Schr\"odinger-Pauli Equation for the Standard Model Extension CPT-Violating Dirac Equation

It is instructive to investigate the non-relativistic limit of the simplest Standard Model Extension (SME) CPT-violating Dirac-like equation but with minimal coupling to the electromagnetic fields. In this limit, it becomes an intuitive Schr"odinger-Pauli-like equation. This is comparable to the free particle treatment as explored by Kostelecky and Lane, but this exercise only considers the $a$ and $b$ CPT-violating terms and $\vec{p}\mathrm{/}m$ terms to first order. Several toy systems are discussed.