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

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 $\mathrm{\infty }$. Moreover, in this situation, the point $\mathrm{\infty }$ 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.