×

Analytic Bergman operators in the semiclassical limit. (English) Zbl 1469.32003

The authors construct an approximate Bergman projection by means of analytic microlocal analysis using what they call the Bargmann-Bergman quantization (Brg quantization). Then they use \(L^2\)-analysis of these operators to show that the exact Bergman projection coincides with the microlocally constructed one up to an exponentially small error in terms of the semiclassical parameter. Then they use the pseudodifferential calculus in analytic classes of symbols developed by the second author [Astérisque 95, 1–166 (1982; Zbl 0524.35007)], to obtain the estimates for the asymptotics of the Bergman kernel showing that the Bergman kernel admits lower and upper bounds of the form \(k^n e^{-kd^2(x,y)/C}.\) They apply this procedure to the Brg quantization on weighted spaces of germs of holomorphic functions, where the weight is an analytic, strictly plurisubharmonic function, which is inspired by the well-known exact formula when the weight is quadratic, see for instance [M. Zworski, Semiclassical analysis. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1252.58001)]. In addition they use this approach also for the Bergman projection for holomorphic line bundles over compact complex manifolds equipped with Hermitian metrics.

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32W25 Pseudodifferential operators in several complex variables
58J40 Pseudodifferential and Fourier integral operators on manifolds
70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, I, Comm. Pure Appl. Math. 14 (1961), no. 3, 187-214. Mathematical Reviews (MathSciNet): MR157250
Zentralblatt MATH: 0107.09102
Digital Object Identifier: doi:10.1002/cpa.3160140303
· Zbl 0107.09102 · doi:10.1002/cpa.3160140303
[2] F. A. Berezin, General concept of quantization, Comm. Math. Phys. 40 (1975), no. 2, 153-174. Mathematical Reviews (MathSciNet): MR411452
Zentralblatt MATH: 1272.53082
Digital Object Identifier: doi:10.1007/BF01609397
Project Euclid: euclid.cmp/1103860463
· Zbl 1272.53082 · doi:10.1007/BF01609397
[3] R. J. Berman, Sharp asymptotics for Toeplitz determinants and convergence towards the Gaussian free field on Riemann surfaces, Int. Math. Res. Not. IMRN 2012, no. 22, 5031-5062. Mathematical Reviews (MathSciNet): MR2997048
Zentralblatt MATH: 1262.60043
Digital Object Identifier: doi:10.1093/imrn/rnr229
· Zbl 1262.60043 · doi:10.1093/imrn/rnr229
[4] R. J. Berman, B. Berndtsson, and J. Sjöstrand, A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Mat. 46 (2008), no. 2, 197-217. · Zbl 1161.32001
[5] B. Berndtsson, “Bergman kernels related to Hermitian line bundles over compact complex manifolds” in Explorations in Complex and Riemannian Geometry, Contemp. Math. 332, Amer. Math. Soc., Providence, 2003, 1-17. Zentralblatt MATH: 1038.32003
· Zbl 1038.32003
[6] B. Berndtsson, “An introduction to things \(\overline{\partial } \)” in Analytic and Algebraic Geometry, IAS/Park City Math. Ser. 17, Amer. Math. Soc., Providence, 2010, 7-76. Zentralblatt MATH: 1227.32039
· Zbl 1227.32039
[7] T. Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéaire positif, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 1, 117-130. · Zbl 0685.32015
[8] L. Boutet de Monvel and V. Guillemin, The Spectral Theory of Toeplitz Operators, Ann. of Math. Stud. 99, Princeton Univ. Press, Princeton, 1981. Zentralblatt MATH: 0469.47021
· Zbl 0469.47021
[9] L. Boutet de Monvel and P. Krée, Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier (Grenoble) 17 (1967), no. 1, 295-323. · Zbl 0195.14403
[10] L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Astérisque 34-35 (1976), 123-164. · Zbl 0344.32010
[11] J. Bros and D. Iagolnitzer, “Tuboïdes et structure analytique des distributions, II: Support essentiel et structure analytique des distributions” in Séminaire Goulaouic-Lions-Schwartz 1974-1975: Équations aux dérivées partielles linéaires et non linéaires, École Polytech., Paris, no. 18, 1975. Zentralblatt MATH: 0333.46029
