We study eigenvalues of the Dirac operator L with canonical form. Under the assumption that the potential is o(1) near infinity, the essential spectrum of L is R. We prove that L has no embedded eigenvalues if the potential is controlled by some Coulomb potential near infinity. And for any real number, we construct Coulomb type potentials so that such number is an eigenvalue of the corresponding Dirac operator L. We also construct potentials so that the corresponding Dirac operator L has any prescribed set (finitely or countably many) of eigenvalues.

This work was completed in an ongoing High School and Undergraduate Research Program "STODO" (Spectral Theory Of Differential Operators) at Texas A & M University. We would like to thank Wencai Liu for managing the program, introducing this project and many inspiring discussions. The authors are also grateful to the anonymous referee, whose comments led to an improvement of our manuscript. This work was partially supported by NSF DMS-2015683 and DMS-2000345.