scholarly journals The discretized harmonic oscillator: Mathieu functions and a new class of generalized Hermite polynomials

2003 ◽  
Vol 44 (5) ◽  
pp. 1913-1936 ◽  
Author(s):  
M. Aunola
Keyword(s):  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Mathieu Functions ◽  
New Class

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

The quantum relativistic harmonic oscillator: generalized Hermite polynomials

1991 ◽  
Vol 156 (7-8) ◽  
pp. 381-385 ◽  
Author(s):  
V. Aldaya ◽  
J. Bisquert ◽  
J. Navarro-Salas
Keyword(s):  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Relativistic Harmonic Oscillator

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.

TWO-VARIABLE HERMITE POLYNOMIALS AS TIME-EVOLUTIONAL TRANSITION AMPLITUDE FOR DRIVEN HARMONIC OSCILLATOR

2007 ◽  
Vol 21 (08) ◽  
pp. 475-480 ◽  
Author(s):  
HONG-YI FAN ◽  
TENG-FEI JIANG
Keyword(s):  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Transition Amplitude ◽  
Initial Number ◽  
Quantum Harmonic Oscillator ◽  
Physical Explanation ◽  
Final State ◽  
Number State

For the two-variable Hermite polynomials Hm,n(β,β*) we find its new physical explanation in the dynamics of a linear forced quantum harmonic oscillator (or a dispaced oscillator), i.e. Hm,n(β,β*) can be explained as the time-evolutional transition amplitude from an initial number state |m〉 at t0 to a final state |n〉 at t of the dispaced oscillator. Two new properties of the time-evolutional operator for driven oscillator are revealed.

scholarly journals Deformed u(2) Algebra as the Symmetry Algebra of the Planar Anisotropic Quantum Harmonic Oscillator with Rational Ratio of Frequencies

1997 ◽  
Vol 12 (19) ◽  
pp. 3335-3346 ◽  
Author(s):  
Dennis Bonatsos ◽  
P. Kolokotronis ◽  
D. Lenis ◽  
C. Daskaloyannis
Keyword(s):  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Symmetry Algebra ◽  
Quantum Harmonic Oscillator ◽  
General Applicability ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Superintegrable Systems ◽  
Frequency Ratios ◽  
Special Case

The symmetry algebra of the two-dimensional anisotropic quantum harmonic oscillator with rational ratio of frequencies is identified as a deformation of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are determined using algebraic methods of general applicability to quantum superintegrable systems. For labelling the degenerate states an "angular momentum" operator is introduced, the eigenvalues of which are roots of appropriate generalized Hermite polynomials. The cases with frequency ratios 1:n correspond to generalized parafermionic oscillators, while in the special case with frequency ratio 2:1 the resulting algebra corresponds to the finite W algebra [Formula: see text].

scholarly journals On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 88
Author(s):  
Elchin I. Jafarov ◽  
Aygun M. Mammadova ◽  
Joris Van der Jeugt
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Short Communication ◽  
Jacobi Polynomials ◽  
Direct Limit ◽  
Hermite Polynomials ◽  
Limit Relation ◽  
Exactly Solvable ◽  
Transformation Formulas ◽  
Oscillator Models

In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials.

scholarly journals Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

2021 ◽  
Author(s):  
Misha V. Feigin ◽  
Martin A. Hallnäs ◽  
Alexander P. Veselov
Keyword(s):  
Hermite Polynomials ◽  
Higher Order ◽  
Symmetric Polynomials ◽  
Invariant Extension ◽  
Cherednik Algebra ◽  
New Class ◽  
Harmonic Term ◽  
Rational Cherednik Algebra ◽  
Conceptual Explanation ◽  
Moser System

AbstractLassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $${\mathcal {A}}$$ A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $${\mathcal {A}}$$ A -Hermite polynomials. These polynomials form a linear basis in the space of $${\mathcal {A}}$$ A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type $$A_N$$ A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues.

scholarly journals Feed-Forward Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator

2006 ◽  
Vol 10 (4) ◽  
pp. 567-577
Author(s):  
Gerasimos Rigatos ◽  
Keyword(s):  
Neural Networks ◽  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Basis Functions ◽  
System Modelling ◽  
General Category ◽  
Quantum Harmonic Oscillator ◽  
Feed Forward ◽  
Change Of Scale ◽  
Feed Forward Neural Networks

The paper introduces feed-forward neural networks where the hidden units employ orthogonal Hermite polynomials for their activation functions. The proposed neural networks have some interesting properties: (i) the basis functions are invariant under the Fourier transform, subject only to a change of scale, (ii) the basis functions are the eigenstates of the quantum harmonic oscillator, and stem from the solution of Schrödinger’s diffusion equation. The proposed feed-forward neural networks belong to the general category of nonparametric estimators and can be used for function approximation, system modelling and image processing.

scholarly journals Quantization of the one-dimensional free harmonic oscillator as an example of the application of the node theorem and the Macdonald-Hylleraas-Undheim theorem

2020 ◽  
Vol 22 (1) ◽  
pp. 87-90
Author(s):  
Kunle Adegoke ◽  
A. Olatinwo
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Schrodinger Equation ◽  
Hermite Polynomials ◽  
Traditional Methods ◽  
One Dimensional ◽  
Energy Eigenvalues ◽  
The One

Using heuristic arguments alone, based on the properties of the  wavefunctions, the energy eigenvalues and the corresponding eigenfunctions of the one-dimensional harmonic oscillator are obtained. This approach is considerably simpler and is perhaps more intuitive than the traditional methods of solving a differential equation and manipulating operators. Keywords: Time-independent Schrödinger equation, MacDonald-Hylleraas-Undheim theorem, Node theorem, Hermite polynomials,  energy eigenvalues

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