Highest weight irreducible representations favored by nuclear forces within SU(3)-symmetric fermionic systems

The consequences of the attractive, short-range nucleon-nucleon (NN) interaction on the wave functions of nuclear models bearing the SU(3) symmetry are reviewed. The NN interaction favors the most symmetric spatial SU(3) irreducible representation (irrep), which corresponds to the maximal spatial overlap among the fermions. The consideration of the highest weight (hw) irreps in nuclei and in alkali metal clusters, leads to the prediction of a prolate to oblate shape transition beyond the mid–shell region. 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. This Hamiltonian conserves the SU(3) symmetry and treats the nucleus as a rigid rotator.

Separability of the Planar 1/ρ2 Potential in Multiple Coordinate Systems

With a number of special Hamiltonians, solutions of the Schrödinger equation may be found by separation of variables in more than one coordinate system. The class of potentials involved includes a number of important examples, including the isotropic harmonic oscillator and the Coulomb potential. Multiply separable Hamiltonians exhibit a number of interesting features, including “accidental” degeneracies in their bound state spectra and often classical bound state orbits that always close. We examine another potential, for which the Schrödinger equation is separable in both cylindrical and parabolic coordinates: A z-independent V∝1/ρ2=1/(x2+y2) in three dimensions. All the persistent, bound classical orbits in this potential close, because all other orbits with negative energies fall to the center at ρ=0. When separated in parabolic coordinates, the Schrödinger equation splits into three individual equations, two of which are equivalent to the radial equation in a Coulomb potential—one equation with an attractive potential, the other with an equally strong repulsive potential.

Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

Hamiltonian mechanics plays an important role in the development of nonlinear science. This paper aims for a fractional Hamiltonian system of variable order. Several issues are discussed, including differential equation of motion, Noether symmetry, and perturbation to Noether symmetry. As a result, fractional Hamiltonian mechanics of variable order are established, and conserved quantity and adiabatic invariant are presented. 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.