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

We investigate the problem of similarity to a self-adjoint operator for $J$-positive Sturm-Liouville operators $L=\frac{1}ω(-\frac{d^2}{dx^2}+q)$ with $2π$-periodic coefficients $q$ and $ω$. 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 $J$-positive differential operators with the singular critical point 0. Also, we extend the Beals and Parfenov regularity conditions for the critical point $\infty$ to the case of operators with periodic coefficients.