Suppose we have a photon with energy less than the necessary to excited the hydrogen atom from ground state to first excited state. Now suppose we make this photon to incise into a hydrogen atom in the ground state, does this mean that cross section for absorption of the photon is zero?
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It means that the cross-section would be very small. Technically, it would not be zero for at least two reasons - one is that all transitions have a width that means that the probability just gets smaller and smaller the further away from the resonant energy you are, but it is not actually zero. Secondly it is possible for the photon to interact with the bound electron/proton system as if it were a classical oscillator - this is known as Rayleigh scattering. The photon is not absorbed, it is elastically scattered. However, the cross-section for Rayleigh scattering is very small compared with the cross-section for resonant absorption of a photon of the right energy, and it decreases as frequency to the 4th power.
