Intrinsic Contractivity of Feynman-Kac Semigroups for Symmetric Jump Processes with Infinite Range Jumps

Let $(X_t{)}_{t\ge 0}$ be a symmetric strong Markov process generated by non-local regular Dirichlet form $(D,\text{D}(D))$ as follows \begin{equation*} \begin{split} & D(f,g)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)\big(g(x)-g(y)\big) J(x,y),dx,dy, \quad f,g\in \D(D) \end{split} \end{equation*} where $J(x,y)$ is a strictly positive and symmetric measurable function on ${\mathbb{R}}^d\times {\mathbb{R}}^d$. We study the intrinsic hypercontractivity, intrinsic supercontractivity and intrinsic ultracontractivity for the Feynman-Kac semigroup $$T_t^V(f)(x)={\text{Ee}}^x(\exp (-{\int }_0^{t}V(X_s)ds)f(X_t)),x\in {\mathbb{R}}^d,f\in L^2({\mathbb{R}}^d;dx)\mathrm{.}$$ In particular, we prove that for $$J(x,y)\asymp \mathrm{\mid }x-y{\mathrm{\mid }}^{-d-\alpha }{\text{I}}_{\{\mathrm{\mid }x-y\mathrm{\mid }\le 1\}}+e^{-\mathrm{\mid }x-y\mathrm{\mid }}{\text{I}}_{\{\mathrm{\mid }x-y\mathrm{\mid }>1\}}$$ with $\alpha \in (0,2)$ and $V(x)=\mathrm{\mid }x{\mathrm{\mid }}^{\lambda }$ with $\lambda >0$, $(T_t^V{)}_{t\ge 0}$ is intrinsically ultracontractive if and only if $\lambda >1$; and that for symmetric $\alpha$-stable process $(X_t{)}_{t\ge 0}$ with $\alpha \in (0,2)$ and $V(x)={\log }^{\lambda }(1+\mathrm{\mid }x\mathrm{\mid })$ with some $\lambda >0$, $(T_t^V{)}_{t\ge 0}$ is intrinsically ultracontractive (or intrinsically supercontractive) if and only if $\lambda >1$, and $(T_t^V{)}_{t\ge 0}$ is intrinsically hypercontractive if and only if $\lambda \ge 1$. Besides, we also investigate intrinsic contractivity properties of $(T_t^V{)}_{t\ge 0}$ for the case that ${\mathrm{lim\hspace{0.17em}inf}}_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}V(x)<\mathrm{\infty }$.