= Modified Helmholtz closed spectral one-form
{title2=$W=e^{-ikz-\lambda\bar z/(ik)}[(q_z+ikq)dz-(q_{\bar z}+\lambda q/(ik))d\bar z]$}
For $k\ne0$, differentiation gives $dW=-2e^{-ikz-\lambda\bar z/(ik)}(q_{z\bar z}-\lambda q)dz\wedge d\bar z$. Thus the <differential form> is closed precisely for the <modified Helmholtz equation> $\Delta q-4\lambda q=0$. Integrating around an oriented domain gives a <global relation>. The primitive $\mu e^{-ikz-\lambda\bar z/(ik)}$ also produces compatible first-order equations whose spectral jumps reconstruct $q$.
Back to article page
