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. There is a self-adjoint operator in some Krein space associated naturally With this expression. We characterize the local definitizability of this operator in a neighbourhood of infinity. Moreover, in this situation, the point infinity 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 results obtained are applied to several classes of Sturm-Liouville operators.