The orthonormal basis for symmetric irreducible representations of O(5) × SU(1, 1) and its application to the interacting boson model

Raising and Lowering Operators for Orbital Angular Momentum Quantum Numbers

“…How to construct the shifting operators that can raise and lower the spin angular momentum quantum numbers is an interesting topic. For spin angular momentum operator S, we do not have relation N · S = 0 (if S represents spin half, N · S = cos θ sin θe −iϕ sin θe iϕ − cos θ = 0), so the relations after (10) have to be all revisited carefully and generalized.…”

“…In 1980, Szpikowski and Góźdź pointed out in passing in the Appendix A of their paper [10] an operator O in the interacting boson nuclear model can diminish both l and m in |lm with m = l as |l, l to another one |l , l , where the operator O is a polynomial of terms containing powers of L + , L − and the tensor operators T (k) where superscript k denotes the rank under rotational transformations. In 1994, with help of the tensor operators T , Shanker [9] showed that the raising and lowering operator A ± = A x ± iA y constructed form the Lenz vector operator A = (p × L − L × p)/2 − r acting on spherical hydrogen atom eigenstates |nlm happens to be A ± |nll = D ± ll |n, (l ± 1), (l ± 1) , where coefficients D ± ll are constants depending on l. Burkardt and Leventhal in 2004 demonstrated that the same relation A ± |nll = D ± ll |n, (l ± 1), (l ± 1) can be obtained without resorting to the tensor operator [1].…”

Partial dynamical symmetries

“…, λ. The multiplicity label n ∆ in the O(5) ⊃ O(3) reduction, counts the maximum number of d-boson triplets coupled to L = 0 [29]. The eigenstates |[N] n d (τ )n ∆ LM are obtained with a Hamiltonian with U(5) DS which, for one-and two-body interactions, can be transcribed in the form…”

Construction of basis vectors for symmetric irreducible representations of O(5) $ \supset$ O(3)

“…A non-trivial simplest case is to construct symmetric irreducible representations (irreps) of the O(5) group in the O(3) basis for identical d-bosons useful in the nuclear collective model [4,5] and the interacting boson model for nuclei [6]. Because of its physical importance, there have been a lot of attempts to construct the O(5) ⊃ O(3) basis vectors [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Most notably, Rowe, Hecht, and many others in a series of papers [23][24][25][26] established the vector-coherent-state (VCS) representations of O(5) ⊃ O(3) and constructed the O(5) spherical harmonics [27].…”