Counting operator analysis of the discrete spectrum of some model Hamiltonians

The first step in the counting operator analysis of the spectrum of any model Hamiltonian H is the choice of a Hermitean operator M in such a way that the third commutator with H is proportional to the first commutator. Next one calculates operators R and R which share some of the properties of creation and annihilation operators, and such that M becomes a counting operator. The spectrum of H is then decomposed into multiplets, not determined by the symmetries of H, but by those of a reference Hamiltonian Href , which is defined by Href = H − R − R, and which commutes with M . Finally, we introduce the notion of stable eigenstates. It is shown that under rather weak conditions one stable eigenstate can be used to construct another one.