= Solution
With a nonconstant <potential energy>, momentum no longer diagonalizes the <Hamiltonian operator>. The operator derivation must use the energy eigenfunctions of the full Hamiltonian, a <Dyson series>, or a time-sliced <Trotter product formula>. In the classical derivation, the straight paths are replaced by every solution of the nonlinear <Euler-Lagrange equation> with the specified endpoints. The <semiclassical propagator> becomes a sum
$$
K(q_f,T;q_i,0)\simeq
\sum_{q_{\rm cl}}
\left(\frac{1}{2\pi i\hbar}
\left|-\frac{\partial^2S_{\rm cl}}{\partial q_f\partial q_i}\right|\right)^{1/2}
e^{iS_{\rm cl}/\hbar-i\pi\mu_{\rm cl}/2},
$$
where the prefactor is the <Van Vleck determinant> and $\mu_{\rm cl}$ is a <Maslov index>. Unlike a quadratic theory, the classical-path sum is generally only an asymptotic approximation: the exact <path integral> includes fluctuations of every order. On the circle, the sum must still include all <winding number> sectors.
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