Detecting level crossings without looking at the spectrum

TL;DR

This paper introduces an algebraic method to detect eigenvalue crossings and avoided crossings in physical systems by locating roots of a polynomial, applicable to atoms and molecules without detailed spectral knowledge.

Contribution

The authors present a novel algebraic approach to identify level crossings without spectral analysis, useful for atomic and molecular systems in external fields.

Findings

Method applied to atoms in magnetic fields, revealing new invariants.

Enables detection of molecular curve crossings without Born-Oppenheimer potentials.

Applicable to systems where spectral data is limited or unavailable.

Abstract

In many physical systems it is important to be aware of the crossings and avoided crossings which occur when eigenvalues of a physical observable are varied using an external parameter. We have discovered a powerful algebraic method of finding such crossings via a mapping to the problem of locating the roots of a polynomial in that parameter. We demonstrate our method on atoms and molecules in a magnetic field, where it has implications in the search for Feshbach resonances. In the atomic case our method allows us to point out a new class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. In the case of molecules, it enables us to find curve crossings with practically no knowledge of the corresponding Born-Oppenheimer potentials.