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Replacing the canonical pair q and p of the harmonic oscillator (HO) by thelocally and symplectically equivalent pair angle phi and action variable Iimplies a qualitative change of the global topological structure of theassociated phase spaces: the pair (q,p) is an element of a topologicallytrivial plane R^2 whereas the pair (phi,I>0) is an element of a topologicallynon-trivial, infinitely connected, punctured plane R^2-{0}, which has the groupSO(1,2) as its "canonical" group. Due to its infinitely many covering groupsthe resulting ("symplectic") spectrum of the associated quantum Hamiltonian isgiven by {hbar omega (n+b), n =0,1,...; b in (0,1]}, in contrast to the"orthodox" spectrum {hbar omega (n+1/2)}.  The potentially most important implications concern the vibrations ofdiatomic molecules in the infrared, e.g. those of molecular hydrogen H_2. Thosesymplectic spectra of the HO may provide a simultaneous key to two outstandingastrophysical puzzles, namely the nature of dark (vacuum) energy and that ofdark matter: To the former because the zero-point energy b hbar omega of freeelectromagnetic wave oscillator modes can be extremely small > 0 (b ca.exp(-35) for the measured dark energy density). And a key to the dark matterproblem because the quantum zero-point energies of the Born-Oppenheimerpotentials in which the two nuclei of H_2 or the nuclei of other primordialdiatomic molecules vibrate can be lower, too, and, therefore, may lead tospectrally detuned "dark" H_2 molecules during the "Dark Ages" of the universeand forming WIMPs in the hypothesized sense. All results appear to be insurprisingly good agreement with the LambdaCDM model of the universe. Besideslaboratory experiments the search for 21-cm radio signals from the Dark Ages ofthe universe and other astrophysical observations can help to explore thosehypothetical implications.

Can the zero-point energy of the quantized harmonic oscillator be lower? Possible implications for the physics of "dark energy" and "dark matter"