Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes

Consider the symmetric non-local Dirichlet form (D,\D(D)) given by $$D(f,f)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)^2 J(x,y)\,dx\,dy$$with \D(D) the closure of the set of $C^1$ functions on $\R^d$ with compact support under the norm $\sqrt{D_1(f,f)}$, where $D_1(f,f):=D(f,f)+\int f^2(x)\,dx$ and $J(x,y)$ is a nonnegative symmetric measurable function on $\R^d\times \R^d$. Suppose that there is a Hunt process $(X_t)_{t\ge 0}$ on $\R^d$ corresponding to (D,\D(D)), and that (L,\D(L)) is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman-Kac semigroup $(T_t^V)_{t\ge 0}$ generated by $L^V:=L-V$, where $V\ge 0$ is a non-negative locally bounded measurable function such that Lebesgue measure of the set $\{x\in \R^d: V(x)\le r\}$ is finite for every $r>0$. By using intrinsic super Poincar\'{e} inequalities and establishing an explicit lower bound estimate for the ground state, we present general criteria for the intrinsic ultracontractivity of $(T_t^V)_{t\ge 0}$. In particular, if J(x,y)\asymp|x-y|^{-d-\alpha}\I_{\{|x-y|\le 1\}}+e^{-|x-y|^\gamma}\I_{\{|x-y|> 1\}} for some $\alpha \in (0,2)$ and $\gamma\in(1,\infty]$, and the potential function $V(x)=|x|^\theta$ for some $\theta>0$, then $(T_t^V)_{t\ge 0}$ is intrinsically ultracontractive if and only if $\theta>1$. When $\theta>1$, we have the following explicit estimates for the ground state $\phi_1$ $$c_1\exp\Big(-c_2 \theta^{\frac{\gamma-1}{\gamma}}|x| \log^{\frac{\gamma-1}{\gamma}}(1+|x|)\Big) \le \phi_1(x) \le c_3\exp\Big(-c_4 \theta^{\frac{\gamma-1}{\gamma}}|x| \log^{\frac{\gamma-1}{\gamma}}(1+|x|)\Big) ,$$ where $c_i>0$ $(i=1,2,3,4)$ are constants. We stress that, our method efficiently applies to the Hunt process $(X_t)_{t \ge 0}$ with finite range jumps, and some irregular potential function $V$ such that $\lim_{|x| \to \infty}V(x)\neq\infty$.Comment: 31 page

Coherence scale of coupled Anderson impurities

For two coupled Anderson impurities, two energy scales are present to characterize the evolution from local moment state of the impurities to either of the inter-impurity singlet or the Kondo singlet ground states. The high energy scale is found to deviate from the single-ion Kondo temperature and rather scales as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction when it becomes dominant. We find that the scaling behavior and the associated physical properties of this scale are consistent with those of a coherence scale defined in heavy fermion systems.Comment: 10 pages, 7 figures, extended versio