 Solution

Identical oriented Dirichlet boundary data imply a rotation-invariant solution whenever the Dirichlet problem is unique. For $\lambda \ge 0$, uniqueness follows directly: the difference $h$ of two solutions has zero boundary trace, and integration by parts gives ${\int }_D|\nabla h{|}^2+4\lambda {\int }_D|h{|}^2=0$. Thus $h=0$. The same conclusion holds for a nonresonant negative parameter by invertibility of the Dirichlet Laplacian shift. Then $q(az)=q(z)$, all three normal traces equal a common $q_N(s)$, and ${\Psi }_j=\Psi$.

SetDivide the global relation by $E(-iak)$. Using $1+a+\bar{a}=0$ yields $E(-ik)/E(-iak)=e(\bar{a}k)$ and $E(-i\bar{a}k)/E(-iak)=e(-k)$. Hencewhere the requested known function isFor an explicitly derivative-free form, put $H(k)={\int }_{-\ell /2}^{\ell /2}{e}^{(k+\lambda /k)s}f(s)ds$ and $f_{\ast }=f(-\ell /2)=f(\ell /2)$. An integration by parts givesThe uniqueness qualification cannot be removed for every real $\lambda$: at a Dirichlet Laplacian eigenvalue a homogeneous addition need not preserve rotation invariance. The scalar reduced relation then applies to the rotation-averaged solution, but not necessarily to the originally chosen solution. Part (d) treats this genuine exceptional case.