Coherent states attached to the quantum disc algebra and their associated polynomials

2021 ◽  
pp. 2150078
Author(s):  
H. Fakhri ◽  
M. Refahinozhat
Keyword(s):  
Coherent States ◽  
Jacobi Polynomials ◽  
Hermite Polynomials ◽  
Hermitian Operator ◽  
Fock Representation ◽  
Disc Algebra ◽  
Variable Formula ◽  
Associated Polynomials ◽  
Hermitian Conjugate ◽  
The One

The one-variable [Formula: see text]-coherent states attached to the [Formula: see text]-disc algebra are constructed and used to obtain the [Formula: see text]-Bargmann–Fock realization of its Fock representation. Then, this realization is used to obtain the [Formula: see text]-continuous Hermite polynomials as well as continuous and discrete [Formula: see text]-Hermite polynomials by using a pair of Hermitian canonical conjugate operators and two pairs of the non-Hermitian conjugate operators, respectively. Besides, we introduce a two-variable family of [Formula: see text]-coherent states attached to the Fock representation space of the [Formula: see text]-disc algebra and its opposite algebra and obtain their simultaneous [Formula: see text]-Bargmann–Fock realization. For an appropriate non-Hermitian operator, the latter realization is served to obtain the well-known little [Formula: see text]-Jacobi polynomials used in constructing the [Formula: see text]-disc polynomials.

Modified photon-added coherent states: Generation and relatedentangled states

2008 ◽  
Vol 86 (12) ◽  
pp. 1387-1392 ◽  
Author(s):  
M -L Liang ◽  
J -N Zhang ◽  
B Yuan
Keyword(s):  
Coherent State ◽  
Radiation Field ◽  
Coherent States ◽  
Hermitian Operator ◽  
Field Interaction ◽  
Wave Approximation ◽  
New Type ◽  
The One ◽  
Entangled Coherent States ◽  
03.65 Ud

We construct one new type of quantum state that we call the modified photon-added coherent state (MPACS) of the radiation field. These states are created by repeatedly applying the Hermitian operator (a + a+) to the coherent state m times. It turns out that these states are the superpositions of the coherent and the photon-added coherent states, and have highly nonclassical behavior depending on the excitation m and other parameters. The one-mode and two-mode modified entangled coherent states are also studied. MPACS can be generated through the atom-field interaction under the nonrotating wave approximation. PACS Nos.: 42.50.Dv, 03.65.Ca, 03.65.Ud

A MODEL OF QUANTUM OSCILLATOR WITH DISCRETE SPACE

2008 ◽  
Vol 23 (13) ◽  
pp. 943-952
Author(s):  
I. I. KACHURYK ◽  
A. U. KLIMYK
Keyword(s):  
Real Line ◽  
Coherent States ◽  
Hermite Polynomials ◽  
Heisenberg Algebra ◽  
Discrete Space ◽  
Quantum Oscillator ◽  
Fock Representation ◽  
New Model

We construct a new model of the quantum oscillator, which is related to the discrete q-Hermite polynomials of the second type. The position and momentum operators in the model are appropriate operators of the Fock representation of a deformation of the Heisenberg algebra. These operators have a discrete non-degenerate spectra. These spectra are spread over the whole real line. Coordinate and momentum realizations of the model are constructed. Coherent states are explicitly given.

The symmetric q-oscillator algebra: q-coherent states, q-Bargmann–Fock realization and continuous q-Hermite polynomials with 0 < q < 1

2016 ◽  
Vol 13 (03) ◽  
pp. 1650028 ◽  
Author(s):  
H. Fakhri ◽  
A. Hashemi
Keyword(s):  
Text Analysis ◽  
Coherent States ◽  
Hermite Polynomials ◽  
Representation Space ◽  
Oscillator Algebra ◽  
Fock Representation ◽  
Minimum Uncertainty

The symmetric [Formula: see text]-analysis is used to construct a type of minimum-uncertainty [Formula: see text]-coherent states in the Fock representation space of the symmetric [Formula: see text]-oscillator ∗-algebra with [Formula: see text]. Then, its corresponding [Formula: see text]-Hermite polynomials are derived by using the [Formula: see text]-Bargmann–Fock realization of the symmetric [Formula: see text]-oscillator algebra.

scholarly journals Heat kernels of the discrete Laguerre operators

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Aleksey Kostenko
Keyword(s):  
Hardy Inequality ◽  
Jacobi Polynomials ◽  
Heat Kernels ◽  
The Other ◽  
Heat Semigroup ◽  
Stochastic Completeness ◽  
Markovian Semigroup ◽  
Other Hand ◽  
The One

AbstractFor the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

Nonclassicality of photon-modulated spin coherent states in the Holstein-Primakoff realization

2021 ◽  
Author(s):  
Xiaoyan Zhang ◽  
Jisuo Wang ◽  
Lei Wang ◽  
Xiangguo Meng ◽  
Baolong Liang
Keyword(s):  
Coherent States ◽  
Wigner Distribution ◽  
Hermite Polynomials ◽  
Photon Number ◽  
Spin Ladder ◽  
Bernoulli Distribution ◽  
Ladder Operators ◽  
Spin Number ◽  
Photon Number Distribution ◽  
Spin Coherent States

