<p>In this paper, we study the spectral problem for the one-dimensional weighted Dirac equation. By introducing the rotation number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2056_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\( \rho (\lambda ) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and studying its properties, we prove that for any integer <i>k</i>, the periodic or anti-periodic eigenvalues are the endpoints of the interval <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2056_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="152" /> </InlineMediaObject> <EquationSource Format="TEX">\( \left\{ \lambda \in {\mathbb {R}}: \rho (\lambda ) =-\frac{k}{2} \right\} \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="}" open="{"> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>:</mo> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mfenced> </math></EquationSource> </InlineEquation>. Moreover, we use the spectral parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2056_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\( \lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> to characterize the Lyapunov stability of Dirac equations. Finally, we apply the trace formula to give the estimates of the periodic eigenvalues.</p>