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

Full Text

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},
  MRNUMBER = {MR2238037},
       URL = {http://mfat.imath.kiev.ua/article/?id=343},
}