Abstract Two new photon-modulated spin coherent states (SCSs) are introduced by operating the spin ladder operators J &#177; on the ordinary SCS in the Holstein-Primakoff realization and the nonclassicality is exhibited via their photon number distribution, second-order correlation function, photocount distribution and negativity of Wigner distribution. Analytical results show that the photocount distribution is a Bernoulli distribution and the Wigner functions are only associated with two-variable Hermite polynomials. Compared with the ordinary SCS, the photon-modulated SCSs exhibit more stronger nonclassicality in certain regions of the photon modulated number k and spin number j, which means that the nonclassicality can be enhanced by selecting suitable parameters.

scholarly journals Operational Methods in the Study of Sobolev-Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 124 ◽  
Author(s):  
Nicolas Behr ◽  
Giuseppe Dattoli ◽  
Gérard Duchamp ◽  
Silvia Penson
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Hermite Polynomials ◽  
Umbral Calculus ◽  
Generalized Hypergeometric Functions ◽  
Normal Ordering ◽  
Gamma Functions ◽  
Image Technique ◽  
Operational Methods

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of Sobolev-Jacobi (SJ) polynomials explicitly.

Approach of the continuous q-Hermite polynomials to x-representation of q-oscillator algebra and its coherent states

2019 ◽  
Vol 17 (02) ◽  
pp. 2050021
Author(s):  
H. Fakhri ◽  
S. E. Mousavi Gharalari
Keyword(s):  
Harmonic Oscillator ◽  
Generating Function ◽  
Coherent States ◽  
Finite Interval ◽  
Hermite Polynomials ◽  
Oscillator Algebra ◽  
Text Representation ◽  
Ladder Operators ◽  
Recursion Relations ◽  
Wilson Operator

We use the recursion relations of the continuous [Formula: see text]-Hermite polynomials and obtain the [Formula: see text]-difference realizations of the ladder operators of a [Formula: see text]-oscillator algebra in terms of the Askey–Wilson operator. For [Formula: see text]-deformed coherent states associated with a disc in the radius [Formula: see text], we obtain a compact form in [Formula: see text]-representation by using the generating function of the continuous [Formula: see text]-Hermite polynomials, too. In this way, we obtain a [Formula: see text]-difference realization for the [Formula: see text]-oscillator algebra in the finite interval [Formula: see text] as a [Formula: see text]-generalization of known differential formalism with respect to [Formula: see text] in the interval [Formula: see text] of the simple harmonic oscillator.

scholarly journals COHERENT STATES, FRACTALS AND BRAIN WAVES

2009 ◽  
Vol 05 (01) ◽  
pp. 245-264 ◽  
Author(s):  
GIUSEPPE VITIELLO
Keyword(s):  
Coherent States ◽  
Action Perception ◽  
Self Similarity ◽  
Memory State ◽  
Functional Representation ◽  
Power Spectral ◽  
Power Spectral Densities ◽  
The One ◽  
Power Law Distributions ◽  
Dissipative Model

I show that a functional representation of self-similarity (as the one occurring in fractals) is provided by squeezed coherent states. In this way, the dissipative model of brain is shown to account for the self-similarity in brain background activity suggested by power-law distributions of power spectral densities of electrocorticograms. I also briefly discuss the action-perception cycle in the dissipative model with reference to intentionality in terms of trajectories in the memory state space.

Meaning of Kato's Formulas for Upper and Lower Bounds to Eigenvalues of Hermitian Operators

1971 ◽  
Vol 49 (2) ◽  
pp. 218-223 ◽  
Author(s):  
Dallas T. Hayes
Keyword(s):  
Eigenvalue Problem ◽  
Recent Work ◽  
Lower Bounds ◽  
Trial Function ◽  
Hermitian Operator ◽  
Upper And Lower Bounds ◽  
The One ◽  
Independent Derivation

Using an independent derivation by Kohn, the full meaning of Kato's formulas for upper and lower bounds to eigenvalues of a Hermitian operator is shown. These bounds are the best possible when the only information available on a particular eigenvalue problem is a suitable trial function and an estimate of the neighboring eigenvalues to the one in question. This was asserted by Kato but not proved. A comparison is made of Kato's bounds with those derived in papers by Stevenson and Crawford and by Cohen and Feldmann. Under the conditions which result in Kato's bounds it is shown that the Stevenson–Crawford and Cohen–Feldmann bounds reduce to those of Kato. When more information is available these bounds are an improvement upon Kato's. This makes more precise the recent work of Walmsley and Cohen–Feldmann, whose results appear to prove in general the greater accuracy of the Stevenson–Crawford and Cohen–Feldmann bounds over those of Kato. A general discussion of all three sets of bounds is given in terms of the parameter λ appearing in the Stevenson–Crawford formulation.

scholarly journals Large Quantum Gravity Effects: Backreaction on Matter

1997 ◽  
Vol 12 (32) ◽  
pp. 2407-2413 ◽  
Author(s):  
Rodolfo Gambini ◽  
Jorge Pullin
Keyword(s):  
Quantum Gravity ◽  
Coherent States ◽  
Rapid Loss ◽  
Gravity Effects ◽  
Dimensional Gravity ◽  
The One ◽  
Loss Of Coherence ◽  
Matter Sector ◽  
Matter Fields ◽  
Large Quantum

We re-examine the large quantum gravity effects discovered by Ashtekar in the context of (2+1)-dimensional gravity coupled to matter. We study an alternative one-parameter family of coherent states of the theory in which the large quantum gravity effects on the metric can be diminished, at the expense of losing coherence in the matter sector. Which set of states is the one that occurs in nature will determine if the large quantum gravity effects are actually observable as wild fluctuations of the metric or rapid loss of coherence of matter fields.

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