Casimir energy of smooth compact surfaces

We discuss the formalism of Balian and Duplantier [Balian and Duplantier, Ann. Phys. (NY) 104, 300 (1977); Balian and Duplantier, Ann. Phys. (NY) 112, 165 (1978)] for the calculation of the Casimir energy for an arbitrary smooth compact surface and use it to give some examples: a finite cylinder with hemispherical caps, a torus, an ellipsoid of revolution, a cube with rounded corners and edges, and a drum made of disks and part of a torus. We propose a model function that approximately captures the shape dependence of the Casimir energy.

Figure 1

Casimir term for the torus, as a function of its major radius $R$ (the minor radius is unity).

Figure 2

Casimir term for the drum of radius $R$ and height 2. For $R$ less than 3, the curved region is dominating the energy, which increases because the area of the curved region is increasing. For large $b$ the attraction between the parallel faces is more important, with the result that the Casimir term decreases and goes negative. For $R=10$ the Casimir term is approximately that of a pair of parallel disks with area $\pi R^2$, which would give ${\mathcal{C}}_{\mathrm{approx}}=-{\pi }^3{R}^2/5760=-0.54$.

Figure 3

Casimir term for the ellipsoid with semiaxes $b,1,1$. For large $b$ the Casimir term per unit length is comparable to that for an infinite cylinder. However, for small $b$ (the limit of extremely oblate ellipsoids), the energy is diverging positive, rather than going negative: Apparently the curvature of the rim of the ellipsoidal disk is a more important effect than the attraction between the two nearly parallel faces.

Figure 4

Casimir term for the soft-edged cube, as a function of the exponent $m$.