Criticality for Schrödinger type operators based on recurrent symmetric stable processes

Let μ \mu be a signed Radon measure on R 1 \mathbb {R}^1 in the Kato class and consider a Schrödinger type operator H μ = ( − d 2 / d x 2 ) α 2 + μ \mathcal {H}^{\mu }=(-d^2/dx^2)^{\frac {\alpha }{2}} + \mu on R 1 \mathbb {R}^1 . Let 1 ≤ α > 2 1\leq \alpha >2 and suppose the support of μ \mu is compact. We then construct a bounded H μ \mathcal {H}^{\mu } -harmonic function uniformly lower-bounded by a positive constant if H μ \mathcal {H}^{\mu } is critical. Moreover, we show that there exists no bounded positive H μ \mathcal {H}^{\mu } -harmonic function if H μ \mathcal {H}^{\mu } is subcritical.