Spectral Deformation for Two-Body Dispersive Systems with e.g. the Yukawa Potential

Abstract

We find an explicit closed formula for the k’th iterated commutator ${\text{ad}_{A}^{k}}(H_{V}(\xi ))$ of arbitrary order k ⩾ 1 between a Hamiltonian $H_{V}(\xi )=M_{\omega _{\xi }}+S_{\check V}$ and a conjugate operator $A=\frac{\mathfrak{i}}{2}(v_{\xi}\cdot\nabla+\nabla\cdot v_{\xi})$, where $M_{\omega _{\xi }}$ is the operator of multiplication with the real analytic function ω _ξ which depends real analytically on the parameter ξ, and the operator $S_{\check V}$ is the operator of convolution with the (sufficiently nice) function $\check V$, and v _ξ is some vector field determined by ω _ξ . Under certain assumptions, which are satisfied for the Yukawa potential, we then prove estimates of the form $\| {{\text{ad}_{A}^{k}}(H_{V}(\xi ))(H_{0}(\xi )+\mathfrak{i} )}\|\leqslant C_{\xi }^{k}k!$ where C _ξ is some constant which depends continuously on ξ. The Hamiltonian is the fixed total momentum fiber Hamiltonian of an abstract two-body dispersive system and the work is inspired by a recent result [3] which, under conditions including estimates of the mentioned type, opens up for spectral deformation and analytic perturbation theory of embedded eigenvalues of finite multiplicity.