  There are three different methods used to search the neutrino mass: - The
electron antineutrino mass can probably best be determined by the Triton decay.
- The neutrinoless Double Beta Decay yields information, if the neutrino is a
Dirac or a Majorana particle. It can also determine the Majorana neutrino mass.
- Electron capture of an atomic bound electron by a proton in a nucleus bound
electron plus proton to neutron plus electron-neutrino can give the mass of the
electron neutrino. This contribution summarizes our theoretical work on the
possibility to determine the electron neutrino mass by electron capture. One
expects the largest influence of the neutrino mass on this decay for a small Q
= 2.8 keV for electron capture in Holmium. The energy of the Q value is
distributed to the emitted neutrino and the excitation of the Dy atom. Thus the
energy difference between the Q value and the upper end of the deexcitation
spectrum is the electron neutrino mass. The excitation spectrum of Dy is
calculate by one-, two- and three-electron hole excitations, and by the
shake-off process. The electron wave functions are calculated selfconsistently
by the Dirac-Hartree-Fock approach for the bound and the continuum states. To
extract the neutrino mass from the spectrum one must adjust simultaneously the
neutrino mass, the Q value, the position, the relative strength and the width
of the highest resonance. This fit is only possible, if the background is
reduced relative to the present situation. In case of a drastically reduced
background a fit of the Q-value and the neutrino mass only seems also to be
possible. The analysis presented here shows, that the determination of the
electron neutrino mass by electron capture is difficult, but seems not to be
impossible.
