The <i>q</i>-deformation of quantum mechanics of one degree of freedom

Maximov, Valodia; Odzijewicz, Anatol

doi:10.1063/1.531080

Cited by 13 publications

(8 citation statements)

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“…The intertwining of finite difference operators facilitates remarkably the solution of recurrence problems [172]. The generalization of q-deformed systems [107,108] permits the construction of chains of finite difference Hamiltonians H k with A k H k = q k H k−1 A k , of considerable interest to quantum optics [161,169,170,172,173].…”

Section: The Finite Differencesmentioning

confidence: 99%

Factorization: little or great algorithm?

Mielnik

Rosas-Ortiz

2004

J. Phys. A: Math. Gen.

196

264

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Section: The Finite Differencesmentioning

confidence: 99%

Factorization: little or great algorithm?

Mielnik

Rosas-Ortiz

2004

J. Phys. A: Math. Gen.

196

264

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“…Beware that the µ-deformation should not be confused with the q-deformation of quantum mechanics. See [19] and references therein for a discussion of the latter.…”

Section: Introductionmentioning

confidence: 99%

The Μ-Deformed SEGAL–BARGMANN TRANSFORM IS a HALL TYPE TRANSFORM

Sontz

2009

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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We present an explanation of how the µ-deformed Segal-Bargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echavarría and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of Segal-Bargmann analysis. This explanation relates the µ-deformed Segal-Bargmann transforms to the generalized Segal-Bargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the µ-deformed Segal-Bargmann transform can be understood as Hall type transforms. In particular, we define a µ-deformation of Hall's "Version C" generalized Segal-Bargmann transform which is then shown to be a µ-deformed convolution with a µ-deformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall.Keywords: Segal-Bargmann analysis, heat kernel analysis, µ-deformed quantum mechanics. AMS Subject Classification: primary: 46N50, 47N50, secondary: 46E15, 81S99 the µ-deformed Segal-Bargmann transform. We underline that Version C of this transform is shown to be a µ-deformed convolution with a µ-deformed heat kernel followed by analytic continuation. For those willing to read this result without necessarily being appraised of all the requisite definitions, we recommend looking at Theorem 1.2 right away, which is our main result. This theorem generalizes and is motivated by the particular case µ = 0, which was presented originally by Hall in [13].In particular we follow Hall's terminology by referring to various versions of the Segal-Bargmann transform, called Versions A, B and C in [13]. (We also introduce a Version D, but this is a minor point.) This terminology however can mislead one into thinking that there is one underlying object of which various versions (in the usual sense of this word) are being studied. However, it is true that Versions A and B are closely related, the only difference between them being a unitary change of variables transform. This is also the relation between Versions C and D. See [13] and Theorem 1.2 for exact details about the various domains, ranges and formulas of these Versions.Many of our formulas are analogous to formulas in [13]. Thus our first three versions (A to C) of the µ-deformed Segal-Bargmann transform are Hall type transforms. Our Version D also fits into this pattern. This directly relates for the first time the well studied µ-deformed Segal-Bargmann transform to the seminal work of Hall, which views Segal-Bargmann analysis as a part of heat kernel analysis. (See [13], [14] and [15].) In short, we show that µ-deformed Segal-Bargmann analysis is a part of µ-deformed heat kernel analysis.Let us note there has been much interest and research activity concerning the spaces and their associated structures that we are studying here and concerning related, and even more general, spaces and their associated structures. Besides our work with co-authors ([2], [3], [7], [21], [22], [23] and [32]) and the articles by other researchers which will be referenc...

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“…, c N are fixed after reduction to H c 1 ,...,c N . In the following we assume that c 0 satisfies 20) which is equivalent to the assumption that |c 0 , c 1 , . .…”

Section: )mentioning

confidence: 99%

“…One has A = 1−q α ∞ n=0 q α n δ |z| 2 − q α n 1 − q α d|z| 2 dψ, (7.75) see [20]. Now let us pass to the classical case.…”

Section: )mentioning

confidence: 99%