Model-independent determination of the gap function of nearly localized superconductors

The gap function $\mathrm{\ensuremath{\Delta}}(\ensuremath{\omega})$ carries essential information on both, the pairing glue as well as the pair-breaking processes in a superconductor. Unfortunately, in nearly localized superconductors with a nonconstant density of states in the normal state, the standard procedure for extraction of $\mathrm{\ensuremath{\Delta}}(\ensuremath{\omega})$ cannot be applied. Here we introduce a model-independent method that makes it possible to extract $\mathrm{\ensuremath{\Delta}}(\ensuremath{\omega})$ also in this case. The feasibility of the procedure is demonstrated on the tunneling data for the disordered thin films of TiN. We find an unconventional feature of $\mathrm{\ensuremath{\Delta}}(\ensuremath{\omega})$ which suggests that the electrons in TiN are coupled to a very soft pair-breaking mode.