Harmonic functions for recurrent symmetric $\alpha$-stable processeses with non-local perturbations

Let ${\mathbf M}$ be the recurrent symmetric (relativistic) $\alpha$-stable process on ${\mathbb R}^d$. Let ${\mathcal H}^{\mu + F} (:= {\mathcal H} + \mu + F)$ be a Schr\"odinger type operator with local and non-local perturbations $\mu$ and $F$. If $\mu$ and $F$ satisfy suitable conditions associated with Kato class, we prove the existence of ground state for a Schr\"odinger type operator relating to ${\mathcal H}^{\mu + F}$. Furthermore, we prove the ground state becomes a probabilistically harmonic function of the Schr\"odinger operator generated by ${\mathcal H}^{\mu + F}$.