Counting operator analysis of the discrete spectrum of some model Hamiltonians

Abstract

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 H ref , which is defined by H ref = 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.In the literature many attempts are found to generalise the notion of creation and annihilation operators. Some of these were introduced in the context of Bogoliubov's notion of quasi-particles -see for instance [1]. Others are related to the method developed by Darboux in the nineteenth century to find new solutions of non-linear equations (see for instance [2]). Lowering and raising operators [3] determine recurrence relations and generate a Lie algebra (see for instance [4]).

A multiplet analysis of spectra in the presence of broken symmetries

“…The l = 1 M -multiplets in the example of the hydrogen atom all contain a pair of stable eigenvectors -the proof follows later on. This is not by accident, as shows the next theorem, which improves on a result from [8].…”

“…Several examples have been mentioned throughout the paper. Many of these examples involve the case of γ being real, which was already studied in [7,8]. This not by accident.…”

“…The following definition is taken from [8]. Eigenvectors ψ for which Rψ = 0 or R † ψ = 0 are clearly stable.…”

“…From these studies it is clear that supersymmetry is a rather exceptional phenomenon not present in the more common models of solid state physics. On the other hand it was pointed out in [7,8] that many hamiltonians H satisfy a higher order commutator relation with respect to certain hermitian operators M . In the present paper such an operator M is called a generalised symmetry of the hamiltonian H. This is motivated by showing that when supersymmetry is broken by adding a perturbation term linear in the supercharges then some symmetry operators become a generalised symmetry of the new hamiltonian.…”

“…In [7,8] the ladder operator R and its conjugate R † are defined as operators satisfying [R, M ] = γR. The parameter γ is assumed to be real.…”