Local Resolvent Estimates for N-body Stark Hamiltonians

Abstract

For an N-body Stark Hamiltonian $H = -\Delta/2 - |E|z + V$ , the resolvent estimate $\|\langle x\rangle^{-\sigma'-1/4}(H - \zeta)^{-1}\langle x\rangle^{-\sigma'-1/4}\|_{\boldsymbol{B}(L^2)}\le C$ holds uniformly in $\zeta \in \boldsymbol{C}$ with Re$\zeta \in I$ and Im$\zeta \ne 0$ , where $\sigma' > 0$ , and $I \subset \boldsymbol{R}$ is a compact interval. This estimate is well known as the limiting absorption principle. In this paper, we report that by introducing the localization $\tilde{q}_0(x) = \sqrt{1-z/\langle x\rangle}$ in the configuration space, a refined resolvent estimate $\|\langle x\rangle^{-1/4}\tilde{q}_0(x)(H - \zeta)^{-1}\tilde{q}_0(x)\langle x\rangle^{-1/4}\|_{\boldsymbol{B}(L^2)} \le C$ holds uniformly in $\zeta \in \boldsymbol{C}$ with Re$\zeta \in I$ and Im$\zeta \ne 0$ .