· Zbl 0333.46029
[12] D. Catlin, “The Bergman kernel and a theorem of Tian” in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math., Birkhäuser Boston, Boston, 1999, 1-23. Zentralblatt MATH: 0941.32002
· Zbl 0941.32002
[13] L. Charles, Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys. 239 (2003), no. 1-2, 1-28. Zentralblatt MATH: 1059.47030
Digital Object Identifier: doi:10.1007/s00220-003-0882-9
· Zbl 1059.47030 · doi:10.1007/s00220-003-0882-9
[14] L. Charles, Quantization of compact symplectic manifolds, J. Geom. Anal. 26 (2016), no. 4, 2664-2710. Zentralblatt MATH: 1357.81127
Digital Object Identifier: doi:10.1007/s12220-015-9644-0
· Zbl 1357.81127 · doi:10.1007/s12220-015-9644-0
[15] L. Charles, Analytic Berezin-Toeplitz operators, preprint, arXiv:1912.06819v1 [math.CV]. arXiv: 1912.06819v1
[16] M. Christ, “Off-diagonal decay of Bergman kernels: On a question of Zelditch” in Algebraic and Analytic Microlocal Analysis, Springer Proc. Math. Stat. 269, Springer, Cham, 2018, 459-481. Zentralblatt MATH: 1429.32003
· Zbl 1429.32003
[17] M. Christ, “Upper bounds for Bergman kernels associated to positive line bundles with smooth Hermitian metrics” in Algebraic and Analytic Microlocal Analysis, Springer Proc. Math. Stat. 269, Springer, Cham, 2018, 437-457. Zentralblatt MATH: 1429.32030
· Zbl 1429.32030
[18] L. A. Coburn, M. Hitrik, and J. Sjöstrand, Positivity, complex FIOs, and Toeplitz operators, Pure Appl. Anal. 1 (2019), no. 3, 327-357. · Zbl 1429.32047
[19] A. Deleporte, The Bergman kernel in constant curvature, preprint, arXiv:1812.06648v1 [math.AP]. arXiv: 1812.06648v1
Zentralblatt MATH: 07262585
· Zbl 1452.32021
[20] A. Deleporte, Toeplitz operators with analytic symbols, preprint, arXiv:1812.07202v2 [math.SP]. arXiv: 1812.07202v2
Zentralblatt MATH: 07250100
Digital Object Identifier: doi:10.1007/s00220-020-03791-4
· Zbl 1447.32031 · doi:10.1007/s00220-020-03791-4
[21] J.-P. Demailly, \(L^2\) estimates for the \(\bar{\partial } \)-operator on complex manifolds, preprint, 1996, https://www-fourier.ujf-grenoble.fr/ demailly/manuscripts/estimations_l2.pdf.
[22] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Math. Soc. Lecture Note Ser. 268, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0926.35002
[23] S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666-705. Zentralblatt MATH: 0883.53032
Digital Object Identifier: doi:10.4310/jdg/1214459407
Project Euclid: euclid.jdg/1214459407
· Zbl 0883.53032 · doi:10.4310/jdg/1214459407
[24] J. J. Duistermaat, Fourier Integral Operators, Progr. Math. 130, Birkhäuser Boston, Boston, 1996. Zentralblatt MATH: 0841.35137
· Zbl 0841.35137
[25] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), no. 1, 1-65. Zentralblatt MATH: 0289.32012
Digital Object Identifier: doi:10.1007/BF01406845
· Zbl 0289.32012 · doi:10.1007/BF01406845
[26] Y. Guedes Bonthonneau, N. Raymond, and S. Vũ Ngọc, Exponential localization in 2D pure magnetic wells, preprint, arXiv:1910.09261v1 [math.AP]. arXiv: 1910.09261v1
[27] B. Helffer and J. Sjöstrand, Puits multiples en limite semi-classique, II: Interaction moléculaire, symétries, perturbation, Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 2, 127-212. · Zbl 0595.35031
[28] H. Hezari, Z. Lu, and H. Xu, Off-diagonal asymptotic properties of Bergman kernels associated to analytic Kähler potentials, Int. Math. Res. Not. IMRN 2020, no. 8, 2241-2286. Zentralblatt MATH: 1442.32011
Digital Object Identifier: doi:10.1093/imrn/rny081
· Zbl 1442.32011 · doi:10.1093/imrn/rny081
[29] H. Hezari and H. Xu, Quantitative upper bounds for Bergman kernels associated to smooth Kähler potentials, preprint, arXiv:1807.00204v1 [math.CV]. arXiv: 1807.00204v1
[30] H. Hezari and H. Xu, On a property of Bergman kernels when the Kähler potential is analytic, preprint, arXiv:1912.11478v3 [math.DG]. arXiv: 1912.11478v3
[31] M. Hitrik, A. Mantile, and J. Sjöstrand, Adiabatic evolution and shape resonances, preprint, arXiv:1711.07583v1 [math.PH]. arXiv: 1711.07583v1
