The similarity problem for J-nonnegative Sturm–Liouville operators

Abstract

Sufficient conditions for the similarity of the operator A : = 1 r ( x ) ( − d 2 d x 2 + q ( x ) ) with an indefinite weight r ( x ) = ( sgn x ) | r ( x ) | are obtained. These conditions are formulated in terms of Titchmarsh–Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm–Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm–Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r ( x ) = sgn x and q ∈ L 1 ( R , ( 1 + | x | ) d x ) , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct q ∈ ⋂ γ < 1 L 1 ( R , ( 1 + | x | ) γ d x ) such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q ≡ 0 , we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward–backward” diffusion equations.