= Adjoint eigenfunction
{title2=$L^*g=\overline{\lambda}g$}
An adjoint <eigenfunction> is a nonzero <eigenfunction> of the <adjoint operator> on its adjoint domain. For an <inner product> conjugate-linear in its first argument, $L^*g=\overline\lambda g$ implies $\langle g,(L-\lambda)f\rangle=0$ for every $f$ in the domain of $L$. Consequently a necessary <solvability condition> for $(L-\lambda)f=h$ is $\langle g,h\rangle=0$. If $L-\lambda I$ is a <Fredholm operator>, orthogonality to every element of its adjoint kernel is also sufficient. <Boundary conditions> and weights in a coupled system enter the adjoint domain and must be derived by <integration by parts>; copying the original <eigenfunction> without checking them can give a wrong projection.
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