Perturbative zero-point energy for a cylinder of elliptical section

Vacuum fluctuations of quantum fields caused by the presence of boundaries produce changes in the zero-point energy, which give rise to the Casimir effect. Interest in quantum vacuum manifestations, including this phenomenon, has been propelled by new theoretical and experimental advances [1, 2, 3]. Hypothetical prospects of technological applications make even more desirable the knowledge of fundamental aspects of the theory, such as the value of the vacuum energy or its dependence on any of the problem conditions (even the sign of the effect is hard to predict, although for interactions between dielectric bodies some progress has been made in Ref. [4]). Particularly striking is the modification of the zeropoint energy caused by a change in boundary shape, even at an infinitesimal level. As Casimir energies prove to be very sensitive to purely geometrical modifications, the subject deserves further consideration. This question has already been addressed for a boundary departing from spherical, which has implications for QCD flux tube models [5]. On the other hand, cylindrical boundaries of circular section have been object of attention under a variety of settings: the perfectly conducting case [6, 7, 8], dielectric media with or without light-velocity conservation [9, 10, 11, 12, 13, 14, 15, 16, 17], dispersion [18, 19, 20], semitransparent boundary [21], varying Robin conditions [22], finite temperature [24], coaxial surfaces [25], cosmic strings [26], etc. However, deviations from circular section constitute a largely uncharted land. In the present work, we make an incursion into this territory by considering a perfectly conducting and infinitely long cylindrical surface of elliptical section, which slightly deviates from circular shape. In Sec. 2 the solutions to the Maxwell equations in