Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes

Let $X_t$ be the relativistic $\alpha$-stable process in $\mathbf{R}^d$, $\alpha \in (0,2)$, $d > \alpha$, with infinitesimal generator $H_0^{(\alpha)}= - ((-\Delta +m^{2/\alpha})^{\alpha/2}-m)$. We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup $T_t$ for this process with generator $H_0^{(\alpha)} - V$, $V \ge 0$, $V$ locally bounded. We prove that if $\lim_{\vert x\vert \to \infty} V(x) = \infty$, then for every $t >0$ the operator $T_t$ is compact. We consider the class $\mathcal{V}$ of potentials $V$ such that $V \ge 0$, $\lim_{\vert x\vert \to \infty} V(x) = \infty$ and $V$ is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For $V$ in the class $\mathcal{V}$ we show that the semigroup $T_t$ is IU if and only if $\lim_{\vert x\vert \to \infty} V(x)/\vert x\vert = \infty$. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction $\phi_1$ for $T_t$. In particular, when $V(x) = \vert x\vert^{\beta}$, $\beta > 0$, then the semigroup $T_t$ is IU if and only if $\beta >1$. For $\beta >1$ the first eigenfunction $\phi_1(x)$ is comparable to

$\exp(-m^{1/{\alpha}}\vert x\vert) \, (\vert x\vert + 1)^{(-d - \alpha - 2 \beta -1 )/2}.$