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 Symmetry algebra of the planar anisotropic quantum harmonic oscillator with rational ratio of frequencies

2020 ◽  
Vol 4 ◽  
pp. 153
Author(s):  
Dennis Bonatsos ◽  
C. Daskaloyannis ◽  
P. Kolokotronis ◽  
D. Lenis
Keyword(s):  
Harmonic Oscillator ◽  
Linear Extension ◽  
Symmetry Algebra ◽  
Quantum Harmonic Oscillator ◽  
Dimensional Representation ◽  
General Applicability ◽  
Finite Dimensional Representation ◽  
Energy Eigenvalues ◽  
Finite Dimensional ◽  
Superintegrable Systems

The symmetry algebra of the two-dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are de- termined using algebraic methods of general applicability to quantum superintegrable systems.

scholarly journals Nonlinear extension of the u(3) algebra as the symmetry algebra of the three-dimensional anisotropic quantum harmonic oscillator with rational ratios of frequencies and the Nilsson model

2020 ◽  
Vol 5 ◽  
pp. 14
Author(s):  
Dennis Bonatsos ◽  
C. Daskaloyannis ◽  
P. Kolokotronis ◽  
D. Lenis
Keyword(s):  
Three Dimensional ◽  
Symmetry Algebra ◽  
Atomic Clusters ◽  
Angular Momentum Operator ◽  
Quantum Harmonic Oscillator ◽  
General Applicability ◽  
Superintegrable Systems ◽  
Present Formalism ◽  
Nilsson Model ◽  
Special Case

The symmetry algebra of the N-dimensional anisotropic quantum har- monic oscillator with rational ratios of frequencies is constructed by a method of general applicability to quantum superintegrable systems. The special case of the 3-dim oscillator is studied in more detail, because of its relevance in the description of superdeformed nuclei and nuclear and atomic clusters. In this case the symmetry algebra turns out to be a nonlinear extension of the u(3) algebra. A generalized angular momentum operator useful for labeling the degenerate states is constructed, clarifying the connection of the present formalism to the Nilsson model.

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.

MARTINGALE CHARACTERIZATIONS OF INCREMENT PROCESSES IN A LOCALLY COMPACT GROUP

2003 ◽  
Vol 06 (04) ◽  
pp. 563-595 ◽  
Author(s):  
HERBERT HEYER ◽  
GYULA PAP
Keyword(s):  
Compact Group ◽  
Martingale Problem ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Fourier Theory ◽  
Group A ◽  
Tensor Square ◽  
Special Case

Martingale characterizations and the related martingale problem are studied for processes with independent (not necessarily stationary) increments in an arbitrary locally compact group. In the special case of a compact Lie group, a Lévy-type characterization is given in terms of a faithful finite dimensional representation of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups associated with the increment processes.

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.

STUDY OF A RELATIVISTIC QUANTUM HARMONIC OSCILLATOR BY THE METHOD OF STATE-DEPENDENT DIAGONALIZATION

1996 ◽  
Vol 07 (05) ◽  
pp. 645-653
Author(s):  
H. C. LEE ◽  
K. L. LIU ◽  
C. F. LO
Keyword(s):  
Harmonic Oscillator ◽  
Relativistic Quantum ◽  
Quantum Harmonic Oscillator ◽  
Relativistic Limit ◽  
New Approach ◽  
Energy Eigenvalues ◽  
State Dependent ◽  
Anharmonic Potential ◽  
Relativistic Parameter ◽  
Special Case

We apply the method of State-dependent Diagonalization to study the eigenstates of the relativistic quantum harmonic oscillator in the low relativistic limit. The relativistic corrections of the energy eigenvalues of the quantum harmonic oscillator are evaluated for different values of the relativistic parameter α ≡ ħω0 / m0c2. Unlike the conventional exact diagonalization, this new method is shown to be very efficient for evaluating the energy eigenvalues and eigenfunctions. We have also found that for non-zero α the eigenfunctions of the system become more localized in space and that the ground state of the SHO (i.e., the α = 0 case) turns into a squeezed state. Furthermore, since our system is a special case of the quantum harmonic oscillator with a velocity-dependent anharmonic potential, this new approach should be very useful for investigating the cases with more complicated velocity-dependent anharmonic potentials.

Neural Structures Using the Eigenstates of a Quantum Harmonic Oscillator

2006 ◽  
Vol 13 (01) ◽  
pp. 27-41 ◽  
Author(s):  
Gerasimos Rigatos ◽  
Spyros Tzafestas
Keyword(s):  
Neural Networks ◽  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Feedforward Neural Networks ◽  
Basis Functions ◽  
System Modelling ◽  
Quantum Harmonic Oscillator ◽  
Wave Nature ◽  
Change Of Scale ◽  
The Fourier Transform

The main result of the paper is the use of orthogonal Hermite polynomials as the basis functions of feedforward neural networks. 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 demonstrate the particle-wave nature of information and can be used in nonparametric estimation. Possible applications of the proposed neural networks include function approximation, image processing and system modelling.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

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