High-order harmonic generation spectra of atoms and molecules in intense laser fields show a wide plateau of odd harmonics, with photon energies high up to the soft X-ray regime. The mechanism leading to the absence of other frequencies, so-called hyper-Raman lines or sidebands, is still unclear. By the use of non-Hermitian quantum mechanics for the many-electron atomic/molecular Hamiltonian system, we derive closed-form expressions for the high-order harmonic generation spectra. The analysis of these expressions provides the (already known) conditions for which the hyper-Raman lines are missing and also the conditions for which they are observable. The new information obtained here is proof that in order to observe sidebands it is not sufficient to populate two different Floquet states. When the duration of the laser pulse is sufficiently long, the two resonance Floquet states should have similar widths (inverse lifetimes). For short pulses, the condition for observable sidebands is more complicated, and it does not depend directly on the lifetime of the resonance states. By numerical integration of the conventional time-dependent Schrodinger equation, we provide illustrative examples that confirm our conclusions. Our results can be used to design new experiments that will show the fingerprints of the hyper-Raman lines.