  We apply some ideas of Bombieri and Garrett to construct natural self-adjoint
operators on spaces of automorphic forms whose only possible discrete spectrum
is $\lambda_{s}$ for $s$ in a subset of on-line zeros of an $L$-function,
appearing as a compact period of cuspidal-data Eisenstein series on $GL_{4}$.
These ideas have their origins in results of Hejhal and Colin de Verdi\'ere. In
parallel with the $GL(2)$ case, the corresponding pair-correlation and
triple-correlation results limit the fraction of on-the-line zeros that can
appear in this fashion.
