New aspects of the Casimir energy theory for a piecewise uniform string

The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of two parts I and II each having in general different tension and mass density but adjusted in such a way that the velocity of sound always equals the velocity of light. This model was introduced by Brevik and Nielsen, and the present paper contains new developments of the theory, in particular, a very simple regularization of the energy density. Using the technique introduced by van Kampen, Nijboer, and Schram, the Casimir energy is written as a contour integral, from which the energy can be readily calculated, for arbitrary length $s=\frac{L_{\mathrm{II}}}{L_I}$ and tension $x=\frac{T_I}{T_{\mathrm{II}}}$ ratios. Also, the finite temperature version of the theory is constructed.