Phase Shifts and Resonances in the Dirac Equation

Abstract

We review the analytic results for the phase shifts δ l (k) in non-relativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the non-relativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E = m. Resonances resulting from scattering from a barrier can be explained in terms of the 'crossing' theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extract phase shifts for general short range potentials is introduced and illustrated by considering relativistic scattering from a Gaussian potential well and barrier.

CalculableR-matrix method for the Dirac equation

“…General results are given in Ref. [23], where Eq. (62) for the s-wave particular case however contains several misprints.…”

“…These restrictions can both be overcome with some extra effort. In this case, the components of the external solution at energy E are approximated by the asymptotic form of the solutions [23],…”

“…As an example, I take a particle with mass m = 1 (in unitsh = c = 1) in a well with radius a = 1 and depth V 0 = 4 like in Ref. [23]. Results at various energies are displayed in Table 1.…”

Resonant states in an attractive one-dimensional cusp potential

“…4 shows the behavior of the phase shift δ as a function of the real part of the resonant energy, it can be observed that the phase shift increases surpassing the value π/2 and after reaching a maximum, it monotonically decreases, showing in this way a behavior that has been observed in square well case. [15] …”

Quasi-exact solution to the Dirac equation for the hyperbolic-secant potential

“…Analytic solutions have been obtained for the the square well [2,3], Woods-Saxon potential [4], cusp potential [5], and Hulthén potential [6] as well as asymmetric barriers [7,8], multiple barriers [9], and a class of short-range potentials [10]. The successful isolation of graphene [11] has led to renewed interest in the transmission-reflection problem for the one-dimensional Dirac equation.…”