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\(\zeta \ne 0\) .