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arXiv · 1110.3629

A multiplet analysis of spectra in the presence of broken symmetries

Jan Naudts·Tobias Verhulst
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Abstract

We introduce the notion of a generalised symmetry M of a hamiltonian H. It is a symmetry which has been broken in a very specific manner, involving ladder operators R and R*. In Theorem 1 these generalised symmetries are characterised in terms of repeated commutators of H with M. Breaking supersymmetry by adding a term linear in the supercharges is discussed as a motivating example. The complex parameter gamma which appears in the definition of a generalised symmetry is necessarily real when the spectrum of M is discrete. Theorem 2 shows that gamma must also be real when the spectrum of H is fully discrete and R and R* are bounded operators. Any generalised symmetry induces a partitioning of the spectrum of H in what we call M-multiplets. The hydrogen atom in the presence of a symmetry breaking external field is discussed as an example. The notion of stability of eigenvectors of H relative to the generalised symmetry M is discussed. A characterisation of stable eigenvectors is given in Theorem 3.

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Jan Naudts, Tobias Verhulst. 2011-10-17. A multiplet analysis of spectra in the presence of broken symmetries. https://doi.org/10.1088/1742-6596%2F343%2F1%2F012084

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