Gutzwiller trace formula I am reading A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition, Commmun. Math. Phys, 202, 463-480 (1990).
Gutzwiller trace formula says

where

and $g$ is a $C^\infty$ function.
I want to understand why this theorem is interesting. The first term in (14) come from Weyl's law, and the third term is supposed to be the interesting term that relates closed orbits with eigenvalues. However, the third term is much smaller than the first and second manifold has dimension greater than 3. In other words, it seems that the interesting term is of little importance.
I was wondering what is the correct way to interpret the formula.
 A: First of all, if you are just looking for intuition, you can easily find more accessible accounts than this paper (such as the book by Gutzwiller, or even Wikipedia). 
Also, you can look up the 'Selberg trace formula', which is conceptually very similar but an exact result, in contrast to Gutzwiller's which is asymptotic.
Anyway, the third term is smaller than the first but, as Beenakker noted, it is most important because it is specific for chaotic systems, in distinction to integrable ones which satisfy the Berry-Tabor formula.
The theorem is interesting because it relates periodic orbits, which are a dynamical concept (or geometric in the Selberg context) to eigenvalues of an operator (Laplacian in the Selberg context, Hamiltonian of quantum system in Gutzwiller). It is therefore a beautiful bridge between apparently distinct areas of mathematics. (In some precise sense, the situation is similar to the connection between prime numbers and zeros of the Riemann zeta function).
To the physics community, it is interesting because it sheds light on the interface between classical mechanics and quantum mechanics in the case of chaotic systems, a problem which was first raised by Albert Einstein.
