In the solution of boundary value problems, usually zero
eigenvalue is ignored. This case also happens in calculating the
eigenvalues of matrices, so that we would often like to find the
nonzero solutions of the linear system AX = λX when λ ≠ 0. But λ = 0 implies that detA = 0 for X ≠ 0 and then the rank of matrix A is reduced at least one degree.
This comment can similarly be stated for boundary value problems.
In other words, if at least one of the eigens of equations related
to the main problem is considered zero, then one of the solutions
will be specified in advance. By using this note, first we study a
class of special functions and then apply it for the potential,
heat, and wave equations in spherical coordinate. In this way,
some practical examples are also given.
A two and one-half dimensional (21/2-D) particle simulation study showing the resolution of the zero-frequency modes into finite-frequency modes is presented. The resolution of the electrostatic mode at a nonzero frequency indicates a strong coupling between the longitudinal electric field fluctuation and the magnetic field fluctuation.
The properties of those states which can be obtained by exciting a single particle from the ground state of a nucleus are studied by using a perturbation-theory expansion for the Green's function. The formula obtained differs from the one used in a straightforward shall model calculation in some important respects. By comparing this formula with the condition for a solution of the Hartree-Fock equations to give a minimum of the expectation value of the energy, derived in an earlier paper, it is shown that it is always possible to choose an independent particle wave function which makes all collective modes stable in the random phase approximation. A study of the formal properties of solutions of the equation shows that the wave functions should be normalized using an indefinite metric: different eigenfunctions are then orthogonal. The eigenfunctions form a complete set, if there is no degeneracy in the solution of the Hartree-Fock equations, but not if there is a degeneracy. An upper bound for the lowest energy level is established, and the rule for calculating matrix elements of one-particle operators between the ground state and an excited state is given. It is shown to follow from the self-consistent field condition that the energy-weighted sum rules are preserved in this approximation. A discussion of spurious states is given, and it is shown that they separate out and have zero energy. A vector introduced in our earlier discussion of the cranking model is shown to be ind
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