  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 $\mathbb{S}^3$, 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.
