Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 328 / 3 / ii

Write $z=x+iy$ and $\overline{z}=x-iy$. A convenient spectral parameter for a linear boundary value problem is any $\lambda \in \mathbb{C}\setminus \{0\}$. Define the modified Helmholtz adjoint plane wavewhereIndeed $A^2+B^2=k^2$, so $(\Delta -k^2)v_{\lambda }=0$. Equivalently, $4{\partial }_z{\partial }_{\overline{z}}{v}_{\lambda }=k^2{v}_{\lambda }$. The exclusion of zero is only the coordinate singularity of this spectral parametrization; it still supplies a whole complex one-parameter family. The conjugate companion isIt also solves the adjoint equation.