A unified solution of the specific-heat–phonon spectrum inversionproblem

In the specific-heat–phonon spectrum inversion problem (SPI), Chen's solution with modified Möbius inversion formula (N. X. Chen Phys. Rev. Lett. 64 (1990) 1193) was novel and of great interest. Meanwhile, Dai's exact solution formula with a parameter s for canceling the divergence has succeeded in obtaining a series of exact solutions and was employed to obtain a phonon spectrum from real specific data of YBCO. In this paper we will show that, by using an integral representation of inverse Laplace transformations and some properties of the Riemann zeta-function, Chen's solution can be derived from Dai's formula, without necessarily using the Möbius inversion formula. Furthermore, the unique existence theorem and convergence of the series of Chen's solution were also obtained. It is also shown that Dai's parameter s and the asymptotic behavior control condition are of crucial importance in the derivation.