Systematics of strength function sum rules

TL;DR

This paper investigates the energy dependence of sum rules for transition strength functions, revealing that they vary with initial state energy and challenging the generalized Brink-Axel hypothesis, with explanations provided through spectral distribution theory.

Contribution

It demonstrates the secular dependence of non-energy-weighted sum rules on initial energy and explains this behavior using spectral distribution theory, providing new insights into transition strength functions.

Findings

Sum rules show strong secular dependence on initial energy.

The generalized Brink-Axel hypothesis does not generally hold.

Spectral distribution theory explains the observed systematics.

Abstract

Sum rules provide useful insights into transition strength functions and are often expressed as expectation values of an operator. In this letter I demonstrate that non-energy-weighted transition sum rules have strong secular dependences on the energy of the initial state. Such non-trivial systematics have consequences: the simplification suggested by the generalized Brink-Axel hypothesis, for example, does not hold for most cases, though it weakly holds in at least some cases for electric dipole transitions. Furthermore, I show the systematics can be understood through spectral distribution theory, calculated via traces of operators and of products of operators. Seen through this lens, violation of the generalized Brink-Axel hypothesis is unsurprising: one \textit{expects} sum rules to evolve with excitation energy. Furthermore, to lowest order the slope of the secular evolution can be traced to a component of the Hamiltonian being positive (repulsive) or negative (attractive).