content_image/354678.jpg content_image/354682.jpg content_image/354681.jpg content_image/354679.jpg content_image/354677.jpg content_image/354676.jpg content_image/354675.jpg content_image/354680.jpg content_image/354674.jpg
## Appendix C: Calculating the L¨ uscher term for an ellipse 

1. Elliptical coordinates and Mathieu functions 

Let us consider an ellipse as the boundary contour. To calculate the determinant of the Laplace operator with this boundary conditions, we parametrize the surface by elliptical (Lam´e) coordinates 

or 

Here σ ∈ [0 , 2 π ) and τ ∈ [0 , τ$_{0}$ ]. The boundary is approached for τ = τ$_{0}$ = arctanh b/a and foci are at ( ± h, 0) with 

The elliptic coordinates are conformal: 

so the area in the conformal gauge reads 

The Laplacian reads in elliptic coordinates as 

The Laplace equation separates in elliptic coordinates to the Mathieu equations 

<img src='content_image/354678.jpg'>

<img src='content_image/354682.jpg'>

<img src='content_image/354681.jpg'>

<img src='content_image/354679.jpg'>

<img src='content_image/354677.jpg'>

<img src='content_image/354676.jpg'>

<img src='content_image/354675.jpg'>

<img src='content_image/354680.jpg'>

where α is a separation constant and q is related to the eigenvalue λ by 

<img src='content_image/354674.jpg'>

A complete set of solutions for F and G is given, respectively, by the Mathieu functions ce$_{m}$ ( σ, q ), se$_{m}$ ( σ, q ) and the modified Mathieu functions Ce$_{m}$ ( τ, q ), Se$_{m}$ ( τ, q ) of integral order m . 

