  We consider the indefinite Sturm-Liouville differential expression
\[\mathfrak{a}(f) := - \frac{1}{w}\left( \frac{1}{r} f' \right)',\] where
$\mathfrak{a}$ is defined on a finite or infinite open interval $I$ with $0\in
I$ and the coefficients $r$ and $w$ are locally summable and such that $r(x)$
and $(\operatorname{sgn} x) w(x)$ are positive a.e. on $I$. With the
differential expression $\mathfrak{a}$ we associate a nonnegative self-adjoint
operator $A$ in the Krein space $L^2_w(I)$, 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 $L^2_{|w|}(I)$.
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.
