  This article deals with boundary integral equation preconditioning for the
multiple scattering problem. The focus is put on the single scattering
preconditioner, corresponding to the diagonal part of the integral operator,
for which two results are proved. Indeed, after applying this geometric
preconditioner, it appears that, firstly, every direct integral equations
become identical to each other, and secondly, that the indirect integral
equation of Brakhage-Werner becomes equal to the direct integral equations, up
to a change of basis. These properties imply in particular that the convergence
rate of a Krylov subspaces solver will be exactly the same for every
preconditioned integral equations. To illustrate this, some numerical
simulations are provided at the end of the paper.
