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We present a new necessary condition for similarity of indefiniteSturm-Liouville operators to self-adjoint operators. This condition isformulated in terms of Weyl-Titchmarsh $m$-functions. Also we obtain necessaryconditions for regularity of the critical points 0 and $\infty$ of$J$-nonnegative Sturm-Liouville operators. Using this result, we constructseveral examples of operators with the singular critical point zero. Inparticular, it is shown that 0 is a singular critical point of the operator$-\frac{(\sgn x)}{(3|x|+1)^{-4/3}} \frac{d^2}{dx^2}$ acting in the Hilbertspace $L^2(\R, (3|x|+1)^{-4/3}dx)$ and therefore this operator is not similarto a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouvilleoperator of type $(\sgn x)(-d^2/dx^2+q(x))$ with the same properties.Comment: 24 pages, LaTeX2e <2003/12/01

Indefinite Sturm–Liouville operators with the singular critical point zero