Let $(M,g)$ be a compact smooth Riemannian manifold without boundary, and let $\Delta_g \geq 0$ denote the nonnegative Laplace-Beltrami operator on complex-valued functions. Let

\begin{align*} P := \sqrt{\Delta_g} \end{align*}

be defined by the spectral theorem. If

\begin{align*} 0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 \leq \cdots \end{align*}

is the spectrum of $\Delta_g$, repeated with multiplicity, define the half-wave trace distribution

\begin{align*} \Theta \in \mathcal{D}'(\mathbb{R}) \end{align*}

by

\begin{align*} \Theta(t) := \operatorname{Tr}(e^{-itP}) = \sum_{j=0}^{\infty} e^{-it\sqrt{\lambda_j}} \end{align*}

in the sense of distributions.

Let

\begin{align*} G^t:S^*M \to S^*M \end{align*}

denote the geodesic flow on the unit cosphere bundle. If $t_0 \in \mathbb{R}\setminus\{0\}$ and $\Theta$ is not smooth in any open neighbourhood of $t_0$, then there exists $\rho \in S^*M$ such that

\begin{align*} G^{t_0}(\rho)=\rho. \end{align*}

Equivalently, $|t_0|$ is the length of a closed geodesic on $M$.

Conversely, fix $t_0 \in \mathbb{R}\setminus\{0\}$ and define

\begin{align*} F_{t_0}:=\{\rho \in S^*M : G^{t_0}(\rho)=\rho\}. \end{align*}

Assume that $F_{t_0}$ is clean, meaning that $F_{t_0}$ is a smooth submanifold of $S^*M$ and, for every $\rho \in F_{t_0}$,

\begin{align*} T_\rho F_{t_0}=\ker(dG^{t_0}_\rho-I). \end{align*}

Then each connected component $C$ of $F_{t_0}$ determines, after microlocal localization near $C$, a Duistermaat-Guillemin local contribution to $\Theta$ near $t=t_0$. This contribution is a classical conormal oscillatory distribution at $t_0$, with principal coefficient equal to the standard clean wave-trace density of Duistermaat and Guillemin, including the Maslov factor and the linearized Poincare-map data. If this principal clean wave-trace coefficient is nonzero for $C$, then the localized contribution from $C$ is genuinely singular at $t=t_0$. The singular contributions from different connected components may cancel in the unlocalized total trace.