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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 $\mathrm{diag}(-1,1)$. 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.

For two assemblies of congruent Euclidean tiles, let $M_s,N_s$ encode the gluing or boundary reflection at each labelled face. An invertible constant matrix $C$ satisfying $C{M}_s=N_{s}C$ carries tile restrictions of Laplacian eigenfunctions bijectively to those on the second assembly. Boundary values satisfy $M_{s}u=u$ and outward normal derivatives satisfy $M_{s}v=-v$, so the intertwining identities preserve matching and boundary conditions. Use diagonal $-1$ for a Dirichlet boundary condition and $+1$ for a Neumann boundary condition.

A triangular wave is a continuous piecewise linear periodic function whose slope alternates between two values. The even $2\pi$-periodic extension of $bx$ from $[0,\pi ]$ is $P(s)=b|s-2k\pi |$ for $(2k-1)\pi \le s\le (2k+1)\pi$. Its Fourier cosine series contains a constant term and odd modes with coefficients proportional to the inverse square of the mode number. Such extensions describe reflection under Neumann boundary conditions.

The spatial integration-by-parts term in the Nambu-Goto phase-space action pairs endpoint displacement with momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Free endpoint displacements require zero flux. Fixed normal coordinates instead set those variations to zero, while tangential variations retain a Neumann boundary condition. Boundary conditions must also be preserved by the equations of motion. In a boundary-preserving gauge with $u=0$, mixed Neumann boundary conditions and Dirichlet boundary conditions implement a planar D-brane.

Let a positive-measure part ${\Gamma }_D$ of the boundary of a bounded connected domain carry a homogeneous Dirichlet boundary condition, and let the complementary part carry a homogeneous Neumann boundary condition. Onthe weak Poisson problem isThe Poincare inequality with a partial Dirichlet boundary makes the left side coercive, so the Lax-Milgram theorem gives a unique weak solution.

On $(0,L)$, the differential operator $\mathcal{L}={\partial }_x^2+\beta {\partial }_x=e^{-\beta x}{\partial }_x(e^{\beta x}{\partial }_x)$ with homogeneous Neumann boundary conditions is self-adjoint in the weighted inner product with weight $e^{\beta x}$. Its eigenfunctions and nonnegative decay rates areTheir squared norms are $N_0=(e^{\beta L}-1)/\beta$ and $N_k=L[1+{\beta }^2/(4q_k^2)]/2$. The Sturm-Liouville eigenfunction expansion givesThis kernel acts against the measure $e^{\beta y}dy$, not Lebesgue measure alone. It is symmetric in $x,y$; the transition density against Lebesgue measure is $e^{\beta y}{K}_{\beta }(x,y,t)$.