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 Schrödinger 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 Schrödinger type operator relating to ${\mathcal{H}}^{\mu +F}$. Furthermore, we prove the ground state becomes a probabilistically harmonic function of the Schrödinger operator generated by ${\mathcal{H}}^{\mu +F}$.