Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a Berry phase -like term stemming from spin or other internal degrees of freedom. The index k distinguishes the primitive periodic orbits : the shortest period orbits of a given set of initial conditions. T k is the period of the primitive periodic orbit and S k is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action n S k and a period which is an integral multiple n of the primitive period. Hence, every repetition of a periodic orbit is another periodic orbit. These repetitions are separately classified by the intermediate sum over the indices n . α α n k is the orbit's Maslov index. The amplitude factor, 1 / sinh ⁡ ⁡ ( χ χ n k / 2 ) , represents the square root of the density of neighboring orbits. Neighboring trajectories of an unstable periodic orbit diverge exponentially in time from the periodic orbit. The quantity χ χ n k characterizes the instability of the orbit. A stable orbit moves on a torus in phase space, and neighboring trajectories wind around it. For stable orbits, sinh ⁡ ⁡ ( χ χ n k / 2 ) becomes sin ⁡ ⁡ ( χ χ n k / 2 ) , where χ χ n k is the winding number of the periodic orbit. χ χ n k = 2 π π m , where m is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because sin ⁡ ⁡ ( χ χ n k / 2 ) = 0 at a classical bifurcation. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo- absorption spectrum.