Geometry & Topology

Central limit theorem for spectral partial Bergman kernels

Steve Zelditch and Peng Zhou

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Partial Bergman kernels Πk,E are kernels of orthogonal projections onto subspaces SkH0(M,Lk) of holomorphic sections of the k th power of an ample line bundle over a Kähler manifold (M,ω). The subspaces of this article are spectral subspaces {ĤkE} of the Toeplitz quantization Ĥk of a smooth Hamiltonian H:M. It is shown that the relative partial density of states satisfies Πk,E(z)Πk(z)1A where A={H<E}. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface A; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1 and 0 of 1A. Such “Erf” asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and the central limit theorem.

Article information

Source
Geom. Topol., Volume 23, Number 4 (2019), 1961-2004.

Dates
Received: 30 August 2017
Revised: 17 April 2018
Accepted: 30 September 2018
First available in Project Euclid: 16 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.gt/1563242523

Digital Object Identifier
doi:10.2140/gt.2019.23.1961

Mathematical Reviews number (MathSciNet)
MR3981005

Zentralblatt MATH identifier
07094911

Subjects
Primary: 32A60: Zero sets of holomorphic functions 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] 81Q50: Quantum chaos [See also 37Dxx]

Keywords
Toeplitz operator partial Bergman kernel interface asymptotics

Citation

Zelditch, Steve; Zhou, Peng. Central limit theorem for spectral partial Bergman kernels. Geom. Topol. 23 (2019), no. 4, 1961--2004. doi:10.2140/gt.2019.23.1961. https://projecteuclid.org/euclid.gt/1563242523


Export citation

References

  • R J Berman, Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math. 131 (2009) 1485–1524
  • P Bleher, B Shiffman, S Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000) 351–395
  • T Can, P J Forrester, G Téllez, P Wiegmann, Singular behavior at the edge of Laughlin states, Phys. Rev. B 89 (2014) art. id. 235137, 7 pages
  • L Charles, Berezin–Toeplitz operators, a semi-classical approach, Comm. Math. Phys. 239 (2003) 1–28
  • I Daubechies, Coherent states and projective representation of the linear canonical transformations, J. Math. Phys. 21 (1980) 1377–1389
  • H Delin, Pointwise estimates for the weighted Bergman projection kernel in $\mathbb{C}^n$, using a weighted $L^2$ estimate for the $\bar\partial$ equation, Ann. Inst. Fourier (Grenoble) 48 (1998) 967–997
  • M Dimassi, J Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge Univ. Press (1999)
  • M R Douglas, S Klevtsov, Bergman kernel from path integral, Comm. Math. Phys. 293 (2010) 205–230
  • G B Folland, Harmonic analysis in phase space, Annals of Mathematics Studies 122, Princeton Univ. Press (1989)
  • L Hörmander, The analysis of linear partial differential operators, IV: Fourier integral operators, Grundl. Math. Wissen. 275, Springer (1985)
  • N Lindholm, Sampling in weighted $L^p$ spaces of entire functions in ${\mathbb C}^n$ and estimates of the Bergman kernel, J. Funct. Anal. 182 (2001) 390–426
  • Z Lu, B Shiffman, Asymptotic expansion of the off-diagonal Bergman kernel on compact Kähler manifolds, J. Geom. Anal. 25 (2015) 761–782
  • X Ma, G Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics 254, Birkhäuser, Basel (2007)
  • A Melin, J Sjöstrand, Fourier integral operators with complex-valued phase functions, from “Fourier integral operators and partial differential equations” (J Chazarain, editor), Lecture Notes in Math. 459, Springer (1975) 120–223
  • L Boutet de Monvel, V Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies 99, Princeton Univ. Press (1981)
  • L Boutet de Monvel, J Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegő, from “Journées: Équations aux dérivées partielles de Rennes”, Astérisque 34–35, Soc. Math. France, Paris (1976) 123–164
  • R Paoletti, Scaling asymptotics for quantized Hamiltonian flows, Internat. J. Math. 23 (2012) art. id. 1250102, 25 pages
  • R Paoletti, Local scaling asymptotics in phase space and time in Berezin–Toeplitz quantization, Internat. J. Math. 25 (2014) art. id. 1450060, 40 pages
  • F T Pokorny, M Singer, Toric partial density functions and stability of toric varieties, Math. Ann. 358 (2014) 879–923
  • D Robert, Autour de l'approximation semi-classique, Progress in Mathematics 68, Birkhäuser, Boston (1987)
  • J Ross, M Singer, Asymptotics of partial density functions for divisors, J. Geom. Anal. 27 (2017) 1803–1854
  • Y A Rubinstein, S Zelditch, The Cauchy problem for the homogeneous Monge–Ampère equation, I: Toeplitz quantization, J. Differential Geom. 90 (2012) 303–327
  • B Shiffman, S Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002) 181–222
  • E M Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton Univ. Press (1993)
  • X-G Wen, Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons, Oxford Univ. Press (2004)
  • P Wiegmann, Nonlinear hydrodynamics and fractionally quantized solitons at the fractional quantum Hall edge, Phys. Rev. Lett. 108 (2012) art. id. 206810, 5 pages
  • S Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997) 305–363
  • S Zelditch, P Zhou, Interface asymptotics of partial Bergman kernels on $S^1$–symmetric Kaehler manifolds, preprint (2016) To appear in J. Symplectic Geom.
  • S Zelditch, P Zhou, Central limit theorem for toric \kahler manifolds, preprint (2018)
  • S Zelditch, P Zhou, Interface asymptotics of partial Bergman kernels around a critical level, preprint (2018)
  • S Zelditch, P Zhou, Pointwise Weyl law for partial bergman kernels, from “Algebraic and analytic microlocal analysis” (M Hitrik, D Tamarkin, B Tsygan, S Zelditch, editors), Springer Proc. Math. Stat. 269, Springer (2018) 589–634