Spectral quasi-clustering estimates for certain semiregular systems

We show a quasi-clustering result for a subclass of the class of Semiregular Metric Globally Elliptic Systems (SMGES) including certain quantum optics models (such as Jaynes-Cummings and its generalizations) which describe light-matter interaction. More precisely, we show that for the class of systems with polynomial coefficients we consider, the spectrum concentrates within the union of intervals (not necessarily disjoint, but at most intersecting in an a priori finite number) centered at a sequence determined in terms of invariants of the (total) symbol and width decreasing as the centers go to infinity.