Hearing the Edges: Recovering a 3D Rectangular Box from Dirichlet Eigenvalues

We investigate whether the geometric parameters of a three-dimensional domain can be recovered from the Dirichlet spectrum of the Laplacian. As a controlled benchmark, we consider rectangular boxes, about which the eigenvalues are explicitly known and the Weyl coefficients can be computed in closed form. Exploiting the short-time asymptotics of the heat trace, we extract the leading Weyl coefficients from finite spectral data and show how they encode volume, surface area, and the third spectral Weyl term. These coefficients uniquely determine the side lengths of the box via an explicit cubic reconstruction formula. Numerical experiments based on several thousand eigenvalues demonstrate that the method is stable, accurate, and robust with respect to spectral truncation. The box setting thus provides a stringent validation of the proposed inverse spectral methodology and serves as a foundation for its extension to smooth curved domains, such as triaxial ellipsoids, where explicit spectral formulas are no longer available.