Berezin Quantization and Reproducing Kernels on Complex Domains

Engliš, Miroslav

doi:10.1090/s0002-9947-96-01551-6

Cited by 82 publications

(72 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another important application of Rawnsley's -function is to study whether a Berezin quantization can be established on some Kähler manifolds. In recent years, Berezin quantization has attracted a lot of attention and has been deeply studied by mathematicians and physicists, see, e.g., Cahen-Gutt-Rawnsley [6], Engliš [19], Loi-Mossa [26] and Zedda [32]. Roughly, a quantization is a construction of a quantum system from the classical mechanics of a system.…”

Section: Remark 13 Notice That Ifmentioning

confidence: 99%

See 1 more Smart Citation

Balanced metrics and Berezin quantization on Hartogs triangles

2020

Annali di Matematica

View full text Add to dashboard Cite

In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by $$\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}$$ Ω n = { ( z 1 , … , z n ) ∈ C n : | z 1 | < | z 2 | < ⋯ < | z n | < 1 } which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric $$g(\nu )$$ g ( ν ) associated with the Kähler potential $$\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)-\nu _n\ln (1-\vert z_n\vert ^2)$$ Φ n ( z ) : = - ∑ k = 1 n - 1 ν k ln ( | z k + 1 | 2 - | z k | 2 ) - ν n ln ( 1 - | z n | 2 ) on $$\varOmega _n$$ Ω n . As main contributions, on one hand we compute the explicit form for Bergman kernel of weighted Hilbert space, and then, we obtain the necessary and sufficient condition for the metric $$g(\nu )$$ g ( ν ) on the domain $$\varOmega _n$$ Ω n to be a balanced metric. On the other hand, by using the Calabi’s diastasis function, we prove that the Hartogs triangles admit a Berezin quantization.

show abstract

Section: Remark 13 Notice That Ifmentioning

confidence: 99%

“…In fact, by using the Rawnsley's -function and the Calabi's diastasis function, Englis̆ [19] gave a sufficient condition for a Kähler manifold ( , g) to admit a Berezin quantization.…”

Section: Remark 13 Notice That Ifmentioning

confidence: 99%

Balanced metrics and Berezin quantization on Hartogs triangles

2020

Annali di Matematica

View full text Add to dashboard Cite

show abstract

“…The reproducing kernel for the Hilbert space of holomorphic, square integrable functions with respect to the scalar product of the type (2.3) was calculated via CS as the scalar product K M (z,w) = (ez, ew) [5,7,9]. On the other side, let us consider the weighted Hilbert space H f of square integrable holomorphic functions on M , with weight e −f [34]…”

Section: Balanced Metric and Berezin Quantization Via Coherent Statesmentioning

confidence: 99%

“…With Theorem 2.2 and the recalled definitions, we can formulate the The proof of the Theorem 2.4 is based on the fundamental conjecture for homogeneous bounded domains [32] and the asymptotic of the Bergman kernel [34].…”

Section: Balanced Metric and Berezin Quantization Via Coherent Statesmentioning

confidence: 99%

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

Berceanu¹

2016

SIGMA

View full text Add to dashboard Cite

show abstract

“…These domains are interesting both from the mathematical and the physical point of view (see for example [12] and [19] for the study of some Riemannian properties of g F and the Berezin quantization of (D F , g F ), [10] and [21] for the construction of global symplectic coordinates on these domains and [26] for the construction of Kähler-Einstein metrics on Hartogs type domains on symmetric spaces).…”

Section: Séminaire De Théorie Spectrale Et Géométrie (Grenoble)mentioning

confidence: 99%

Canonical metrics on some domains of ℂ n

Zuddas¹

2009

Séminaire De Théorie Spectrale Et Géométrie

View full text Add to dashboard Cite

The study of the existence and uniqueness of a preferred Kähler metric on a given complex manifold M is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kähler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains D in C n , the so-called Hartogs domains, which can be equipped with a natural Kaehler metric g. We show that if g is a Kähler-Einstein, constant scalar curvature, extremal or a soliton metric then (D, g) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space. If D is bounded, we also show the same assertion under the assumption that g is a scalar multiple of the Bergman metric. The results we present are proved in papers joint with A. Loi and A. J. Di Scala ([11], [20]). 1. Canonical metrics: existence and uniqueness 1.1. The compact case Let M be a complex manifold, with complex structure J. Let g be a Kähler metric on M , i.e. a J-invariant Riemannian metric such that the associated Kähler form ω, defined by ω(v, w) = g(Jv, w), is closed. In this talk we are interested in those Kähler metrics on M which are canonical in the sense that they are minima of natural geometric functionals on M or arise as limits of important geometric flows like the Kähler-Ricci flow. In order to give the precise definitions, let us fix some notations. Let Ric g denote the Ricci tensor of (M, g) and let ρ ω be its Ricci form defined by ρ ω (v, w) = Ric g (Jv, w) (we will omit the subscript ω when the context is clear). If, in a chart endowed with complex coordinates z 1 , .

show abstract

Berezin Quantization and Reproducing Kernels on Complex Domains

Cited by 82 publications

References 33 publications

Balanced metrics and Berezin quantization on Hartogs triangles

Balanced metrics and Berezin quantization on Hartogs triangles

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

Canonical metrics on some domains of ℂ n

Contact Info

Product

Resources

About