Indefinite Sturm-Liouville operators in polar form

We consider the indefinite Sturm-Liouville differential expression $$\mathfrak{a}(f):=-\frac{1}{w}{\left(\frac{1}{r}{f}^{\mathrm{\prime }}\right)}^{\mathrm{\prime }},$$ where $\mathfrak{a}$ is defined on a finite or infinite open interval $I$ with $0\in I$andthecoefficients$r$and$w$arelocallysummableandsuchthat$r(x)$and$(\operatorname{sgn} x) w(x)$arepositivea\mathrm{.}e\mathrm{.}on$I$. With the differential expression $\mathfrak{a}$ we associate a nonnegative self-adjoint operator $A$ in the Krein space $L_w^2(I)$, which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of $I$ with the positive and the negative semi-axis. For the operator $A$ we derive conditions in terms of the coefficients $w$ and $r$ for the existence of a Riesz basis consisting of generalized eigenfunctions of $A$ and for the similarity of $A$ to a self-adjoint operator in a Hilbert space $L_{\mathrm{\mid }w\mathrm{\mid }}^2(I)$. These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces, which are couplings of two symmetric operators acting in Hilbert spaces.