An alternative and somewhat systematic definition of the vector spherical harmonics, in analogy with the commonly used scalar spherical harmonics, is presented. The new set of vector spherical harmonics satisfies the properties of orthogonality and completeness. and is compared with other existing definitions of vector spherical harmonics. Some applications to problems in magnetostatics are illustrated.

A variational analysis of the spiked harmonic oscillator Hamiltonian operator -d 2 1dx 2 +x 2 + 1(1 + 1)/x 2 +A lxi-a, where a is areal positive parameter, is reported in this work. The formalism makes use of the functional space spanned by the solutions of the SchrOdinger equation for the linear harmonic oscillator Hamiltonian supplemented by a Dirichlet boundary condition, and a standard procedure for diagonalizing symmetric matrices. The eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system, valid for positive and relatively large values of the coupling parameter A. Additionally, a large coupling perturbative expansion is carried out and the contributions up to fourth-order to the ground state energy are explicitly evaluated. Numerical results are compared for the special case a = 5/2.

A set of vector spherical harmonics which was earlier introduced and shown to be applicable to magnetostatics problems, is employed to decompose and solve Maxwell equations. Expressions for the radiation fields for the transverse electric and magnetic modes, the angular distribution of multipole radiation, and the total power radiated in terms of the set of vector spherical harmonics are also derived. The interesting and important Rayleigh scattering problem of electromagnetic plane waves scattering by a conducting sphere is discussed in some detail within the approach. Numerous other problems particularly well adapted for the vector spherical harmonic approach advocated in the paper are suggested.

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Using a power series expansion and differential calculus, the normal derivative of the magnitude of the static electric field near the surface of a curved conductor is obtained. This provides an alternative way of solving a problem which is usually treated by an application of Gauss’s theorem.