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

Replacing the canonical pair q and p of the harmonic oscillator (HO) by the locally and symplectically equivalent pair angle phi and action variable I implies a qualitative change of the global topological structure of the associated phase spaces: the pair (q,p) is an element of a topologically trivial plane $R^2$ whereas the pair (φ,I > 0) ∈ $S^1$ × $R_{+}$ is an element of a topologically non-trivial, infinitely connected, punctured plane $R^2$-{0}, which has the group SO↑ (1,2) (or its two-fold covering, the symplectic group Sp(2, $R$)) as its "canonical" group. Due to its infinitely many covering groups the resulting ("symplectic") spectrum of the associated quantum Hamiltonian H=ωÎ is given by {$\hslash$ω (n + b), n = 0, 1, ..., b ∈ (0:1], e.g. b = 1/s; s ∈ $N$ and large}, in contrast to the "orthodox" spectrum {{$\hslash$ω (n + 1/2)}. The deeper reason for the difference is that for the description of the periodic orbit {p = p(q)} one covering of $S^1$ suffices, whereas one generally needs many coverings for the time evolution φ(t). And this, in turn, can lead to a lowering of the zero-point energies. Several theoretical and possible experimental implications of the "symplectic" spectra of the HO are discussed: The potentially most important implications concern the vibrations of diatomic molecules in the infrared, e.g. those of molecular hydrogen H${}_2$. Those symplectic spectra of the HO may provide a simultaneous key to two outstanding astrophysical puzzles, namely the nature of dark (vacuum) energy and that of dark matter: To the former because the zero-point energy b hbar omega of free electromagnetic wave oscillator modes can be extremely small > 0 (b $\approx$ exp (-35) for the measured dark energy density). And a key to the dark matter problem because the quantum zero-point energies of the Born-Oppenheimer potentials in which the two nuclei of H${}_2$ or the nuclei of other primordial diatomic molecules vibrate can be lower, too, and, therefore, may lead to spectrally detuned "dark" H${}_2$ molecules during the "Dark Ages" of the universe and forming WIMPs in the hypothesized sense. All results appear to be in surprisingly good agreement with the $\Lambda$CDM model of the universe. Besides laboratory experiments the search for 21-cm radio signals from the Dark Ages of the universe and other astrophysical observations can help to explore those hypothetical implications.