High orders of Weyl series: resurgence for odd balls

Abstract

The semiclassical Weyl series for the d-dimension, unit-radius sphere quantum billiard is studied. A conjecture of Berry and Howls (1994 447 527-55) on the late terms of such series for two-dimensional billiards is seen to survive for general (integer) dimension. The conjecture postulates a leading-order, factorial-on-power approximation for the late terms of the Weyl series in terms of the length of a periodic orbit of the classical system. The expansions manifest a difference between odd and even dimensions. The dominating orbit is the diametral, length-4 path in the even-dimension spheres, echoing the known result for the circle billiard. However, when d is odd, it is the next-longest orbit. This surprise can be traced to an `accidental' symmetry in a postulated hyperasymptotic remainder term. Higher-order asymptotic correction terms are found confirming the resurgent link of the Weyl series to the low orders of the oscillatory periodic orbit corrections. From the structure of the latter, it is possible to make further conjectures on the late terms of the periodic orbit corrections themselves. A factorial-on-power behaviour is also found, but now involving the differences between p-bounce orbits and associated whispering-gallery modes.