Central limit theorem for spectral partial Bergman kernels

Zelditch, Steve¹ ; Zhou, Peng²

¹ Department of Mathematics, Northwestern University, Evanston, IL, United States
² Department of Mathematics, Northwestern University, Evanston, IL, United States, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

Geometry & topology, Tome 23 (2019) no. 4, pp. 1961-2004.

Résumé

Partial Bergman kernels ${\Pi }_{k,E}$ are kernels of orthogonal projections onto subspaces ${\mathcal{S}}_k\subset H^0\left(M,L^k\right)$ of holomorphic sections of the $k^{\text{ th}}$ power of an ample line bundle over a Kähler manifold $\left(M,\omega \right)$. The subspaces of this article are spectral subspaces $\left\{{Ĥ}_k\le E\right\}$ of the Toeplitz quantization ${Ĥ}_k$ of a smooth Hamiltonian $H:M\to ℝ$. It is shown that the relative partial density of states satisfies ${\Pi }_{k,E}\left(z\right)∕{\Pi }_k\left(z\right)\to 1_{\mathcal{A}}$ where $\mathcal{A}=\left\{H<E\right\}$. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface $\partial \mathcal{A}$; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values $1$ and $0$ of $1_{\mathcal{A}}$. 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.

Détail

DOI : 10.2140/gt.2019.23.1961

Classification : 32A60, 32L10, 81Q50
Keywords: Toeplitz operator, partial Bergman kernel, interface asymptotics

Bibliographie

[1] R J Berman, Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math. 131 (2009) 1485 | DOI

[2] P Bleher, B Shiffman, S Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000) 351 | DOI

[3] T Can, P J Forrester, G Téllez, P Wiegmann, Singular behavior at the edge of Laughlin states, Phys. Rev. B 89 (2014) | DOI

[4] L Charles, Berezin–Toeplitz operators, a semi-classical approach, Comm. Math. Phys. 239 (2003) 1 | DOI

[5] I Daubechies, Coherent states and projective representation of the linear canonical transformations, J. Math. Phys. 21 (1980) 1377 | DOI

[6] H Delin, Pointwise estimates for the weighted Bergman projection kernel in Cn, using a weighted L2 estimate for the ∂ equation, Ann. Inst. Fourier (Grenoble) 48 (1998) 967 | DOI

[7] M Dimassi, J Sjöstrand, Spectral asymptotics in the semi-classical limit, 268, Cambridge Univ. Press (1999) | DOI

[8] M R Douglas, S Klevtsov, Bergman kernel from path integral, Comm. Math. Phys. 293 (2010) 205 | DOI

[9] G B Folland, Harmonic analysis in phase space, 122, Princeton Univ. Press (1989) | DOI

[10] L Hörmander, The analysis of linear partial differential operators, IV : Fourier integral operators, 275, Springer (1985) | DOI

[11] N Lindholm, Sampling in weighted Lp spaces of entire functions in Cn and estimates of the Bergman kernel, J. Funct. Anal. 182 (2001) 390 | DOI

[12] Z Lu, B Shiffman, Asymptotic expansion of the off-diagonal Bergman kernel on compact Kähler manifolds, J. Geom. Anal. 25 (2015) 761 | DOI

[13] X Ma, G Marinescu, Holomorphic Morse inequalities and Bergman kernels, 254, Birkhäuser (2007) | DOI

[14] A Melin, J Sjöstrand, Fourier integral operators with complex-valued phase functions, from: "Fourier integral operators and partial differential equations" (editor J Chazarain), Lecture Notes in Math. 459, Springer (1975) 120 | DOI

[15] L Boutet De Monvel, V Guillemin, The spectral theory of Toeplitz operators, 99, Princeton Univ. Press (1981) | DOI

[16] 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 (1976) 123

[17] R Paoletti, Scaling asymptotics for quantized Hamiltonian flows, Internat. J. Math. 23 (2012) | DOI

[18] R Paoletti, Local scaling asymptotics in phase space and time in Berezin–Toeplitz quantization, Internat. J. Math. 25 (2014) | DOI

[19] F T Pokorny, M Singer, Toric partial density functions and stability of toric varieties, Math. Ann. 358 (2014) 879 | DOI

[20] D Robert, Autour de l’approximation semi-classique, 68, Birkhäuser (1987) | DOI

[21] J Ross, M Singer, Asymptotics of partial density functions for divisors, J. Geom. Anal. 27 (2017) 1803 | DOI

[22] Y A Rubinstein, S Zelditch, The Cauchy problem for the homogeneous Monge–Ampère equation, I : Toeplitz quantization, J. Differential Geom. 90 (2012) 303

[23] B Shiffman, S Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002) 181 | DOI

[24] E M Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, 43, Princeton Univ. Press (1993)

[25] 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) | DOI

[26] P Wiegmann, Nonlinear hydrodynamics and fractionally quantized solitons at the fractional quantum Hall edge, Phys. Rev. Lett. 108 (2012) | DOI

[27] S Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997) 305 | DOI

[28] S Zelditch, P Zhou, Interface asymptotics of partial Bergman kernels on S1–symmetric Kaehler manifolds, preprint (2016)

[29] S Zelditch, P Zhou, Central limit theorem for toric manifolds, preprint (2018)

[30] S Zelditch, P Zhou, Interface asymptotics of partial Bergman kernels around a critical level, preprint (2018)

[31] S Zelditch, P Zhou, Pointwise Weyl law for partial bergman kernels, from: "Algebraic and analytic microlocal analysis" (editors M Hitrik, D Tamarkin, B Tsygan, S Zelditch), Springer Proc. Math. Stat. 269, Springer (2018) 589 | DOI