Geometrical properties of Maslov indices in the semiclassical trace formula for the density of states.

Abstract

We show that in the trace formula of Gutzwiller [J. Math. Phys. 8, 1979 (1967); 10, 1004 (1969); 11, 1791 (1970); 12, 343 (1971)] and Balian and Bloch [Ann. Phys. (N.Y.) 60, 401 (1970); 63, 592 (1971); 69, 76 (1972); 85, 514 (1974)], applied to systems of two degrees of freedom, the Maslov index arising in the contribution from each periodic orbit is equal to twice the number of times the stable and unstable manifolds wind around the periodic orbit. As a consequence, we find that the Maslov index of a periodic orbit is equal to the Maslov index defined by either its stable or its unstable manifold. In this way it becomes apparent that the Maslov index occurring in the trace formula is an intrinsic property of the periodic orbit, being independent of the coordinates used to find it. In contrast to the case of torus quantization applied to integrable systems, where only even Maslov indices appear, we find that odd Maslov indices can arise in the trace formula of chaotic systems. These odd Maslov indices arise in the contributions of periodic orbits that are hyperbolic with reflection.