Indefinite Sturm - Liouville operators with the singular critical point zero

We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh m-functions. Also we obtain necessary conditions for regularity of the critical points 0 and∞ of J-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular critical point zero. In particular, it is shown that 0 is a singular critical point of the operator − (sgnx) (3|x|+1)−4/3 d dx2 acting in the Hilbert space L2(R, (3|x|+1)−4/3dx) and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type (sgnx)(−d2/dx2 + q(x)) with the same properties.