Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

Details Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

2007-07-05

arxiv.org/abs/0707.0865v1

Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 3, 483-503

Mathematics

Spectral Theory

21 pages, LaTeX

Scientific paper

10.1017/S0308210507000686

We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of $\infty$. Moreover, in this situation, the point $\infty$ is a regular critical point. We construct an operator $A=(\sgn x)(-d^2/dx^2+q)$ with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.

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