We prove the following. For any complex valued L p -function b(x), 2 ≤ p < ∞ or L ∞ -function with the norm b|L ∞ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d 2 /dx 2 + x 2 + b(x) in L 2 (R 1 ) is discrete and eventually simple. Its SEAF (system of eigen-and associated functions) is an unconditional basis in L 2 (R).2000 Mathematics Subject Classification. 47E05, 34L40, 34L10.