Commutation technique for an exciton photocreated close to a metal.

Recently, we have derived the changes in the absorption spectrum of an exciton when this exciton is photocreated close to a metal. The resolution of this problem - which has similarities with Fermi edge singularities - has been made possible by the introduction of "exciton diagrams" in which the exciton metal-electron vertex has been essentially guessed. The validity of this procedure relied on a dreadful calculation based on standard free electron and free hole diagrams, with the semiconductor-metal interaction included at second order only, and its guessed extention to higher orders. Using a commutation technique similar to the one we recently introduced to deal with interacting excitons, we are now able to prove that this exciton diagram procedure we proposed is indeed valid for this problem at any order in the interaction