| A new approach to the problem of X-ray edge singularities and peaks of many-body origin observed in the optical spectra is presented. We first establish the analogy between a system of one hole interacting with many electrons and the polaron, where one electron interacts with many phonons. We approach both cases as an eigenvalue problem for the time independent Schrodinger equation, obtain expressions for the many-body eigenstates in a convenient basis and use them to understand the behavior of the system. In the case of a localized hole with infinite mass, we diagonalize the Hamiltonian exactly using first a wavefunction and then a noncanonical transformation technique. We use the expressions for the eigenstates to obtain the exact analytic formulas for the optical spectra, accurate over the whole frequency range. We then apply our techniques to the case of a finite mass valence hole, which is realized in modulation doped quantum wells or wires. In this case we approximately diagonalize the Hamiltonian by treating exactly the most important terms leading to nonperturbative behavior. The terms neglected are identified and can be treated in perturbation theory. Our method provides a clear physical picture and valuable insight into the accurate treatment of many-body Hamiltonians. |
