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The excitation spectrum of the frustrated spin-$1/2$ Heisenberg chain isreexamined using variational and exact diagonalization calculations. We showthat the overlap matrix of the short-range resonating valence bond states basiscan be inverted which yields tractable equations for single and two spinonsexcitations. Older results are recovered and new ones, such as the bond-statedispersion relation and its size with momentum at the Majumdar-Ghosh point arefound. In particular, this approach yields a gap opening at $J_2=0.25J_1$ andan onset of incommensurability in the dispersion relation at $J_2=9/17J_1$ [asin S. Brehmer \emph{et al.}, J. Phys.: Condens. Matter \textbf{10}, 1103(1998)]. These analytical results provide a good support for the understandingof exact diagonalization spectra, assuming an independent spinons picture.

Spinon excitation spectra of the $J_1$-$J_2$ chain from analytical calculations in the dimer basis and exact diagonalization