GutzwillerTraceFormula

The Gutzwiller trace formula can be written as follows:

$$\rho(E) = \rho_{\text{avg}}(E) + \rho_{\text{osc}}(E)$$

Here, $\rho(E)$ is the density of states at energy $E$, $\rho_{\text{avg}}(E)$ represents the smooth or average part of the density of states, and $\rho_{\text{osc}}(E)$ corresponds to the oscillating part of the density of states, which contains information about the periodic orbits of the classical system.

The oscillating part of the formula, $\rho_{\text{osc}}(E)$, can be further expressed as a sum over all periodic orbits:

$$ \rho_{\text{osc}}(E) = \sum_{p} A_p(E) \cos\left[\frac{S_p(E)}{\hbar} - \pi\nu_p\right] $$

In this expression, the sum is taken over all periodic orbits $p$, $A_p(E)$ is the amplitude of the contribution of the $p$-th periodic orbit, $S_p(E)$ is the action of the $p$-th periodic orbit, $\hbar$ is the reduced Planck constant, and $\nu_p$ is the Maslov index, which is related to the number of conjugate points along the periodic orbit.