= Solution
For the <quantum rotor>, the transition kernel is the <path integral>
$$
K(\theta_f,\theta_i;T)=\int_{\theta(0)=\theta_i}^{\theta(T)=\theta_f\ ({\rm mod}\ 2\pi)}\mathcal D\theta\,e^{iS[\theta]/\hbar},\qquad S[\theta]=\frac\Lambda2\int_0^T\dot\theta^2\,dt.
$$
The integral includes all paths on the circle, not just paths whose chosen real lifts have the same endpoint difference. Operator evolution and the stated energy-state normalization give
$$
\boxed{K(\theta_f,\theta_i;T)=\langle\theta_f|e^{-iHT/\hbar}|\theta_i\rangle=\frac1{2\pi}\sum_{n\in\mathbb Z}e^{in(\theta_f-\theta_i)}e^{-i\hbar n^2T/(2\Lambda)}.}
$$
The factor $1/(2\pi)$ comes from the completeness relation; it makes the zero-time limit the periodic <Dirac delta distribution>. For real time the oscillatory sum is understood through $T\to T-i0$, or as a distributional limit.
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