  We study eigenvalues of the Dirac operator with canonical form
  \begin{equation}
  L_{p,q} \begin{pmatrix}
  u \\ v
  \end{pmatrix}=
  \begin{pmatrix}
  0 & -1 \\ 1 & 0
  \end{pmatrix}\frac{d}{dt}
  \begin{pmatrix}
  u \\ v
  \end{pmatrix}+\begin{pmatrix}
  -p & q \\ q & p
  \end{pmatrix}\begin{pmatrix}
  u \\ v
  \end{pmatrix},\nonumber
  \end{equation}
  where $ p$ and $q$ are real functions. Under the assumption that
  \begin{equation}
  \limsup_{x\to \infty}x\sqrt{p^2(x)+q^2(x)}<\infty,\nonumber
  \end{equation}
  the essential spectrum of $L_{p,q}$ is $(-\infty,\infty)$.
  We prove that $L_{p,q}$ has no eigenvalues if
  $$\limsup_{x\to \infty}x\sqrt{p^2(x)+q^2(x)}<\frac{1}{2}.$$ Given any $A\geq
\frac{1}{2}$ and any $\lambda\in\R$, we construct functions $p$ and $q$ such
that $\limsup_{x\to \infty}x\sqrt{p^2(x)+q^2(x)}=A$ and $\lambda$ is an
eigenvalue of the corresponding Dirac operator $L_{p,q}$. We also construct
functions $p$ and $q$ so that the corresponding Dirac operator $L_{p,q}$ has
any prescribed set {(finitely or countably many)} of eigenvalues.
