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 ${\mathrm{a}\mathrm{d}}_A^k(H_V(\xi ))$ of arbitrary order $k\ge 1$ between a Hamiltonian $H_V(\xi )=M_{{\omega }_{\xi }}+S_{\check{V}}$ and a conjugate operator $A=\frac{\mathfrak{i}}{2}(v_{\xi }\cdot \mathrm{\nabla }+\mathrm{\nabla }\cdot v_{\xi })$, where $M_{{\omega }_{\xi }}$ is the operator of multiplication with the real analytic function ${\omega }_{\xi }$ which depends real analytically on the parameter $\xi$, and the operator $S_{\check{V}}$ is the operator of convolution with the (sufficiently nice) function $\check{V}$, and $v_{\xi }$ is some vector field determined by ${\omega }_{\xi }$. Under certain assumptions, which are satisfied for the Yukawa potential, we then prove estimates of the form $\lVert$\mathrm{a}\mathrm{d}$A^k(H_V(\xi))(H_0(\xi)+\mathfrak{i})^{-1}\rVert\le C\xi^kk!$where$C_\xi$issomeconstantwhichdependscontinuouslyon$\xi$. 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-M{\o}ller-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.