A multiplet analysis of spectra in the presence of broken symmetries

TL;DR

This paper introduces a new concept of generalized symmetries in Hamiltonians, characterizes them mathematically, and explores their implications for spectral analysis and stability, with applications to systems like the hydrogen atom under external fields.

Contribution

It defines generalized symmetries involving ladder operators, characterizes them via commutators, and applies these ideas to spectral partitioning and stability analysis.

Findings

Generalized symmetries are characterized by repeated commutators.

Spectrum of Hamiltonians can be partitioned into M-multiplets.

Eigenvector stability is linked to the properties of generalized symmetries.

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.