Geodesic
Spectrum of the weighted Laplace operator in unbounded domains
Mathematica Bohemica, Tome 136 (2011) no. 4, pp. 415-427

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MR Zbl
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 $\|u\|^2_{L_{2, s} (\Omega )}= \int _{\Omega } r^{-s} |u|^2 {\rm 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 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 $\|u\|^2_{L_{2, s} (\Omega )}= \int _{\Omega } r^{-s} |u|^2 {\rm 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.
DOI : 10.21136/MB.2011.141701
Classification : 35J15, 35J20, 35J25, 35P05, 35P15
Keywords: Laplace operator; multiplicative perturbation; Dirichlet problem; Friedrichs extension; purely discrete spectra; purely continuous spectra 
Filinovskiy, Alexey. Spectrum of the weighted Laplace operator in unbounded domains. Mathematica Bohemica, Tome 136 (2011) no. 4, pp. 415-427. doi: 10.21136/MB.2011.141701
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