We consider the counter images H_♭(R^d) and H_{0,♭}(R^d) of entire functions with exponential and almost exponential bounds, respectively, under the Bargmann transform, and we characterize them by estimates of powers of the harmonic oscillator. We also consider the Pilipovic ́ spaces S_s(R^d ) and Σ_s(R^d) when 0 < s < 1/2 and deduce their images under the Bargmann transform.