Trace formulas and closed geodesics

Formally, the cosine propagator has trace equal to the sum of cosines of time times eigenfrequency. The series need not converge pointwise, because its terms do not decay with index. Pair it instead with a compactly supported smooth time test function. Repeated integration by parts makes the paired high-frequency contributions decay faster than any fixed power, and Weyl growth then makes their sum well defined. This procedure defines a distribution rather than an ordinary function with a value at every time.

The constant zero modes contribute a smooth constant to that distribution. They affect the complete trace but do not create nonzero-time singularities. A finite frequency cutoff produces a smooth oscillatory approximation, which can exhibit sharp peaks near singular times. Its peak heights depend on the cutoff and cannot be identified with a pointwise value of the limiting distribution. Smoothing in time is part of the measurement protocol, not merely a plotting option.

The wave kernel propagates singularities along geodesics. Taking a trace restricts to matching initial and final spatial points, so closed trajectories become candidates for nonzero singular times. The general Poisson relation places singular support inside zero together with positive and negative lengths of closed geodesics. This inclusion is the safe universal statement. It does not assert that every geometric length necessarily survives as a nonzero singularity after all contributions have been combined.

A more precise trace expansion near a length needs hypotheses on its closed trajectories, such as appropriate clean or nondegenerate fixed-point behavior. Amplitudes depend on linearized return maps, phases, and possible contributions from several trajectories of the same length. Cancellation or degeneracy can obstruct a naive converse to the singular-support inclusion. Accordingly, recovering an entire length spectrum from spectral data requires qualifications; identifying only an allowed set of singular times is a weaker conclusion.

On a circle of length L, the zero frequency is simple and every positive frequency is double. The cosine trace is therefore one plus twice the positive-frequency cosine sum. The distributional Poisson summation identity turns this into an L-weighted comb of delta masses at integer multiples of L. These nonzero times are exactly the lengths of repeated traversals of the circle. The identity includes the zero-time singularity and respects multiplicity automatically.

This exceptionally explicit case should not be mistaken for the general form of every wave-trace singularity. Higher-dimensional manifolds can have different singular orders, families of closed geodesics, and more complicated coefficients. The circle calculation is valuable because every normalization can be checked: frequency spacing, period, multiplicity, and delta-mass weight. It also shows why replacing squared frequencies directly by eigenvalues inside the cosine would predict the wrong propagation times.