## What is/are Isotropic Harmonic?

Isotropic Harmonic - We considered a system of N bosons confined in a three-dimensional isotropic harmonic trap.^{[1]}Single-particle energy levels of nucleons moving in an anisotropic harmonic oscillator potential along with a spin-orbit and an orbit-orbit interaction have been calculated.

^{[2]}Using the approximation of the stepwise variation of the magnetic field, we obtain explicit formulas describing the change of the energy and magnetic moment from the initial equilibrium state, created with the aid of a weak anisotropic harmonic potential.

^{[3]}We work out the cases of anisotropic harmonic potential and the Hall potential as physical applications of the proposed method.

^{[4]}It is described by the isotropic harmonic oscillator Hamiltonian supplemented by a Zeeman type term with a real coupling constant g.

^{[5]}These monopoles also appear during the examination of the duality between isotropic harmonic oscillators and the MICZ-Kepler problems.

^{[6]}Subsequently, the consequences of the use of the hw irreps on the binding energies and two-neutron separation energies in the rare earth region are discussed within the proxy-SU(3) scheme, by considering a very simple Hamiltonian, containing only thethree dimensional (3D) isotropic harmonic oscillator (HO) term and the quadrupole-quadrupole interaction.

^{[7]}We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential.

^{[8]}In this study, we investigate the effects of noncommutative Quantum Mechanics in three dimensions on the energy-levels of a charged isotropic harmonic oscillator in the presence of a uniform magnetic field in the z-direction.

^{[9]}We present the isotropic harmonic signal of a gold mirror of 0.

^{[10]}Exact eigenstate solutions of Schrodinger equation for a generalized oscillator system that includes, as special cases; ring shaped oscillator potential and isotropic harmonic oscillator potential are examined in an analytical treatment by using extended Nikiforov Uvarov method.

^{[11]}The spacetime control action is induced from harmonic fields called nonlinear anisotropic harmonic fields that are defined on Machian state space-time.

^{[12]}By generalizing our automated algebra approach from homogeneous space to harmonically trapped systems, we have calculated the fourthand fifth-order virial coefficients of universal spin-1/2 fermions in the unitary limit, confined in an isotropic harmonic potential.

^{[13]}We have studied the structural properties of the ground state of different two-electron systems under isotropic harmonic confinement (IHC).

^{[14]}Solution method: Having diagonalized the matrix corresponding to the folded-Yukawa single-nucleon Hamiltonian expressed in the basis of the anisotropic harmonic oscillator basis, one then calculates the matrix elements of the one-body angular-momentum as well as the quadrupole and octupole-moments operators.

^{[15]}We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap.

^{[16]}In this work we present a systematic study of the three-dimensional extension of the ring dark soliton, examining its existence, stability, and dynamics in isotropic harmonically trapped Bose-Einst.

^{[17]}The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems.

^{[18]}As examples, the systems of the Three-dimensional Infinite Square Well, the Three-dimensional Harmonic Oscillator and the Two-dimensional Isotropic Harmonic Oscillator are solved.

^{[19]}The underlying symmetries are respectively the spherical symmetry of the 3D isotropic harmonic oscillator and the cylindrical symmetry of the 3D anisotropic harmonic oscillator with two frequencies equal.

^{[20]}Eigenspaces of the quantum isotropic Harmonic Oscillator on have extremally high multiplicites and the eigenspace projections have special asymptotic properties.

^{[21]}Various well-known statistical measures like López-Ruiz, Mancini, Calbet (LMC) and Fisher–Shannon complexity have been explored for confined isotropic harmonic oscillator (CHO) in composite position (r) and momentum (p) spaces.

^{[22]}Regarding an anisotropic Harmonic Oscillator potential we show that the corresponding Wheeler de Witt wave functional of the system is described by Hermit polynomials.

^{[23]}We have presented the single, particle spectrum for a particle in a mean-field of an isotropic harmonic oscillator with l_(→)· s_(→) coupling based on our semiclassical approach.

^{[24]}We study a two-dimensional isotropic harmonic oscillator with a hard-wall confining potential in the form of a circular cavity defined by the radial coordinate $\rho_0$.

^{[25]}We study the dynamics of a quantum or classical particle in a two-dimensional rotating anisotropic harmonic potential.

^{[26]}Here, we consider the classical diamagnetism of a charged particle in an isotropic harmonic potential which follows from the four famous spectra of random classical radiation.

^{[27]}For Bose gases confined in three- and two-dimensional isotropic harmonic potentials, we obtain the higher-order corrections to the usual results of the critical temperatures.

^{[28]}The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal potential and a combination thereof.

^{[29]}In this paper, the system dealt with consisting of an ultra-cold neutral spin-polarized Fermi gas undergoing rotation (or in the so-called synthetic magnetic field) trapped by an anisotropic harmonic potential in a two and three-dimensional space at zero temperature.

^{[30]}Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.

^{[31]}We shall find the dimension of the spaces of holomorphic sections and holomorphic differentials of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps from the sphere and torus to complex projective spaces.

^{[32]}, the fractional Lotka biochemical oscillator model and the fractional isotropic harmonic oscillator model, are discussed to illustrate the results and methods.

^{[33]}We calculate the shape Renyi and generalized Renyi complexity of a noncommutative anisotropic harmonic oscillator in a homogeneous magnetic field.

^{[34]}

## Dimensional Isotropic Harmonic

We considered a system of N bosons confined in a three-dimensional isotropic harmonic trap.^{[1]}We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap.

