Polyanalytic Toeplitz Operators: Isomorphisms, Symbolic Calculus and Approximation of Weyl Operators

Keller, Johannes; Luef, Franz

doi:10.1007/s00041-021-09843-0

Cited by 3 publications

(2 citation statements)

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“…There are several papers devoted to Toeplitz operators acting on spectral subspaces of the Landau Hamiltonian in R 2n (see, for instance, [7, 15, 28-30, 32, 33] and references therein). For constant magnetic fields, such operators are related with the Toeplitz operators acting on Bargmann-Fock type spaces of polyanalytic functions (see, for instance, [1,2,13,16,20,34,36] and references therein). In particular, in [20], quantization schemes defined by polyanalytic Toeplitz operators are discussed.…”

Section: Introductionmentioning

confidence: 99%

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Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

Kordyukov

2022

J. Spectr. Theory

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In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers L p of the prequantum line bundle L asymptotically splits into clusters of size O.p 3=4 / around the points pƒ, where ƒ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigenvalues from the cluster. We show that it provides a Berezin-Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kähler quantization.

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Section: Introductionmentioning

confidence: 99%

“…For constant magnetic fields, such operators are related with the Toeplitz operators acting on Bargmann-Fock type spaces of polyanalytic functions (see, for instance, [1,2,13,16,20,34,36] and references therein). In particular, in [20], quantization schemes defined by polyanalytic Toeplitz operators are discussed.…”

Section: Introductionmentioning

confidence: 99%