Uniform Observability for Mixed Parity Sectors of an Equilateral Rhombus

Abstract: We study frequency-uniform $L<sup>2$ observability for Laplace eigenfunctions in the mixed Dirichlet-Neumann parity sectors of the equilateral $60<sup>\circ$-$120<sup>\circ$ rhombus. After reflection across the short diagonal, these sectors correspond to the DDN and NND problems on an equilateral half-triangle $T$, where $D$ and $N$ denote Dirichlet and Neumann boundary conditions, respectively. The mixed reflection signs do not close to a scalar character; their exact scalar resolution is given instead by a genus-two arithmetic translation surface with two $4π$ cone points. We introduce an admissible class of nonempty open observation sets $W\subset T$ whose lift contains a fixed punctured neighborhood of the cone set and intersects every regular closed orbit in each completely periodic direction; no geometric hitting condition is imposed in uniquely ergodic directions. For every such $W$, there exists a constant $c_W&gt;0$, depending $W$ but independent of the eigenvalue $λ$, such that every DDN or NND eigenfunction $u$ with $-Δu=λu$ satisfies [ |u|{L^2(W)}^2 \ge c_W |u|{L^2(T)}^2. ] In particular, complements of sufficiently small generic interior holes form an explicit family of admissible observations. The proof combines exact unfolding, semiclassical propagation away from the observed cone neighborhoods, Veech dichotomy, and invariant-measure disintegration in periodic cylinders. We also obtain a reflection-overlap observability theorem for the full rhombus and a strict fixed-frequency cross-parity angle.