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

Abstract

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. Mathematics subject classification (2010): 47E05, 34B24, 34B09, 34L10, 47B50.