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Eigenvalues of the Laplace–Beltrami Operator on the Surface of a Torus 231
Fig. 4. Eigenfunctions corresponding to eigenvalues λ 4 to λ 15 (in order left to right,
top to bottom).
4
Eigenvalues as Function of R/ρ
3.5
3
2.5
λ j
2
1.5
1
0.5
0
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Ratio R/ρ
Fig. 5. Bifurcation diagram of 14 leading nontrivial eigenvalues as functions of the
ratio R/ρ. Solid curves are double eigenvalues and dashed curves are singletons.
We note that eigenfunctions associated with single eigenvalues are sheets
that only change sign in the θ direction. Eigenfunctions corresponding to
double eigenvalues change sign in both the θ and φ directions. We studied the
eigenvalue trajectories plotted as functions of the aspect ratio R/ρ and noted
that crossings of these curves provided instances of quadruple eigenvalues
and also of triple eigenvalues. Results of this study are shown graphically in
Figure 5.