Displaying similar documents to “Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results.”

A note on Berezin-Toeplitz quantization of the Laplace operator

Alberto Della Vedova (2015)

Complex Manifolds

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Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.

Semiclassical spectral estimates for Toeplitz operators

David Borthwick, Thierry Paul, Alejandro Uribe (1998)

Annales de l'institut Fourier

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Let X be a compact Kähler manifold with integral Kähler class and L → X a holomorphic Hermitian line bundle whose curvature is the symplectic form of X . Let H ∈ C ∞ ( X , ℝ ) be a Hamiltonian, and let T k be the Toeplitz operator with multiplier H acting on the space ℋ k = H 0 ( X , L ⊗ k ) . We obtain estimates on the eigenvalues and eigensections of T k as k → ∞ , in terms of the classical Hamilton flow of H . We study in some detail the case when X is an integral coadjoint orbit of a Lie group.

A Reproducing Kernel and Toeplitz Operators in the Quantum Plane

Stephen Bruce Sontz (2013)

Communications in Mathematics

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We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which...

Index and dynamics of quantized contact transformations

Steven Zelditch (1997)

Annales de l'institut Fourier

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Quantized contact transformations are Toeplitz operators over a contact manifold ( X , α ) of the form U χ = Π A χ Π , where Π : H 2 ( X ) → L 2 ( X ) is a Szegö projector, where χ is a contact transformation and where A is a pseudodifferential operator over X . They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine ind ( U χ ) when the principal symbol is unitary, or equivalently...

Toeplitz Quantization for Non-commutating Symbol Spaces such as S U q ( 2 )

Stephen Bruce Sontz (2016)

Communications in Mathematics

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Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group S U q ( 2 ) is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples...

Deformation quantization and Borel's theorem in locally convex spaces

Miroslav Engliš, Jari Taskinen (2007)

Studia Mathematica

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It is well known that one can often construct a star-product by expanding the product of two Toeplitz operators asymptotically into a series of other Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin-Toeplitz quantization. We show that one can obtain in a similar way in fact any star-product which is equivalent to the Berezin-Toeplitz star-product, by using instead of Toeplitz operators other suitable mappings from compactly supported smooth...