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 = 1/ω(- d²/dx² +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 ∞ to the case of operators with periodic coefficients.