Dm. Ming et al., Exact solution of the specific-heat-phonon spectrum inversion from the Mobius inverse formula, PHYS REV E, 62(3), 2000, pp. R3019-R3022
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 spec
trum with the correct cut-off factor and frequency dependence from the Mobi
us formula. The main difficulty arises from the fact that the Mobius functi
on mu(n) is not completely known for large n in practice. In this paper, so
me exact solutions of SPI are obtained by using the Mobius inversion formul
a, most importantly the Debye spectrum as a special case, and the problem o
f the unknown Mobius function mu(n) for large n is avoided. It is shown tha
t the Mobius inversion formula can be useful for exact solutions to spectra
l inversion problems.