[32] M. Hitrik and J. Sjöstrand, Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions, I, Ann. Henri Poincaré 5 (2004), no. 1, 1-73. · Zbl 1059.47056
[33] M. Hitrik and J. Sjöstrand, “Two minicourses on analytic microlocal analysis” in Algebraic and Analytic Microlocal Analysis, Springer Proc. Math. Stat. 269, Springer, Cham, 2018, 483-540. · Zbl 1418.32003
[34] M. Hitrik, J. Sjöstrand, and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for nonselfadjoint operators, Amer. J. Math. 129 (2007), no. 1, 105-182. Zentralblatt MATH: 1172.35085
Digital Object Identifier: doi:10.1353/ajm.2007.0001
· Zbl 1172.35085 · doi:10.1353/ajm.2007.0001
[35] L. Hörmander, \(L^2\) estimates and existence theorems for the \(\bar{\partial }\) operator, Acta Math. 113 (1965), 89-152. · Zbl 0158.11002
[36] L. Hörmander, Fourier integral operators, I, Acta Math. 127 (1971), no. 1-2, 79-183. · Zbl 0212.46601
[37] M. Kashiwara, “Analyse micro-locale du noyau de Bergman” in Séminaire Goulaouic-Schwartz (1976/1977): Équations aux dérivées partielles et analyse fonctionnelle, École Polytech., Palaiseau, no. 8, 1977. · Zbl 0445.32020
[38] Y. A. Kordyukov, On asymptotic expansions of generalized Bergman kernels on symplectic manifolds, Algebra i Analiz 30 (2018), no. 2, 163-187. · Zbl 1410.58009
[39] Y. Le Floch, A Brief Introduction to Berezin-Toeplitz Operators on Compact Kähler Manifolds, CRM Short Courses, Springer, Cham, 2018. Zentralblatt MATH: 06908788
· Zbl 1452.32002
[40] X. Ma and G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, Progr. Math. 254, Birkhäuser, Basel, 2007. Zentralblatt MATH: 1135.32001
· Zbl 1135.32001
[41] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer, New York, 2002. Zentralblatt MATH: 0994.35003
· Zbl 0994.35003
[42] A. Melin and J. Sjöstrand, Bohr-Sommerfeld quantization conditions for non-selfadjoint operators in dimension 2, Astérisque 284 (2003), 181-244. · Zbl 1061.35186
[43] D. H. Phong and J. Sturm, “Lectures on stability and constant scalar curvature” in Current Developments in Mathematics, 2007, Int. Press, Somerville, 2009, 101-176. Zentralblatt MATH: 1188.53081
· Zbl 1188.53081
[44] O. Rouby, Bohr-Sommerfeld quantization conditions for non-selfadjoint perturbations of selfadjoint operators in dimension one, Int. Math. Res. Not. IMRN 2018, no. 7, 2156-2207. Zentralblatt MATH: 07013452
· Zbl 1510.81071
[45] B. Simon, The classical limit of quantum partition functions, Comm. Math. Phys. 71 (1980), no. 3, 247-276. Zentralblatt MATH: 0436.22012
Digital Object Identifier: doi:10.1007/BF01197294
Project Euclid: euclid.cmp/1103907536
· Zbl 0436.22012 · doi:10.1007/BF01197294
[46] J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95, Soc. Math. France, Paris, 1982. · Zbl 0524.35007
[47] J. Sjöstrand, “Function spaces associated to global \(I\)-Lagrangian manifolds” in Structure of Solutions of Differential Equations (Katata/Kyoto, 1995), World Scientific, River Edge, 1996, 369-423. · Zbl 0889.46027
[48] J. Song and S. Zelditch, Bergman metrics and geodesics in the space of Kähler metrics on toric varieties, Anal. PDE 3 (2010), no. 3, 295-358. Zentralblatt MATH: 1282.35428
Digital Object Identifier: doi:10.2140/apde.2010.3.295
Project Euclid: euclid.apde/1513731079
· Zbl 1282.35428 · doi:10.2140/apde.2010.3.295
[49] G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), no. 1, 99-130. · Zbl 0706.53036
[50] S. Zelditch, Szegö kernels and a theorem of Tian, Int. Math. Res. Not. IMRN 1998, no. 6, 317-331. Zentralblatt MATH: 0922.58082
Digital Object Identifier: doi:10.1155/S107379289800021X
· Zbl 0922.58082 · doi:10.1155/S107379289800021X
[51] S. Zelditch and P. Zhou, Central limit theorem for spectral partial Bergman kernels, Geom. Topol. 23 (2019), no. 4, 1961-2004. Zentralblatt MATH: 1434.32013
Digital Object Identifier: doi:10.2140/gt.2019.23.1961
Project Euclid: euclid.gt/1563242523
· Zbl 1434.32013 · doi:10.2140/gt.2019.23.1961
[52] M.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.