Casimir theory for the piecewise uniform string: Division into 2Npieces

The Casimir energy for the transverse oscillations of a piecewise uniform string in Minkowski space is calculated. The string consists in general of 2N pieces of equal length, of alternating type I and type II material, endowed with different tensions ${\mathit{T}}_{\mathrm{I}}$, ${\mathit{T}}_{\mathrm{II}}$ and mass densities ${\mathrm{\rho }}_{\mathrm{I}}$, ${\mathrm{\rho }}_{\mathrm{II}}$ but adjusted in such a way that the velocity of sound always equals the velocity of light. This string model (with N=1) was introduced by the present authors in 1990. In the present paper, the governing equations are formulated in the case of arbitrary integers N, although detailed calculations are restricted to the case N=2, i.e., a four-piece string. The Casimir energy is regularized using (i) the contour integration technique introduced by van Kampen, Nijboer, and Schram, and (ii) the Hurwitz ζ-function technique introduced by Li, Shi, and Zhang. The energy is found as a function of the tension ratio x=${\mathit{T}}_{\mathrm{I}}$/${\mathit{T}}_{\mathrm{II}}$. The finite temperature version of the theory is also given.