Singularities of the Wave Trace near Cluster Points of the Length Spectrum

The significance of 2π is that it is a cluster point of the length spectrum from the left (t < 2π), as described in more detail below. In brief, the length spectrum is the closure of the set of lengths of periodic geodesics. The number 2π represents the length of the (generalized) geodesic that follows the circumference of the circle. The number 8 is the geodesic that follows a diameter of the circle four times. It is well known that there are no geodesics of lengths in between 2π and 8 and that h(t) is smooth in 2π < t < 8. The content of the theorem is that h is smooth from the right up to the endpoint 2π. The same proof applies to every cluster point 2π` of the length spectrum of geodesic flow on the disk. We recall now the relationship between h(t) and the wave equation. Consider the initial value problem for the wave equation, (∂t −∆)u(t, x) = 0, x ∈ D, t > 0 u(t, x) = 0, x ∈ ∂D, t > 0 u(0, x) = f(x), x ∈ D ∂tu(0, x) = g(x), x ∈ D The solution is