Open Access

# A spectral analysis of some indefinite differential operators

### Abstract

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 $\frac{{\rm sgn}\, x}{|x|^\alpha}\left( -\frac{d^2}{dx^2}+c \delta \right), \, \alpha>-1$.

### Article Information

 Title A spectral analysis of some indefinite differential operators Source Methods Funct. Anal. Topology, Vol. 12 (2006), no. 2, 157-169 MathSciNet MR2238037 Copyright The Author(s) 2006 (CC BY-SA)

### Authors Information

A. S. Kostenko
Department of Mathematics, Donets'k National University, 24 Universitets'ka, Donets'k, 83055, Ukraine

### Citation Example

A. S. Kostenko, A spectral analysis of some indefinite differential operators, Methods Funct. Anal. Topology 12 (2006), no. 2, 157-169.

### BibTex

@article {MFAT343,
AUTHOR = {Kostenko, A. S.},
TITLE = {A spectral analysis of some indefinite differential operators},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {12},
YEAR = {2006},
NUMBER = {2},
PAGES = {157-169},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=343},
}