Hearing the Symmetries of Crystal Lattices from the Integrated Acoustic Spectrum

Let $C$ be a crystal and $\phi$ be a periodic realization of it in $\mathbb{R}^n$, also let $L$ be the lattice group of $\phi(C)$ which preserves the covering space nature of crystal lattices. In this article, firstly, we define the concept of acoustic spectrum of the crystal lattice C and then provide the algebraic formalism of the question of finding the frequencies of the torus $\mathbb{R}^n/L$, when the set of acoustic spectrum is known. An answer for crystals with uniform atomic force constants is given.