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We prove that the $1-d$ quantum harmonic oscillator is stable under spatiallylocalized, time quasi-periodic perturbations on a set of Diophantinefrequencies of positive measure. This proves a conjecture raised byEnss-Veselic in their 1983 paper \cite{EV} in the general quasi-periodicsetting. The motivation of the present paper also comes from construction ofquasi-periodic solutions for the corresponding nonlinear equation.

Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations