Berezin-Toeplitz Operators, a Semi-Classical Approach

Charles, Laurent

doi:10.1007/s00220-003-0882-9

Cited by 101 publications

(183 citation statements)

References 15 publications

(13 reference statements)

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“…As it is proved in section 5 of [5], the assumption (v α , v α ) = 1 implies that (v α ) is an admissible sequence. In the same way a sequence of sections O(k −∞ α ) for the L 2 norm is negligible.…”

Section: 3mentioning

confidence: 76%

“…Hence if k is sufficiently large, the spectrum of (V * k V k ) is a subset of ( , ∞) where > 0 does not depend on k. Applying proposition 12 of [5], we obtain that (V * k V k ) − 1 2 is a Toeplitz operator. Now…”

Section: 3mentioning

confidence: 86%

“…[2], [3], [4], [5]). The purpose of this article is to quantize the Lagrangian manifolds of M , by generalising the ansatz for the Schwartz kernel of a Toeplitz operator that we proposed in [5]. We will apply this to produce quasimodes of Toeplitz operators and deduce the Bohr-Sommerfeld conditions.…”

mentioning

confidence: 99%

“…In the usual semi-classical theory, the semi-classical observables are the pseudodifferential operators with a small parameter . The Schwartz kernel of these operators is of the form 1 2π n e i −1 (x−y).ξ a(x, ξ, )|dξ| (2) Our main result in [5] was to give a similar expression for the Schwartz kernel of a Toeplitz operator:…”

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confidence: 99%

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Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Charles¹

2003

Communications in Partial Differential Equations

104

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Abstract. This article is devoted to the quantization of the Lagrangian submanifold in the context of geometric quantization. The objects we define are similar to the Lagrangian distributions of the cotangent phase space theory. We apply this to construct quasimodes for the Toeplitz operators and we state the Bohr-Sommerfeld conditions under the usual regularity assumption. To compare with the Bohr-Sommerfeld conditions for a pseudodifferential operator with small parameter, the Maslov index, defined from the vertical polarization, is replaced with a curvature integral, defined from the complex polarization. We also consider the quantization of the symplectomorphisms, the realization of semi-classical equivalence between two different quantizations of a symplectic manifold and the microlocal equivalences.Let (M, ω) be a symplectic compact manifold of dimension 2n endowed with a prequantization bundle, that is a complex line bundle L → M with a Hermitian structure h and a covariant derivation ∇ whose curvature is ω. To quantize these data, we assume that M is endowed with a complex structure J which is integrable and compatible with −iω. The quantum space H k is defined as the space of the holomorphic sections of L k → M . k is any positive integer and the semi-classical limit is k → ∞. The quantum semi-classical observables are the Berezin-Toeplitz operators (cf. [2], [3], [4], [5]). The purpose of this article is to quantize the Lagrangian manifolds of M , by generalising the ansatz for the Schwartz kernel of a Toeplitz operator that we proposed in [5]. We will apply this to produce quasimodes of Toeplitz operators and deduce the Bohr-Sommerfeld conditions. Let us state this last result in the case M is 2-dimensional. Consider the Toeplitz operatorand f 1 are some functions of C ∞ (M ) and M f0+k −1 f1 is the multiplication operator by f 0 + k −1 f 1 .Assume that E 0 is a regular value of the principal symbol f 0 of (T k ) and that f −1 0 (E 0 ) is connected. Then if E belongs to some neighborhood U of E 0 , the level set f −1 (E) = Λ E is a circle.Theorem 0.1. For all sequences (E α , k α ) of U × N,where

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Section: 3mentioning

confidence: 76%

Section: 3mentioning

confidence: 86%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Charles¹

2003

Communications in Partial Differential Equations

104

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“…Loi [43,44] refined Engliš' recursive formula and gave a new proof of Engliš' asymptotic expansion. See also [10] for related works. As shown in [43] (also cf.…”

Section: Local and Global Bergman Kernels: An Explicit Computationmentioning

confidence: 99%

Bergman kernel and Kähler tensor calculus

2013

Pure and Applied Mathematics Quarterly

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Fefferman [22] initiated a program of expressing the asymptotic expansion of the boundary singularity of the Bergman kernel for strictly pseudoconvex domains explicitly in terms of boundary invariants. Hirachi, Komatsu and Nakazawa [30] carried out computations of the first few terms of Fefferman's asymptotic expansion building partly on Graham's work on CR invariants and Nakazawa's work on the asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains. In this paper, we prove a formula for coefficients in Nakazawa's asymptotic expansion as explicit summations over strongly connected graphs, and a formula expressing partial derivatives of Kähler metrics (resp. functions) as summations over rooted trees encoding covariant derivatives of curvature tensors (resp. functions). These formulae shall be useful in studying general patterns of Fefferman's asymptotic expansion.

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Soft Riemann‐Hilbert problems and planar orthogonal polynomials

Hedenmalm

2023

Comm Pure Appl Math

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Riemann‐Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, for example, in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix‐valued Riemann‐Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest descent method. For orthogonal polynomials on the line or on the circle with respect to exponentially varying weights, this led to a strong asymptotic expansion in the given parameters. For orthogonal polynomials with respect to exponentially varying weights in the plane, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), based on the technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a certain matrix ‐problem (Its, Takhtajan), which we refer to as a soft Riemann‐Hilbert problem. Here, we use this perspective to offer a simplified approach based not on foliations but instead on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the soft Riemann‐Hilbert problem. This allows the problem to decompose into a hierarchy of scalar Riemann‐Hilbert problems along the interface (the free boundary for a related obstacle problem). Inspired by microlocal analysis, the method allows for control of the solution in such a way that for real‐analytic weights, the asymptotics holds in the L2 sense with error in a fixed neighborhood of the closed exterior of the interface, for some constant , where . Here, m is the degree of the polynomial, and in terms of pointwise asymptotics, the expansion dominates the error term in the exterior domain and across the interface (by a distance proportional to ). In particular, the zeros of the orthogonal polynomial are located in the interior of the spectral droplet, away from the droplet boundary by a distance at least proportional to .

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Berezin-Toeplitz Operators, a Semi-Classical Approach

Cited by 101 publications

References 15 publications

Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Bergman kernel and Kähler tensor calculus

Soft Riemann‐Hilbert problems and planar orthogonal polynomials

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