Spectral deformation for two-body dispersive systems with e.g. the Yukawa potential

We find an explicit closed formula for the $k$'th iterated commutator $\mathrm{ad}_A^k(H_V(ξ))$ of arbitrary order $k\ge1$ between a Hamiltonian $H_V(ξ)=M_{ω_ξ}+S_{\check V}$ and a conjugate operator $A=\frac{\mathfrak{i}}{2}(v_ξ\cdot\nabla+\nabla\cdot v_ξ)$, where $M_{ω_ξ}$ 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 $\lVert\mathrm{ad}_A^k(H_V(ξ))(H_0(ξ)+\mathfrak{i})^{-1}\rVert\le C_ξ^kk!$ 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 [Engelmann-Møller-Rasmussen, 2015] which, under conditions including estimates of the mentioned type, opens up for spectral deformation and analytic perturbation theory of embedded eigenvalues of finite multiplicity.

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