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 where α is a separation constant and q is related to the eigenvalue λ by 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 . 16 q = λ h 2 4 . (C8) d $^{2}$G d τ 2 − ( α − 2 q cosh 2 τ ) G = 0 . (C7b) d $^{2}$F d σ 2 + ( α − 2 q cos 2 σ ) F = 0 , (C7a) Δ = 1 h 2 ( sinh 2 τ + sin 2 σ ) ( ∂ 2 ∂τ 2 + ∂ 2 ∂σ 2 ) . (C6) A = 1 2 ∫ τ 0 0 d τ ∫ 2 π 0 d σ ( ∂X ∂τ · ∂X ∂τ + ∂X ∂σ · ∂X ∂σ ) (C5) d s 2 = h 2 ( sinh 2 τ + sin 2 σ ) ( d τ 2 + d σ $^{2}$) , (C4) h = √$_{a}$ 2 − b $^{2}$. (C3) x$_{1}$ + i x$_{2}$ = h cosh ( τ + i σ ) . (C2) x$_{1}$ = h cosh τ cos σ , x$_{2}$ = h sinh τ sin σ (C1)