HIGH ORDERS OF THE WEYL EXPANSION FOR QUANTUM BILLIARDS - RESURGENCE OF PERIODIC-ORBITS, AND THE STOKES PHENOMENON

A formalism is developed for calculating high coefficients c(r) of the Weyl (high energy) expansion for the trace of the resolvent of the La place operator in a domain B with smooth boundary partial derivative B . The c(r) are used to test the following conjectures. (a) The sequenc e of c(r) diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. (b) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Mobius strip w hich is the space of chords of B. (c) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its l east term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, tha t is across the negative energy axis (Stokes line). These conjectures are motivated: by recent results in asymptotics. They survive tests fo r the circle billiard: and fora family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes c omplex. For some systems which are not smooth billiards (e.g. a partic le on a ring, or in a billiard where partial derivative B is a polygon ), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only t he complex geodesics are accessible from the Weyl series.