High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon

Abstract

A formalism is developed for calculating high coefficients c r of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂ B The c r are used to test the following conjectures. ( a ) The sequence 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 Möbius strip which is the space of chords of B . ( c ) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂ 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 the complex geodesics are accessible from the Weyl series.