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High orders of Weyl series: resurgence for odd balls

High orders of Weyl series: resurgence for odd balls
High orders of Weyl series: resurgence for odd balls
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.
0305-4470
1487-1506
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Trasler, S.A.
37dd480a-1862-44fc-8642-ec3e69a317d0
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Trasler, S.A.
37dd480a-1862-44fc-8642-ec3e69a317d0

Howls, C.J. and Trasler, S.A. (1999) High orders of Weyl series: resurgence for odd balls. Journal of Physics A: Mathematical and General, 32 (8), 1487-1506. (doi:10.1088/0305-4470/32/8/015).

Record type: Article

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.

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Published date: 1999
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 29206
URI: http://eprints.soton.ac.uk/id/eprint/29206
ISSN: 0305-4470
PURE UUID: 40e3397c-1515-4591-a827-268687134907
ORCID for C.J. Howls: ORCID iD orcid.org/0000-0001-7989-7807

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Date deposited: 21 Jul 2006
Last modified: 20 Jul 2019 01:08

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Contributors

Author: C.J. Howls ORCID iD
Author: S.A. Trasler

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