Quantum scars of classical closed orbits in phase space

Abstract

The way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x = (q, p) through the spectral Wigner function W(x; E, $\epsilon$). This is a sum over Wigner functions of eigenstates within a range $\epsilon$ of energy E. In the classical limit, W is concentrated on the energy surface and smoothly distributed over it. Closed orbits provide oscillatory corrections (scars) for which explicit semiclassical formulae are calculated. Each scar is a fringe pattern decorating the orbit. As x moves off the energy surface the fringes form an Airy pattern with spacing of order h$^\frac{2}{3}$. As x moves off the closed orbit the fringes form a complex gaussian with spacing h$^\frac{1}{2}$.