Spectrum-generating algebra

Bohm, A., Ne eman, Y., and Barut, A. O. (1988), Dynamical Groups and Spectrum Generating Algebras, World Scientific, Singapore. 

Dothan, Y. (1970), Finite-Dimensional Spectrum-Generating Algebras, Phys. Rev. D 2, 2944. 

This equation was solved for the hydrogen atom by Vladimir Fock [2,3]. The solution in p space revealed the four-dimensional symmetry responsible for the degeneracy of states with the same n but different I quantum numbers in the hydrogen atom. This is a fine example where the momentum-space perspective led to fresh and deep insight. Fock s work spawned much further research on dynamical groups and spectrum-generating algebras. 

The operators J and J 2 are constant within a given irreducible representation of the spectrum-generating algebra they are not of interest here. The effective coupling is given by the scalar product which... 

To begin with, we generalize the spectrum-generating algebra (3.1), introduced for two interacting oscillators, by considering, for N interacting oscillators, the product... 

Figure 22 Schematic representation of the Dirac spectrum and of the Schrodinger spectrum generated in the algebraic approximation...

In this section we shall discuss in some detail the formalism needed to apply the so(4, 2) algebraic methods to problems whose unperturbed Hamiltonian is hydrogenic. First a scaling transformation is applied to obtain a new Hamiltonian whose unperturbed part is just the so(2, 1) generator T3, which has a purely discrete spectrum. Next we use the scaled hydrogenic eigenfunctions of T3 as a basis for the expansion of the exact wave function. This discrete basis is complete with respect to the expansion of bound-state wave functions whereas the usual bound-state eigenfunctions do not form a complete set continuum functions must also be included to ensure completeness (cf. Section VI,A)-... 

QED provides a framework for describing the role of the negative energy states in the Dirac theory and the divergences which arise in studies of electrodynamic interactions.83-85 In Section 2.3 it will be shown how the use of the algebraic approximation to generate a discrete representation of the Dirac spectrum has opened the way for the transcription of the rules of QED into practical algorithms for the study of many-electron systems. 

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