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Rovibrational quantum states in the $X^1\Sigma_g^+$ electronic ground stateof H$_2$ are prepared in the $v=13$ vibrational level up to its highest boundrotational level $J=7$, and in the highest bound vibrational level $v=14$ (for$J=1$) by two-photon photolysis of H$_2$S. These states are laser-excited in asubsequent two-photon scheme into $F^1\Sigma_g^+$ outer well states, where theassignment of the highest ($v,J$) states is derived from a comparison ofexperimentally known levels in \F, combined with \emph{ab initio} calculationsof \X\ levels. The assignments are further verified by excitation of$F^1\Sigma_g^+$ population into autoionizing continuum resonances which arecompared with multi-channel quantum defect calculations. Precisionspectroscopic measurements of the $F-X$ intervals form a test for the \emph{abinitio} calculations of ground state levels at high vibrational quantum numbersand large internuclear separations, for which agreement is found.

Photolysis production and spectroscopic investigation of the highest vibrational states in H$_2$ (X$^1\Sigma_g^+$ $v=13,14$)