Charged hydrogenic problem in a magnetic field: Non-commutative translations, unitary transformations, and coherent states

An operator formalism is developed for a description of charged electron-hole complexes in magnetic fields. A novel unitary transformation of the Hamiltonian that allows one to partially separate the center-of-mass and internal motions is proposed. We study the operator algebra that leads to the appearance of new effective particles, electrons and holes with modified interparticle interactions, and their coherent states in magnetic fields. The developed formalism is used for studying a two-dimensional negatively charged magnetoexciton $X^-$. It is shown that Fano-resonances are present in the spectra of internal $X^-$ transitions, indicating the existence of three-particle quasi-bound states embedded in the continuum of higher Landau levels.