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

TL;DR

This paper derives explicit formulas for iterated commutators of a two-body dispersive Hamiltonian with a conjugate operator, enabling spectral deformation analysis for systems like the Yukawa potential.

Contribution

It provides a closed-form expression for higher-order commutators and establishes factorial bounds, facilitating spectral analysis of dispersive two-body systems.

Findings

Explicit formula for iterated commutators of the Hamiltonian.

Factorial bounds on commutator norms under certain assumptions.

Application to spectral deformation and perturbation theory for embedded eigenvalues.

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.