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In this letter, based on Hamiltonian identity we propose a generalized expression of the second hypervirial for an arbitrary central potential wave functions in higher dimensions D and then present a useful general recurrence relation among the off-diagonal multipolar matrix elements. We show that this new proposed recurrence formula is very powerful in deriving the recurrence relations among the hydrogenic matrix elements. Some important recurrence relations can be simply obtained. It is interesting to find that the selection rule is independent of the central potential V(r) for f = r^k with k = 0, 2 and that the diagonal matrix elements of V′(r) are independent of the system eigenvalues for k = 1. The application of the general recurrence relation to the isotropic harmonic oscillator case is discussed briefly.

At least one neutrino has a mass of about 50 meV or larger. However, the absolute mass scale for the neutrino remains unknown. Studies of double beta decay offer hope for determining the absolute mass scale. Furthermore, the critical question "Is the neutrino its own antiparticle?" is unanswered. In particular, zero-neutrino double beta decay (0 νββ) can address the issues of lepton number conservation, the particle–antiparticle nature of the neutrino, and its mass. A summary of the recent progress in 0 νββ, and the related technologies will be discussed in the context of the future 0 νββ program.

The exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained in two dimensions. We construct the ladder operators directly from the radial wave functions and find that these operators satisfy the commutation relations of an SU(1, 1) group. We obtain the explicit expressions of the matrix elements for some related functions ρ and with ρ = r². We also explore another symmetry between the eigenvalues E(r) and E(ir) by substituting r→ir.

The eigenvalues and eigenfunctions of the Schrödinger equation with a non-relativistic electron in a uniform magnetic field are presented. A realization of the creation and annihilation operators for the radial wave-functions is carried out. It is shown that these operators satisfy the commutation relations of an SU(1,1) group. Closed analytical expressions are evaluated for the matrix elements of different functions ρ² and .

The tensor properties of the $Sp(12,R)$ algebra generators are determined in respect to the reduction chain $Sp(12,R)\supset U(6)\supset U_p(3)\otimes U_n(3)\supset U(3)\supset SO(3)$, which defines a shell-model coupling scheme of the proton–neutron symplectic model (PNSM). They are further used to calculate the matrix elements of the basic $Sp(12,R)$ operators of the PNSM in the space of fully symmetric representations in the $U(6)$-coupled basis using a generalized Wigner–Eckart theorem. The obtained results allow further the matrix elements of any physical operator of interest, such as the relevant transition operators or the collective potential, to be calculated. As an illustration, the matrix elements of the basic irreducible tensor terms which appear in the $U(6)$ decomposition of the long-range full major-shell mixing proton–neutron quadrupole–quadrupole interaction are presented.

A method for computing integrals of polynomial functions on compact symmetric spaces is given. Those integrals are expressed as sums of functions on symmetric groups.

The differential equation for the product of Legendre polynomials provides a recursive algorithm for calculation succeeding multipole matrix elements with them. Non-multipole matrix elements as, for instance, particular Clebsh-Gordon coefficients also can be computed on the basis of this algorithm.