### Moments of the product $F$ distribution.

Nadarajah, Saralees, Kotz, Samuel (2006)

Applied Mathematics E-Notes [electronic only]

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Nadarajah, Saralees, Kotz, Samuel (2006)

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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.