CMC tori of revolution in S³: additional data on the spectra of their Jacobi operators

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We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit 3-sphere S³, confirming that every such torus has Morse index at least five, and showing that other known lower bounds for this Morse index are close to optimal.