Bohr-Sommerfeld quantization of periodic orbits.

We show, that the canonical invariant part of h̄ corrections to the Gutzwiller trace formula and the Gutzwiller-Voros spectral determinant can be computed by the Bohr-Sommerfeld quantization rules, which usually apply for integrable systems. We argue that the information content of the classical action and stability can be used more effectively than in the usual treatment. We demonstrate the improvement of precision on the example of the three disk scattering system. Typeset using REVTEX 1 Gutzwiller trace formula for chaotic systems is often presented as the counterpart of the Bohr-Sommerfeld (BS) quantization of integrable systems [1]. Consequently, corrections of the trace formula proportional with powers of h̄ are usually associated with quantum corrections [3]. In this letter we would like to show, that a part of h̄ corrections is not connected to deep quantum effects and they can be calculated with some precision from purely semiclassical BS arguments. Then we propose a new trace formula and spectral determinant which is more precise than the usual trace formula although it uses only the linear stability and action as an input data, just like the original trace formula. First we would like to introduce periodic orbits from an unusual point of view. Chaotic and integrable systems on the level of periodic orbits are in fact not as different from each other as we might think. If we start orbits in the neighborhood of a periodic orbit and look at the picture on the Poincaré section we can see a regular pattern. For stable periodic orbits the points form small ellipses around the center and for unstable orbits they form hyperbola. The motion close to a periodic orbits is regular in both cases. This is due to the fact, that we can linearize the Hamiltonian close to a periodic orbit, and linear systems are always integrable. Based on Poincaré’s idea, Arnold and coworkers have shown [4], that the Hamiltonian close to a periodic orbit can be brought to a very practical form. One has to introduce new coordinates: one which is parallel with the orbit (x‖) and others which are orthogonal. In the orthogonal directions we get linear equations. These equations with x‖ dependent rescaling can be transformed into normal coordinates so that we get tiny oscillators, or inverse oscillators, in the new coordinates with constant, frequencies. In the new coordinates, the Hamiltonian is H0(x‖, p‖, xn, pn) = 1 2 p‖ + U(x‖) + d−1