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We investigate the problem of similarity to a self-adjoint operator for$J$-positive Sturm-Liouville operators$L=\frac{1}{\omega}(-\frac{d^2}{dx^2}+q)$ with $2\pi$-periodic coefficients $q$and $\omega$. It is shown that if 0 is a critical point of the operator $L$,then it is a singular critical point. This gives us a new class of $J$-positivedifferential operators with the singular critical point 0. Also, we extend theBeals and Parfenov regularity conditions for the critical point $\infty$ to thecase of operators with periodic coefficients.

The similarity problem for indefinite Sturm-Liouville operators with periodic coefficients