stable orbits, becomes , where is the winding number of the periodic orbit. , where is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because 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. Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a function of action. 2) There are an infinite number of periodic orbits, and the convergence properties of periodic-orbit theory are unknown. This difficulty is also present when applying periodic-orbit theory to regular systems. 3) Long-period orbits are difficult to compute because most trajectories are unstable and sensitive to roundoff errors and details of the numerical integration. Gutzwiller applied the trace formula to approach the anisotropic Kepler problem (a single particle in a potential with an anisotropic mass tensor) semiclassically. He found agreement with quantum computations for low lying (up to ) states for small anisotropies by using only a small set of easily computed periodic orbits, but the agreement was poor for large anisotropies. The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties surrounding long-period orbits to try to find the density of states (energy levels), one can use standard quantum mechanical perturbation theory to compute eigenvalues (energy levels) and use the Fourier transform to look for the periodic modulations of the spectrum which are the signature of periodic orbits. Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform. Rough sketch on how to arrive at the Gutzwiller trace formula Start with the semiclassical approximation of the time-dependent Green's function (the Van Vleck propagator). Realize that for caustics the description diverges and use the insight by Maslov (approximately Fourier transforming to momentum space (stationary phase approximation with h a small parameter) to avoid such points and afterwards transforming back to position space can cure such a divergence, however gives a phase factor). Transform the Greens function to energy space to get the energy dependent Greens function (again approximate Fourier transform using the stationary