Skip to main content
SpringerLink
Log in

Local Scaling Asymptotics for the Gutzwiller Trace Formula in Berezin–Toeplitz Quantization

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Let f be a smooth function on a quantizable compact Kähler manifold, with the special property that the associated Hamiltonian flow acts by isometries. We consider a naturally associated Berezin–Toeplitz operator, which generates a unitary action on the Hardy space of the quantization. In this setting, we produce local scaling asymptotics in the semiclassical regime for a Berezin–Toeplitz version of the Gutzwiller trace formula, in the spirit of the near-diagonal scaling asymptotics of Szegö and Toeplitz kernels. More precisely, we consider an analogue of the ‘Gutzwiller–Toeplitz kernel’  previously introduced in this setting by Borthwick, Paul, and Uribe, and study how it asymptotically concentrates along the appropriate classical loci defined by the dynamics, with an explicit description of the exponential decay along normal directions. These local scaling asymptotics probe into the concentration behavior of the eigenfunctions of the quantized Hamiltonian flow. When globally integrated, they yield the analogue of the Gutzwiller trace formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We shall generally use the same notation for an operator and its Schwartz kernel.

  2. These should not be confused with the ‘vertical’  and ‘horizontal’  distributions defined above on M along \(M_E\).

References

  1. Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142, 351–395 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of zeros on symplectic manifolds. Random Matrices Appl. 40, 31–69 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler manifolds and gl(N), \(N\rightarrow \infty \) limits. Commun. Math. Phys. 165(2), 281–296 (1994)

    Article  MATH  Google Scholar 

  5. Borthwick, D., Paul, T., Uribe, A.: Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122(2), 359–402 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borthwick, D., Paul, T., Uribe, A.: Semiclassical spectral estimates for Toeplitz operators. Ann. Inst. Fourier (Grenoble) 48(4), 1189–1229 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boutet de Monvel, L., Guillemin, V.: The spectral theory of Toeplitz operators. Ann. Math. Stud. 99, 3–160 (1981)

    MathSciNet  MATH  Google Scholar 

  8. Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. Astérisque 34–35, 123–164 (1976)

    MATH  Google Scholar 

  9. Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds. I. Geometric interpretation of Berezin’s quantization. J. Geom. Phys. 7(1), 45–62 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Charles, L.: Berezin–Toeplitz operators, a semi-classical approach. Commun. Math. Phys. 239(1–2), 1–28 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Charles, L.: Quantization of compact symplectic manifolds. http://arxiv.org/abs/1409.8507

  12. Christ, M.: Slow off-diagonal decay for Szegö kernels associated to smooth Hermitian line bundles. Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001). Contemporary Mathematics, vol. 320, pp. 77–89. American Mathematical Society, Providence, RI (2003)

  13. Duistermaat, J.J.: Fourier integral operators. Reprint of the 1996 edition based on the original lecture notes published in 1973. Modern Birkhäuser Classics. Birkhäuser/Springer, New York (2011)

  14. Guillemin, V.: Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35(1), 85–89 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Second Edition, Springer Study edn. Springer, Berlin (1990)

    MATH  Google Scholar 

  16. Karabegov, A., Schlichenmaier, M.: Identification of Berezin–Toeplitz deformation quantization. J. Reine Angew. Math. 540, 49–76 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Ma, X., Marinescu, G.: Holomorphic Morse Inequalities and Bergman Kernels. Progress in Mathematics 254. Birkhauser Verlag, Basel (2007)

    MATH  Google Scholar 

  18. Ma, X., Marinescu, G.: Generalized Bergman kernels on symplectic manifolds. Adv. Math. 217(4), 1756–1815 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ma, X., Zhang, W.: Bergman kernels and symplectic reduction. Astérisque No. 318 (2008)

  20. McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford Mathematical Monographs, 2nd edn. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  21. Melin, A., Sjöstrand, J.: Fourier integral operators with complex-valued phase functions. Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974). Lecture Notes in Mathematics, vol. 459, pp. 120–223. Springer, Berlin (1975)

  22. Paoletti, R.: Szegö kernels, Toeplitz operators, and equivariant fixed point formulae. J. Anal. Math. 106, 209236 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Paoletti, R.: Local asymptotics for slowly shrinking spectral bands of a Berezin–Toeplitz operator. Int. Math. Res. Not. 5, 1165–1204 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Paoletti, R.: Asymptotics of Szegö kernels under Hamiltonian torus actions. Isr. J. Math. 191(1), 363–403 (2011). doi:10.1007/s11856-011-0212-4

    Article  MATH  Google Scholar 

  25. Paoletti, R.: Local trace formulae and scaling asymptotics for general quantized Hamiltonian flows. J. Math. Phys. 53, 023501 (2012). doi:10.1063/1.3679660

    Article  MathSciNet  MATH  Google Scholar 

  26. Paoletti, R.: Scaling asymptotics for quantized Hamiltonian flows. Int. J. Math. 23(10), 1250102 (2012). 25 pp

    Article  MathSciNet  MATH  Google Scholar 

  27. Schlichenmaier, M.: Berezin–Toeplitz quantization for compact Kähler manifolds. A review of results. Adv. Math. Phys., Art. ID 927280 (2010)

  28. Shiffman, B., Zelditch, S.: Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544, 181–222 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Zelditch, S.: Index and dynamics of quantized contact transformations. Ann. Inst. Fourier (Grenoble) 47(1), 305–363 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

I am indebted to the referee for various precious comments, remarks, and corrections, and for proposing some valuable improvements to the presentation.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi 55, 20125, Milan, Italy

    Roberto Paoletti

Authors
  1. Roberto Paoletti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Paoletti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paoletti, R. Local Scaling Asymptotics for the Gutzwiller Trace Formula in Berezin–Toeplitz Quantization. J Geom Anal 28, 1548–1596 (2018). https://doi.org/10.1007/s12220-017-9878-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-017-9878-0

Keywords

Mathematics Subject Classification

Search

Navigation