We obtain the general formulation which can handle the rotational preionization spectrum of $H_2$ in the region above its ${H_2}^+$ ionization threshold, ($^2{sum}_g^+$, ${ u}^+=0$, $N^+=0$) converging toward its rotationally excited (${ u}^+=0$, $N^+=2$) limit and perturbed by the vibrationally excited levels $7p{pi}$ ${ u}=1$ and $5p{pi}$ ${ u}^=2$. The formulation is based on phase-shifted multichannel quantum-defect theory. With this formulation, resonance structures are analyzed in detail.