It is shown how unbound electron wave functions can be expanded in a suitably chosen localized basis sets for any desired range of energies. In particular, we focus on the use of gaussian basis sets, commonly used in first-principles codes. The possi
ble usefulness of these basis sets in a first-principles description of field emission or scanning tunneling microscopy at large bias is illustrated by studying a simpler related phenomenon: The lifetime of an electron in a H atom subjected to a strong electric field.
