ON THE ANALYSIS OF THE EIGENVALUES OF THE DIRAC EQUATION WITH A 1/r POTENTIAL IN D DIMENSIONS

The Dirac equation is investigated in D+1 dimensional space-time. The radial equations of this quantum system are obtained and solved exactly by the confluent hypergeometric equation approach. The energy levels E(n,l,D) are analytically presented. For the continuous dimension D as proposed by Nieto, the dependences of the energy difference ΔE(n,l,D) for D and D-1 on the dimension D are demonstrated as three different kinds of change rules. The dependences of the energy E(n,l,D) on the dimension D are also discussed. It is found that the energies E(n,l,D) (l≠0) are almost independent of the quantum number l for a large D, while E(n,0,D) first decreases and then increases with the increasing dimension D. The dependences of the energies E(n,l,ξ) on the potential strength ξ are also studied for the given dimension D=3. We find that the energies E(n,l,ξ) decrease with ξ≤l+1.

On the dispersion relation for trapped internal waves

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp = 1/CpO + k/ωmax + ∈(k) where cp is the wave phase speed for a particular mode, CpO is the phase speed at k = 0, ωmax is the maximum possible wave angular frequency and ωmax ≤ Nmax where Nmax is the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) = o(k) for k large, and is bounded for finite k. In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp ≈ (∫∞0N2(y)ydy)-½ + k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functions N2(x) which is taken to be a class of realvalued functions of a real variable x where O ≤ x ∞ such that N2(x) = O(e-βx) as x → ∞ and 1/β is an arbitrary length scale. It is demonstrated that N2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax = Nmax. More generally, it is demonstrated that when k → ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that if N2(x) is O(e-βx) as x → ∞ and has a maximum value N2max then a sufficient condition for 1/cp ∼ k/Nmax to hold for large k for the lowest mode is that N2(t)/t is convex for O ≤ t ≤ 1 where t = e-βx.