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