  The pulsed-field-ionization zero-kinetic-energy photoelectron spectrum of
H$_2$ has been recorded in the vicinity of the dissociative-ionization
threshold following three-photon excitation via selected rotational levels of
the B $^1\Sigma_u^+$ ($v=19$) and $\bar{\rm H}$ $^1\Sigma_g^+$ ($v=11$)
intermediate states. The spectra consist of transitions to bound levels of the
X$^+$ $^2\Sigma_g^+$ state of H$_2^+$ with $v^+$ in the range 14-19 and $N^+$
in the range 0-9, of the A$^+$ $^2\Sigma_u^+$ state with $v^+=0$ and $N^+=0-2$,
and of shape resonances corresponding to the X$^+\ (v^+=17,N^+=7)$ and X$^+\
(v^+=18,N^+=4)$ quasibound levels. Calculations of the level structure of
H$_2^+$ have been carried out and the influence of adiabatic, nonadiabatic,
relativistic and radiative corrections on the positions of these levels, and in
the case of the shape resonances also on their widths, has been investigated.
Different methods of calculating the widths and profiles of the shape
resonances have been tested for comparison with the experimental observations.
Slow oscillations of the dissociative-ionization yield have been observed and
reflect, in first approximation, the Franck-Condon factors of the $\bar{\text
H}$ $\rightarrow$ X$^+$, A$^+$ bound - free transitions.
