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 $\mathrm{ad}_A^k(H_V(\xi))$ of arbitrary order $k\ge1$ 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 $\omega_\xi$ 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 $\lVert\mathrm{ad}_A^k(H_V(\xi))(H_0(\xi)+\mathfrak{i})^{-1}\rVert\le C_\xi^kk!$ where $C_\xi$ is some constant which depends continuously on $\xi$. 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.