Casimir energy for a three-piece relativistic string

The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of three pieces I, II, III of equal length, endowed with different tensions and mass densities, but adjusted in such a way that the velocity of sound always equals the velocity of light. In this sense the string forms a relativistic mechanical system. In the present paper the string is subjected to the following analysis: the dispersion function is derived and the zero-point energy is regularized using (i) a contour integration technique, being most convenient for the generalization of the theory to the case of finite temperatures, and (ii) the Hurwitz $\zeta$-function technique, being usually the most compact method when the purpose is to calculate the Casimir energy numerically at $T=0\mathrm{}$. The energy, being always nonpositive, is shown graphically in some cases as functions of the tension ratios I/II and II/III. The generalization to finite temperature theory is also given.