A unified solution of the specific-heat–phonon spectrum inversion problem

¹ Group of Quantum Statistics and Methods of Theoretical Physics Surface Physics Laboratory, Department of Physics, Fudan University Shanghai 200433, PRC
² Department of Chemistry, New York University - New York, NY 10003, USA
³ Department of Physics, Brigham Young University - Provo, Utah 84602, USA
⁴ Department of Physics, Shanghai University - Shanghai 200436, PRC

Received: 2 October 2002
Accepted: 7 January 2003

Abstract

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.