciences, Moscow State Technical University, Moskva, 2-nd Baumanskaya ul. 5, 105005, Russian Federation, e-mail: flnv@yandex.ru

Abstract: We investigate the spectral properties of the differential operator $-r^s \Delta$, $s\ge0$ 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 $\|u\|^2_{L_{2, s} (\Omega)}= \int_{\Omega} r^{-s} |u|^2 d x$, 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.

Keywords: Laplace operator, multiplicative perturbation, Dirichlet problem, Friedrichs extension, purely discrete spectra, purely continuous spectra

Classification (MSC 2010): 35J20, 35J25, 35P15

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