Isospectrality for quantum toric integrable systems

Laurent Charles; Álvaro Pelayo; San Vũ Ngoc

Displaying similar documents to “Isospectrality for quantum toric integrable systems”

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 $Π : H^{2} (X) \to L^{2} (X)$ 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 operators, Kähler manifolds, and line bundles.

Foth, Tatyana (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

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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 $L \to X$ a holomorphic Hermitian line bundle whose curvature is the symplectic form of $X$ . Let $H \in C^{\infty} (X, ℝ)$ be a Hamiltonian, and let $T_{k}$ be the Toeplitz operator with multiplier $H$ acting on the space $ℋ_{k} = H^{0} (X, L^{\otimes k})$ . We obtain estimates on the eigenvalues and eigensections of $T_{k}$ as $k \to \infty$ , 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.

Quantum ergodicity of C*-dynamical systems

Steven Zelditch (1994-1995)

Séminaire de théorie spectrale et géométrie

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Spectral invariants for coupled spin-oscillators

San Vũ Ngọc (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

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This text deals with in a semiclassical setting. Given a quantum system, the haunting question is “What interesting quantities can be discovered on the spectrum that can help to characterize the system ?” The general framework will be semiclassical analysis, and the issue is to recover the classical dynamics from the quantum spectrum. The coupling of a spin and an oscillator is a fundamental example in physics where some nontrivial explicit calculations can be done.

Fourier-like kernels in geometric quantization

K. Gawędzki

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CONTENTSI. Introduction............................................................................................................................................... 5II. Preliminary notions................................................................................................................................ 7III. Geometric quantization.........................................................................................................................12 A. Elements of...

Berezin-Toeplitz quantization for compact Kähler manifolds. A review of results.

Schlichenmaier, Martin (2010)

Advances in Mathematical Physics

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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...