In summary, the paper discusses the idea of quantum systems emitting energy due to spacetime expansion. However, the paper has not been published in a peer-reviewed journal and makes false assumptions about the behavior of our actual universe. Its claims are not supported by available physical evidence and therefore, the paper's argument is not valid.f
This paper doesn't look like it's been published in a peer-reviewed journal. (If you can find a reference that it has, please post it.)
Assumption A at the bottom of p. 2 is obviously false for our actual universe, since what the paper calls "metric expansion" is a feature of the FRW models which are homogeneous and isotropic, and our universe is very, very far from being homogeneous and isotropic on any length scales smaller than tens to hundreds of millions of light years. The paper's claim on p. 3 that assumption A is "not contradicted by available physical evidence" is simply wrong.
The above by itself, I suspect, would be sufficient for a peer-reviewed journal to reject this paper, since it does not claim to be simply an investigation of a mathematical hypothesis but rather an investigation of something which could be true of our actual universe.
Assumptions B and C on p. 3 of the paper could be taken as correct with an appropriate interpretation of the words they use; but unfortunately that is not the interpretation that the paper gives them. The "contraction" of bound systems in comoving coordinates is not a physical effect, it's a coordinate effect. And bound systems, such as galaxy clusters, galaxies, stars, and planets, while they technically can only radiate energy in finite sized quanta (since that is the case for any system), radiate amounts of energy in the course of formation that are so many orders of magnitude larger than the size of the energy quanta they radiate that the continuous approximation, i.e., the classical approximation, is more than good enough and there is no need to consider any quantum specific properties of radiation in order to analyze their behavior.
So the whole basis of the paper's argument appears to me to be wrong.