Complete Lagrangian subspace characterization of self-adjoint boundary conditions

Let $E_q$ be the fiber of the Hermitian vector bundle over a boundary point $q$, let $A^n$ be the self-adjoint normal principal-symbol operator on $E_q$, and let $S_q\subseteq E_q$ be a subbundle. A subspace $S\subseteq E_q$ is complete Lagrangian relative to $A^n$ when

$$S=S^\#,$$

where

$$S^\#=\{\phi\in E_q:(\phi\mid A^n\chi)=0\ \forall\chi\in S\}.$$

Complete Lagrangian boundary-condition conjecture. The boundary condition

$$\psi(q)\in S_q\quad\forall q\in\partial\mathcal{Q}$$

can occur in a self-adjoint extension of $H$ from $C_c^\infty(E|_{\mathcal{Q}^\circ})$ in $L^2(E)$ if and only if $S_q$ is a complete Lagrangian subspace of $E_q$ relative to $A^n$ for every $q\in\partial\mathcal{Q}$. This characterizes the reflecting boundary conditions compatible with self-adjointness; the paper presents it as the right condition for such boundary conditions.