Relativistic Comparison Theorems

Comparison theorems are established for the Dirac and Klein--Gordon equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive central potentials in d dimensions that support discrete Dirac eigenvalues E^{(1)}_{k_d\nu} and E^{(2)}_{k_d\nu}. We prove that if V^{(1)}(r) \leq V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered E^{(1)}_{k_d\nu} \leq E^{(2)}_{k_d\nu}. This result generalizes an earlier more restrictive theorem that required the wave functions to be node free. For the the Klein--Gordon equation, similar reasoning also leads to a comparison theorem provided in this case that the potentials are negative and the eigenvalues are positive.Comment: 6 page

Solutions to the 1d Klein-Gordon equation with cutoff Coulomb potentials

In a recent paper by Barton (J. Phys. A40, 1011 (2007)), the 1-dimensional Klein-Gordon equation was solved analytically for the non-singular Coulomb-like potential V_1(|x|) = -\alpha/(|x|+a). In the present paper, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cutoff Coulomb potential V_2(|x|) = -\alpha/|x|, ~|x| > a, and otherwise V_2(|x|) = -\alpha/a.Comment: 8 pages, 4 figure

Semirelativistic stability of N-boson systems bound by 1/r pair potentials

We analyze a system of self-gravitating identical bosons by means of a semirelativistic Hamiltonian comprising the relativistic kinetic energies of the involved particles and added (instantaneous) Newtonian gravitational pair potentials. With the help of an improved lower bound to the bottom of the spectrum of this Hamiltonian, we are able to enlarge the known region for relativistic stability for such boson systems against gravitational collapse and to sharpen the predictions for their maximum stable mass.Comment: 11 pages, considerably enlarged introduction and motivation, remainder of the paper unchange

Eigenvalue bounds for a class of singular potentials in N dimensions

The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963] for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to N-dimensions. In particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential V(x) = x^2 + lambda/x^alpha, alpha > 0, lambda > 0, and is valid for all discrete eigenvalues, arbitrary angular momentum ell, and spatial dimension N.Comment: 10 pages (plain tex with 2 ps figures). J.Phys.A:Math.Gen.(In Press

Supersymmetric analysis for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions

A supersymmetric analysis is presented for the d-dimensional Dirac equation with central potentials under spin-symmetric (S(r) = V(r)) and pseudo-spin-symmetric (S(r) = - V(r)) regimes. We construct the explicit shift operators that are required to factorize the Dirac Hamiltonian with the Kratzer potential. Exact solutions are provided for both the Coulomb and Kratzer potentials.Comment: 12 page

Positron Tunnelling through the Coulomb Barrier of Superheavy Nuclei

We study beams of medium-energy electrons and positrons which obey the Dirac equation and scatter from nuclei with $Z > 100.$ At small distances the potential is modelled to be that of a charged sphere. A large peak is found in the probability of positron penetration to the origin for $Z \approx 184.$ This may be understood as an example of Klein tunnelling through the Coulomb barrier: it is the analogue of the Klein Paradox for the Coulomb potential.Comment: 3 figures, to be published in Physics Letters

Asymptotic iteration method for eigenvalue problems

An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y'' = lambda(x) y' + s(x) y is introduced, where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to Schroedinger type problems, including some with highly singular potentials, are presented.Comment: 14 page

Study of a class of non-polynomial oscillator potentials

We develop a variational method to obtain accurate bounds for the eigenenergies of H = -Delta + V in arbitrary dimensions N>1, where V(r) is the nonpolynomial oscillator potential V(r) = r^2 + lambda r^2/(1+gr^2), lambda in (-infinity,\infinity), g>0. The variational bounds are compared with results previously obtained in the literature. An infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues.Comment: 16 page

Spectral characteristics for a spherically confined -1/r + br^2 potential

We consider the analytical properties of the eigenspectrum generated by a class of central potentials given by V(r) = -a/r + br^2, b>0. In particular, scaling, monotonicity, and energy bounds are discussed. The potential $V(r)$ is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical boundary of radius R. With the aid of the asymptotic iteration method, several exact analytic results are obtained which exhibit the parametric dependence of energy on a, b, and R, under certain constraints. More general spectral characteristics are identified by use of a combination of analytical properties and accurate numerical calculations of the energies, obtained by both the generalized pseudo-spectral method, and the asymptotic iteration method. The experimental significance of the results for both the free and confined potential V(r) cases are discussed.Comment: 16 pages, 4 figure

Gravitating semirelativistic N-boson systems

Analytic energy bounds for N-boson systems governed by semirelativistic Hamiltonians of the form H=\sum_{i=1}^N(p_i^2 + m^2)^{1/2} - sum_{1=i<j}^N v/r_{ij}, with v>0, are derived by use of Jacobi relative coordinates. For gravity v=c/N, these bounds are substantially tighter than earlier bounds and they are shown to coincide with known results in the nonrelativistic limit.Comment: 7 pages, 2 figures It is now proved that the reduced Hamiltonian is bounded below by the simple N/2 Hamiltonia