  We present formulas for the nuclear and electronic spin relaxation times due
to the hyperfine interaction for nanostructed systems and show that the times
depend on the square of the local density of electronic states at the nuclear
position. A drastic sensitivity (orders of magnitude) of the electronic and
nuclear spin coherence times to small electric fields is predicted for both
uniformly distributed nuclear spins and for $\d$-doped layers of specific
nuclei. This sensitivity is robust to nuclear spin diffusion.
