Note: Effect of localization on mean-field density of state near jamming

We discuss the effects of the localized modes on the density of state $D(\omega)$ by introducing the probability distribution function of the proximity to the marginal stability. Our theoretical treatment reproduces the numerical results in finite dimensions near the jamming point., in particular, successfully captures the novel $D(\omega)\sim \omega^4$ scaling including its pressure dependence of the pre-factor.