# Can you calculate the static lattice/potential energy given a phonon dispersion curve?

Let's say I determine a material's dispersion relation from experiments. Would it then be possible to use the information contained within the dispersion relation to calculate the material's static lattice energy?

No, it is impossible. The reason is very simple if one recalls the meaning of phonon dispersion curves. They give the set of possible frequencies, $$\omega_s({\bf k})$$, for the eigenmodes of the $$s-$$th dispersion branch.
The energy density of a harmonic crystal, at a temperature $$T$$ is: $$u = u_{static} + \frac{1}{V}\sum_{{\bf k},s} \frac{1}{2} \hbar \omega_s({\bf k}) + \frac{1}{V}\sum_{{\bf k},s} \frac{ \hbar \omega_s({\bf k})}{e^{ \frac{ \hbar \omega_s({\bf k})}{k_B T}}-1},$$ where the second term is independent on $$T$$ and corresponds to the so-called zero-point contribution. $$u_{static}$$ is the energy of a (static) perfect lattice.
It is clear that the whole information on phonons is contained in the second and third terms. The first term is the (static) potential energy of a set of particles at the equilibrium positions in the lattice. Even though both $$u_{static}$$ and $$\omega_s({\bf k})$$ are obtained from the same potential energy as a function of the atomic positions, they contain different information and one cannot obtain one of them from knowledge of the other.