Symmetry in Nonlinear Mathematical Physics - 2009

Tobias Verhulst (Universiteit Antwerpen, Belgium)

Analysis of the spectrum of some Hamiltonians based on a generalization of the Dolan-Grady condition

We study the spectrum of model Hamiltonians H by choosing Hermitean operators M in such a way that the third commutator with H is proportional to the first commutator. This is a generalization of the Dolan-Grady condition. From this, one calculates the generalized creation and annihilation operators R and R. The operators M then act like counting operators.

We then prove two theorems showing how the eigenvalues and eigenvectors of H can be constructed using R and R. A third theorem states that energy levels within a multiplet cannot cross each other as functions of the parameters of the model, even though the classification of the energy levels in multiplets is not based on the symmetries of H. The emerging picture is that of the decomposition of the spectrum of H into multiplets, not determined by the symmetries of H but by those of the reference Hamiltonian Href = HRR.

We use our theorems to calculate explicitly the spectrum of some simple models and classify the eigenstates. These models include for example the Jaynes-Cummings model, the one dimensional Hubbard model and Witten's supersymetric model.

1) Verhulst T., Anthonis B. and Naudts J.: Analysis of the N=4 Hubbard ring using counting operators, Phys. Lett. A 373 (2009), 2109-2113, arXiv:0811.3077.
2) Verhulst T., Naudts J. and Anthonis B.: Counting operator analysis of the discrete spectrum of some model Hamiltonians, submitted to J. Phys. A, arXiv:0811.3073.