^{[2]}As examples, the systems of the Three-dimensional Infinite Square Well, the Three-dimensional Harmonic Oscillator and the Two-dimensional Isotropic Harmonic Oscillator are solved.

^{[3]}We study a two-dimensional isotropic harmonic oscillator with a hard-wall confining potential in the form of a circular cavity defined by the radial coordinate $\rho_0$.

^{[4]}For Bose gases confined in three- and two-dimensional isotropic harmonic potentials, we obtain the higher-order corrections to the usual results of the critical temperatures.

^{[5]}

## Fractional Isotropic Harmonic

Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.^{[1]}, the fractional Lotka biochemical oscillator model and the fractional isotropic harmonic oscillator model, are discussed to illustrate the results and methods.

^{[2]}

## isotropic harmonic oscillator

Single-particle energy levels of nucleons moving in an anisotropic harmonic oscillator potential along with a spin-orbit and an orbit-orbit interaction have been calculated.^{[1]}It is described by the isotropic harmonic oscillator Hamiltonian supplemented by a Zeeman type term with a real coupling constant g.

^{[2]}These monopoles also appear during the examination of the duality between isotropic harmonic oscillators and the MICZ-Kepler problems.

^{[3]}Subsequently, the consequences of the use of the hw irreps on the binding energies and two-neutron separation energies in the rare earth region are discussed within the proxy-SU(3) scheme, by considering a very simple Hamiltonian, containing only thethree dimensional (3D) isotropic harmonic oscillator (HO) term and the quadrupole-quadrupole interaction.

^{[4]}We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential.

^{[5]}In this study, we investigate the effects of noncommutative Quantum Mechanics in three dimensions on the energy-levels of a charged isotropic harmonic oscillator in the presence of a uniform magnetic field in the z-direction.

^{[6]}Exact eigenstate solutions of Schrodinger equation for a generalized oscillator system that includes, as special cases; ring shaped oscillator potential and isotropic harmonic oscillator potential are examined in an analytical treatment by using extended Nikiforov Uvarov method.

^{[7]}Solution method: Having diagonalized the matrix corresponding to the folded-Yukawa single-nucleon Hamiltonian expressed in the basis of the anisotropic harmonic oscillator basis, one then calculates the matrix elements of the one-body angular-momentum as well as the quadrupole and octupole-moments operators.

^{[8]}The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems.

^{[9]}As examples, the systems of the Three-dimensional Infinite Square Well, the Three-dimensional Harmonic Oscillator and the Two-dimensional Isotropic Harmonic Oscillator are solved.

^{[10]}The underlying symmetries are respectively the spherical symmetry of the 3D isotropic harmonic oscillator and the cylindrical symmetry of the 3D anisotropic harmonic oscillator with two frequencies equal.

^{[11]}Eigenspaces of the quantum isotropic Harmonic Oscillator on have extremally high multiplicites and the eigenspace projections have special asymptotic properties.

^{[12]}Various well-known statistical measures like López-Ruiz, Mancini, Calbet (LMC) and Fisher–Shannon complexity have been explored for confined isotropic harmonic oscillator (CHO) in composite position (r) and momentum (p) spaces.

^{[13]}Regarding an anisotropic Harmonic Oscillator potential we show that the corresponding Wheeler de Witt wave functional of the system is described by Hermit polynomials.

^{[14]}We have presented the single, particle spectrum for a particle in a mean-field of an isotropic harmonic oscillator with l_(→)· s_(→) coupling based on our semiclassical approach.

^{[15]}We study a two-dimensional isotropic harmonic oscillator with a hard-wall confining potential in the form of a circular cavity defined by the radial coordinate $\rho_0$.

^{[16]}The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal potential and a combination thereof.

^{[17]}Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.

^{[18]}, the fractional Lotka biochemical oscillator model and the fractional isotropic harmonic oscillator model, are discussed to illustrate the results and methods.

^{[19]}We calculate the shape Renyi and generalized Renyi complexity of a noncommutative anisotropic harmonic oscillator in a homogeneous magnetic field.

^{[20]}

## isotropic harmonic potential

Using the approximation of the stepwise variation of the magnetic field, we obtain explicit formulas describing the change of the energy and magnetic moment from the initial equilibrium state, created with the aid of a weak anisotropic harmonic potential.^{[1]}We work out the cases of anisotropic harmonic potential and the Hall potential as physical applications of the proposed method.

^{[2]}By generalizing our automated algebra approach from homogeneous space to harmonically trapped systems, we have calculated the fourthand fifth-order virial coefficients of universal spin-1/2 fermions in the unitary limit, confined in an isotropic harmonic potential.

^{[3]}We study the dynamics of a quantum or classical particle in a two-dimensional rotating anisotropic harmonic potential.

^{[4]}Here, we consider the classical diamagnetism of a charged particle in an isotropic harmonic potential which follows from the four famous spectra of random classical radiation.

^{[5]}For Bose gases confined in three- and two-dimensional isotropic harmonic potentials, we obtain the higher-order corrections to the usual results of the critical temperatures.

^{[6]}In this paper, the system dealt with consisting of an ultra-cold neutral spin-polarized Fermi gas undergoing rotation (or in the so-called synthetic magnetic field) trapped by an anisotropic harmonic potential in a two and three-dimensional space at zero temperature.

^{[7]}

## isotropic harmonic trap

We considered a system of N bosons confined in a three-dimensional isotropic harmonic trap.^{[1]}We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap.

^{[2]}