  Quasi-localised modes appear in the vibrational spectrum of amorphous solids
at low-frequency. Though never formalised, these modes are believed to have a
close relationship with other important local excitations, including shear
transformations and two-level systems. We provide a theory for their frequency
density, $D_{L}(\omega)\sim\omega^{\alpha}$, that establishes this link for
systems at zero temperature under quasi-static loading. It predicts two regimes
depending on the density of shear transformations $P(x)\sim x^{\theta}$ (with
$x$ the additional stress needed to trigger a shear transformation). If
$\theta>1/4$, $\alpha=4$ and a finite fraction of quasi-localised modes form
shear transformations, whose amplitudes vanish at low frequencies. If
$\theta<1/4$, $\alpha=3+ 4 \theta$ and all quasi-localised modes form shear
transformations with a finite amplitude at vanishing frequencies. We confirm
our predictions numerically.
