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Keywords = Djrbashian kernel

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42 pages, 569 KiB  
Review
Weighted Bergman Kernels and Mathematical Physics
by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito
Axioms 2020, 9(2), 48; https://doi.org/10.3390/axioms9020048 - 29 Apr 2020
Cited by 1 | Viewed by 3561
Abstract
We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω ⊂ C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is [...] Read more.
We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω ⊂ C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form γ = | φ | m on strictly pseudoconvex domains Ω = { φ < 0 } ⊂ C n ) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on ∂ Ω × S 1 ). Several open problems are indicated throughout the survey. Full article
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)
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