= Transplantation by reflection parity
{title2=$C=2^{-1/2}\begin{pmatrix}1&-1\\1&1\end{pmatrix}$}
Reflection across a cut splits <Laplacian eigenfunctions> into odd and even parts. The odd restriction has a <Dirichlet boundary condition> on the cut; the even restriction has a <Neumann boundary condition> there. The displayed matrix intertwines the two-tile swap with $\operatorname{diag}(-1,1)$. A connected uniform-Dirichlet rectangle of width $2a$ is therefore isospectral to the disjoint union of width-$a$ rectangles with all-Dirichlet and three-Dirichlet/one-Neumann conditions. This illustrates why changing boundary conditions can conceal connectedness in the <spectrum>.
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