Quantum integrability of quadratic Killing tensors

Duval, Christian; Valent, Galliano

doi:10.1063/1.1899986

Cited by 44 publications

(77 citation statements)

References 33 publications

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“…An even more difficult task would be to check whether the classical integrability survives to quantization in the sense of [4].…”

Section: Resultsmentioning

confidence: 99%

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

Valent¹

2017

Regul. Chaot. Dyn.

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We present a family of superintegrable (SI) sytems living on a riemannian surface of revolution and which exhibits one linear integral and two integrals of any integer degree larger or equal to 2 in the momenta. When this degree is 2 one recovers a metric due to Koenigs.The local structure of these systems is under control of a linear ordinary differential equation of order n which is homogeneous for even integrals and weakly inhomogeneous for odd integrals. The form of the integrals is explicitly given in the so-called "simple" case (see definition 2).Some globally defined examples are worked out which live either in H 2 or in R 2 .MSC 2010 numbers: 32C05, 81V99, 37E99, 37K25.

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“…An even more difficult task would be to check whether the classical integrability survives to quantization in the sense of [4].…”

Section: Resultsmentioning

confidence: 99%

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

Valent¹

2017

Regul. Chaot. Dyn.

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“…families of different metrics having the same un-parametrized geodesics) have been studied intensively by Matveev, Bolsinov, Topalov [4,20] and other authors. In particular, they showed that these metrics are integrable, can be quantized into integrable SDS of type (p, 0, 0) in our sense, and are essentially the same as the so called (separable Stäckel-) Benenti systems studied by Benenti and many other authors in classical and quantum mechanics, see, e.g., [2,4,6,20]. The metrics (induced from R n ) on multi-dimensional ellipsoids belong to this family of integrable metrics, and so the Brownian motion on a p-dimensional ellipsoid is an integrable SDS of type (p, 0, 0).…”

Section: 4mentioning

confidence: 99%

Reduction and Integrability of Stochastic Dynamical Systems

Zung

Thien

2017

J Math Sci

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Abstract. This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction and integrability. In particular, we show that an SDS which is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS's, and extend the results about the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS's. We also show how integrable SDS's are related to compatible families of integrable Riemannian metrics on manifolds.

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“…Transforming the formulas given in [14] we obtain the integrals (6). Let us compare with Koenigs results 1 , given in [12] p. 378.…”

Section: Part I the Trigonometric Case 1 Local Structurementioning

confidence: 99%

“…The simplest and most natural quantization is certainly Carter's (or minimal) quantization (see [6]). Denoting by a hat the quantum operators and setting = 1, the quantization rules are:…”

Section: Part IV Quantum Aspects 1 Carter Quantizationmentioning

confidence: 99%

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Global structure and geodesics for Koenigs superintegrable systems

Valent¹

2016

Regul. Chaot. Dyn.

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Starting from the framework defined by Matveev and Shevchishin we derive the local and the global structure for the four types of super-integrable Koenigs metrics. These dynamical systems are always defined on non-compact manifolds, namely R 2 and H 2 . The study of their geodesic flows is made easier using their linear and quadratic integrals. Using Carter (or minimal) quantization we show that the formal superintegrability is preserved at the quantum level and in two cases, for which all of the geodesics are closed, it is even possible to compute the discrete spectrum of the quantum hamiltonian.

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Quantum integrability of quadratic Killing tensors

Abstract: Quantum integrability of classical integrable systems given by quadratic Killing tensors on curved configuration spaces is investigated. It is proven that, using a "minimal" quantization scheme, quantum integrability is insured for a large class of classic examples.

Cited by 44 publications

References 33 publications

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

Reduction and Integrability of Stochastic Dynamical Systems

Global structure and geodesics for Koenigs superintegrable systems

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