Criticality for schrödinger type operators based on recurrent symmetric stable processes

Let μ be a signed Radon measure on ℝ¹ in the Kato class andconsider a Schrödinger type operator Hμ = (−d2/dx2)α/2 + μ on R1. Let 1 ≤ α < 2 and suppose the support of μ is compact. We then construct a bounded Hμ-harmonic function uniformly lower-bounded by a positive constant if Hμ is critical. Moreover, we show that there exists no bounded positive Hμ-harmonic function if Hμ is subcritical.