Bloch Solutions of Periodic Dirac Equations in SPPS Form

TL;DR

This paper introduces a new representation of quasi-periodic solutions for periodic Dirac equations using spectral parameter power series, enabling explicit analysis of spectral properties and transformations.

Contribution

It provides the SPPS form of solutions and the Dirac Hill discriminant, extending the method to Darboux transformations and specific solvable potentials.

Findings

SPPS representation of Dirac solutions derived

Discriminant expressed in SPPS form under Darboux transformation

Application to Razavy's quasi-exactly solvable potential

Abstract

We provide the representation of quasi-periodic solutions of periodic Dirac equations in terms of the spectral parameter power series (SPPS) recently introduced by V.V. Kravchenko (2008,2009). We also give the SPPS form of the Dirac Hill discriminant under the Darboux nodeless transformation using the SPPS form of the discriminant and apply the results to one of Razavy's quasi-exactly solvable periodic potentials