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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 $q$ suchthat $\limsup_{x\to \infty}x\sqrt{p^2(x)+q^2(x)}=A$ and $\lambda$ is aneigenvalue of the corresponding Dirac operator $L_{p,q}$. We also constructfunctions $p$ and $q$ so that the corresponding Dirac operator $L_{p,q}$ hasany prescribed set {(finitely or countably many)} of eigenvalues.

Sharp bound for embedded eigenvalues of Dirac operators with decaying potentials