Exact solution of the specific-heat-phonon spectrum inversion from the Mobius inverse formula

The application of the Mobius inversion formula to the specific-heat-phonon spectrum inversion problem (SPI) initially appeared promising [N.X. Chen, Phys. Rev. Lett. 64, 1193 (1990); J. Maddox, Nature (London) 344, 377 (1990)]. However, no one has previously been able to obtain the exact Debye spectrum with the correct cut-off factor and frequency dependence from the Mobius formula. The main difficulty arises from the fact that the Mobius function mu(n) is not completely known for large n in practice. In this paper, some exact solutions of SPI are obtained by using the Mobius inversion formula, most importantly the Debye spectrum as a special case, and the problem of the unknown Mobius function mu(n) for large n is avoided. It is shown that the Mobius inversion formula can be useful for exact solutions to spectral inversion problems.