Definition: Boundary relation, including boundary charts

Choose a subspace $W\subseteq {\mathbb{C}}^d$, an isometric inclusion $V:{\mathbb{C}}^{\dim W}\to {\mathbb{C}}^d$ with range $W$, and a Hermitian operator $H_W$ on ${\mathbb{C}}^{\dim W}$. The boundary condition is

The component of $Nb$ in $W^{\perp }$ is unrestricted.

For $W={\mathbb{C}}^d$, this is

For $W=\{0\}$, it is Friedrichs. The diagonal choice $H_{jj}={\nu }_j{\lambda }_j$ gives $b_j={\lambda }_j{a}_j$. The cases ${\lambda }_j=\infty$ are represented by removing that direction from $W$, not by an ordinary finite matrix entry.