# Vector Models in \(\mathcal{PT}\) Quantum Mechanics

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## Abstract

We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of \(\mathcal {PT}\) quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.

## Keywords

Non-Hermitian quantum mechanics PT quantum mechanics Wigner-Eckhart theorem## Notes

### Acknowledgements

The authors would like to thank Harsh Mathur and Carl Bender for useful conversations.

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© Springer Science+Business Media New York 2013