We predict the existence of a ubiquitous class of long-range molecular Rydberg states, whose Born-Oppenheimer potential curves are oscillatory in nature. These oscillations reflect the nodal structure of the atomic Rydberg state wave functions. The temperature and density of atoms in a Bose-Einstein condensate are particularly favorable for the laser excitation of ultra-long-range vibrational bound states localized at internuclear distances in the range 10(3)- 10(5) a.u. A surprising trilobitelike class of polar homonuclear diatomics should exhibit electric dipole moments in the kilodebye range.

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We identify two qualitatively different mechanisms that control three-body recombination in a spinpolarized gas near zero temperature. A universal curve describes the recombination rate versus the two-body scattering length a. It grows as a 4 for large jaj, with different mechanisms for a , 0 and a . 0. Our calculations document a previously established mechanism that causes K 3 to grow rapidly as the two-body scattering length a increases toward 1`, and a new tunneling mechanism that produces an even stronger enhancement of K 3 as a ! 2`. The expectations based on these two mechanisms can be modified by quantum mechanical interference or resonance effects.