A. S. Kostenko

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Articles: 1

A spectral analysis of some indefinite differential operators

A. S. Kostenko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 12 (2006), no. 2, 157-169

We investigate the main spectral properties of quasi--Hermitian extensions of the minimal symmetric operator $L_{\rm min}$ generated by the differential expression $-\frac{{\rm sgn}\, x}{|x|^{\alpha}}\frac{d^2}{dx^2} \ (\alpha>-1)$ in $L^2(\mathbb R, |x|^{\alpha})$. We describe their spectra, calculate the resolvents, and obtain a similarity criterion to a normal operator in terms of boundary conditions at zero. As an application of these results we describe the main spectral properties of the operator $\frac{{\rm sgn}\, x}{|x|^\alpha}\left( -\frac{d^2}{dx^2}+c \delta \right), \, \alpha>-1$.

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