Planar Dirac Electron in Coulomb and Magnetic Fields: a Bethe ansatz approach

Details Planar Dirac Electron in Coulomb and Magnetic Fields: a Bethe ansatz approach Planar Dirac Electron in Coulomb and Magnetic Fields: a Bethe ansatz approach

2001-05-22

arxiv.org/abs/hep-th/0105223v1

J.Math.Phys. 43 (2002) 43-51

Physics

High Energy Physics

High Energy Physics - Theory

LaTex, 12 pages, no figures

Scientific paper

10.1063/1.1418426

The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is an example of the so-called quasi-exactly solvable models. The solvable parts of its spectrum was previously solved from the recursion relations. In this work we present a purely algebraic solution based on the Bethe ansatz equations. It is realised that, unlike the corresponding problems in the Schr\"odinger and the Klein-Gordon case, here the unknown parameters to be solved for in the Bethe ansatz equations include not only the roots of wave function assumed, but also a parameter from the relevant operator. We also show that the quasi-exactly solvable differential equation does not belong to the classes based on the algebra $sl_2$.

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