We analyze the existences, properties and stabilities of the self-localized solutions of the nonlinear quantum harmonic oscillator (NQHO) using spectral renormalization method (SRM). We show that self-localized single and dual soliton solutions of the NQHO do exist. Additionally we report that single and dual soliton solutions satisfy the necessary Vakhitov and Kolokolov slope condition, at least for the parameter ranges considered. Additionally, we investigate the stability characteristics of the single and dual soliton solutions using a split-step Fourier scheme. We show that single and dual soliton solutions are pulsating during time stepping. We discuss our findings and comment on our results.