Conceptually, the spectral function studies how an injected electron or hole propagates in a given system with energy resolution revealing the system’s electronic structure. By tuning the bias between a scanning tunneling microscope (STM) tip and a sample, electrons or holes of different energies can be injected. The derivative with respect to the applied bias should yield the wished energy resolution. And indeed, the conductance is the spectral function in the Tersoff-Hamann picture of STM. Now, the spectral function contains all possible excitations produced by the injected particle. Then, the STM grants us access to the excitation spectra of our substrate.
In this talk, I will review the tantalizing experiments showing the vibrational spectra of adsorbed molecules, and the theory that permits us to explain the experiments as well as interpret the spectra. But, not only vibrations are detected, magnetic excitations have also been revealed. I will also review experimental data and our simple theory that allows us to have a quantitative account of the observed data. Finally, the spectral function also informs on correlations at the atomic scale. The Kondo effect has been revealed on surfaces by the STM and I will show our efforts to provide simulations of the many Kondo signals discovered on adsorbed molecules.