Periodic orbits and scaled energy spectroscopy

According to Gutzwiller s result (4.1.72) the density of states can be computed as a generalized Fourier sum which contains only classical periodic orbit information. Therefore, using a suitable projection technique it should be possible to invert the transform and extract classical periodic orbit information from the level density. The inversion can be achieved using scaled energy spectroscopy, a technique first introduced by 

Wintgen (1987). This section consists of three parts. In part (a) we derive the trace formula for the one-dimensional heUum atom, a system with an odd-even symmetry. In part (b) we use the classical scaUng properties of the one-dimensional helium atom to apply the scaled energy technique. In part (c) we generalize the technique to apply to autonomous systems without scaling symmetries. 

We use (10.4.25) to obtain the level densities of positive and negative parity states, p+ E) and p- E), respectively, according to 

Evaluating p in stationary phase approximation we get something new. The stationary phase condition is given by 

This technique uses scaling properties of the classical Hamiltonian in order to extract classical periodic orbit information from the fluctuating part of the level density. We illustrate the technique with the help of the positive parity states computed in Section 10.4.1. In order to compute we need the average level density. We solve the empirical formula (10.4.9) in the form 

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