Casimir Effect for the Piecewise Uniform String

The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. In its simplest version the string consists of two parts I and II having in general different tension and mass density, but is always obeying the condition that the velocity of sound is equal to the velocity of light. The model, first introduced by Brevik and Nielsen in 1990, possesses attractive formal properties implying that it becomes easily regularizable by several methods, the most powerful one being the contour integration method. We also consider the case where the string is divided into 2N pieces, of alternating type-I and type-II material. The free energy at finite temperature, as well as the Hagedorn temperature, are found. Finally, we make some remarks on the relationship between this kind of theory and the theory of quantum star graphs, recently considered by Fulling et al.