On the problem of the vanishing discriminant

TL;DR

This paper discusses how the discriminant can become trivial and uninformative in physical models when symmetry remains unchanged, demonstrated through a specific Hermitian Hamiltonian example.

Contribution

It highlights a limitation of using the discriminant in physical problems with unaltered symmetry, illustrating the issue with a concrete matrix example.

Findings

Discriminant can vanish for all parameters due to symmetry degeneracy.

Symmetry invariance can lead to trivial discriminant results.

Illustration with a 6x6 Hermitian matrix example.

Abstract

We show that the straightforward application of the discriminant to some physical problems may yield a trivial useless result. If the symmetry of the model matrix does not change with variations of the model parameter the discriminant may vanish for all values of the parameter due to degeneracy. We illustrate this problem by means of a simple $6\times6$ matrix representation of an Hermitian Hamiltonian operator.