Singularities in the optical spectra of a system involving a Fermi sea of electrons and a localized hole: A method for obtaining many-body wave functions

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Abstract
In this paper, an alternative approach to many-body effects is introduced. We transform the Schrödinger equation into an eigenvalue problem, obtain an expression for the wave functions of the many-body eigenstates in a convenient basis, and apply it to understand the behavior of the system. We use this method to explain the singularities in the optical spectra due to transitions between a localized hole state and a Fermi sea of electrons. Our result, a power-law behavior of the spectra close to the threshold, coincides with that obtained from a Green’s-function treatment of the problem. Our method is based on very simple first principles. It provides a better understanding of what causes the singularities and explains the underlying physical picture in a simple way, without requiring elaborate mathematical techniques. It demonstrates the physics behind the ‘‘parquet’’-diagram analysis and the connection with the independent-boson model. It can also be generalized to understand less clear aspects of the problem, like the unbinding of the Mahan exciton and the transition to the conventional exciton and explain the case of a finite-mass valence hole interacting with a Fermi sea.