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Gutzwiller’s semiclassical trace formula and Maslov-type index theory for symplectic paths

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Abstract

Gutzwiller’s famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with a view toward the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley–Zehnder index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley–Zehnder’s index theory in which we will recall all essential ingredients is the Maslov-type index theory for symplectic paths developed by Long and his collaborators. It would shed new light on the computations and understandings of the semiclassical trace formula. The insights in Gutzwiller’s work also seems plausible for the studies of Hamiltonian systems.

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Acknowledgements

I would like to thank Professor Yiming Long for teaching me the fine points in Maslov-type index theory for symplectic paths, and for many encouragements and supports during the past two decades. Thanks also go to the referee and Pengfei Zhang for their careful readings and valuable suggestions.

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  1. Department of Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China

    Shanzhong Sun

  2. Beijing Center for Mathematics and Information, Interdisciplinary Sciences, Beijing, 100048, People’s Republic of China

    Shanzhong Sun

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  1. Shanzhong Sun
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Correspondence to Shanzhong Sun.

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Dedicated to Professor Paul Henry Rabinowitz with admiration.

The study was partially supported by NSFC (Nos. 10731080, 11131004, 11271269), PHR201106118, the Institute of Mathematics and Interdisciplinary Science at CNU.

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Sun, S. Gutzwiller’s semiclassical trace formula and Maslov-type index theory for symplectic paths. J. Fixed Point Theory Appl. 19, 299–343 (2017). https://doi.org/10.1007/s11784-016-0355-3

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