Casimir energy 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, endowed in general with different tensions and mass densities, although adjusted in such a way that the velocity of sound always equals the velocity of light. The dispersion equation is worked out under general conditions and the frequency spectrum is determined in special cases. When the ratio $L_{\mathrm{II}/L_{\mathrm{I}}}$ between the string lengths is an integer, it is in principle possible to determine the frequency spectrum through solving algebraic equations of increasingly high degree. The Casimir energy relative to the uniform string is in general found to be negative, although in the special case $L_{\mathrm{I}=L_{\mathrm{II}}}$ the energy is equal to zero. Delicate points in the regularization procedure are discussed; in particular, it turns out that a straightforward use of the Riemann ζ-function regularization method leads to an incorrect expression for the Casimir energy.