Girvin-MacDonald-Platzman Collective Mode at General Filling Factors: Magneto-Roton Minimum at Half-Filled Landau Level

The single mode approximation has proved useful for the excitation spectrum at $\nu=1/3$. We apply it to general fractions and find that it predicts $n$ magneto-roton minima in the dispersion of the Girvin-MacDonald-Platzman collective mode for the fractional quantum Hall states at $\nu=n/(2n+1)$, and one magneto-roton minimum for both the composite Fermi sea and the paired composite fermion state. Experimental relevance of the results will be considered.