  We report medium resolution VLT ISAAC K-band spectroscopy of the nuclei of
seven ultraluminous infrared galaxies. After accounting for stellar absorption
features, we have detected several molecular hydrogen (H_2) v=1-0, 2-1, and 3-2
vibrational emission lines, as well as the HI Br\gamma and HeI 2^1P-2^1S
recombination lines. The relative H_2 line intensities show little variation
between the objects, suggesting that the H_2 excitation mechanisms in the
nuclei are similar in all the objects. The 1-0 emissions appear thermalised at
temperatures T\sim1000K. However, the 2-1 and 3-2 emissions show evidence of
being radiatively excited by far-ultraviolet (FUV) photons, suggesting that the
H_2 excitation in the ULIRGs may arise in dense photon dominated regions
(PDRs). We show that the line ratios in the nuclei are consistent with PDRs
with cloud densities between 10^4 to 10^5cm^{-3}, exposed to far ultraviolet
(FUV) radiation fields at least 10^3 times more intense than the ambient FUV
intensity in the local interstellar medium. We have constructed starburst
models for the ULIRGs based on their H_2 properties, as well as on the
intensities of the recombination lines. Our models provide a consistent picture
of young 1-5Myr star clusters surrounded by relatively dense PDRs which are
irradiated by intense FUV fluxes. Comparison to the inner few hundred parsecs
of the Milky Way indicates that the star formation efficiency in ULIRGs is
10--100 times higher than in the Galactic Center.
