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Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula

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Abstract

We use the theory of Berezin–Toeplitz operators of Ma and Marinescu to study the quantum Hamiltonian dynamics associated with classical Hamiltonian flows over closed prequantized symplectic manifolds in the context of geometric quantization of Kostant and Souriau. We express the associated evolution operators via parallel transport in the quantum spaces over the induced path of almost complex structures, and we establish various semi-classical estimates. In particular, we establish a Gutzwiller trace formula for the Kostant–Souriau operator and compute explicitly the leading term. We then describe a potential application to contact topology.

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Acknowledgements

The author wants to thank Pr. Xiaonan Ma for his support and Pr. Leonid Polterovich for helpful discussions. The author also wants to thank the anonymous referees for useful comments and suggestions. This work was supported by the European Research Council Starting grant 757585.

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Correspondence to Louis Ioos.

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Louis Ioos: Partially supported by the European Research Council Starting Grant 757585.

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Ioos, L. Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula. Lett Math Phys (2020). https://doi.org/10.1007/s11005-020-01267-z

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Keywords

  • Geometric quantization
  • Berezin-Toeplitz operators
  • Hamiltonian flows
  • Gutzwiller trace formula
  • Contact topology

Mathematics Subject Classification

  • 53D50
  • 37C27
  • 32A25
  • 57R17
  • 58J20