Asymptotic expansion of polyanalytic Bergman kernels

Abstract

We consider mainly the Hilbert space of bianalytic functions on a given domain in the plane, square integrable with respect to a weight. We show how to obtain the asymptotic expansion of the corresponding bianalytic Bergman kernel for power weights, under the standard condition on those weights. This is known only in the analytic setting, from the work of e.g. Tian, Yau, Zelditch, Catlin, et al. We remark that a bianalytic function may be identified with a vector-valued analytic function, supplied with a locally singular metric on the vectors. We also apply our findings to two bianalytic Bergman metrics introduced here.

40 pages

References in corpus (8)

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• Random normal matrices and Ward identities
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• Toeplitz operators on Bergman spaces of polyanalytic functions
• Coherent States Quantization for Generalized Bargmann Spaces with Formulae for their Attached Berezin Transforms in Terms of the Laplacian on Cn
• A formula representing Magnetic Berezin Transformsas functions of the Laplacian on Cn
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Cited by in corpus (5)

• The polyanalytic Ginibre ensembles
• The Weyl-Heisenberg ensemble: hyperuniformity and higher Landau levels
• Harmonic analysis in phase space and finite Weyl-Heisenberg ensembles
• Equivalence of Palm measures for determinantal point processes governed by Bergman kernels
• Wehrl entropies and Euclidean Landau levels