The spinless relativistic Woods–Saxon problem

Abstract

Motivated by the observation of a recent renewal of rather strong interest in the description of bound states by (semi-)relativistic equations of motion, we revisit, for the example of the Woods-Saxon interactions, the eigenvalue problem posed by the spinless Salpeter equation and recall various elementary knowledge, considerations, and techniques that practitioners seeking solutions to this specific reduction of the Bethe-Salpeter equation may find helpful.

The spinless relativistic Yukawa problem

“…. , d−1, which allows us to localize eigenvalues by upper bounds of increasing tightness [13,14] for rising d. We find and have always found [15][16][17][18][19][20][21][22][23][24] it advantageous to span these finite-dimensional variational trial subspaces by a basis the representations of which are known analytically in both configuration and momentum space (related, of course, by Fourier transformation): in this case, the expectation values of H may be analytically given by evaluating those of T (p) in momentum space and those of V (x) in configuration space. By spherical symmetry, each basis vector factorizes into the product of a radial part and a spherical harmonic Y ℓm (Ω) for angular momentum ℓ and projection m depending on the solid angle Ω.…”

The spinless relativistic Hellmann problem

“…Consequently, this latter finding is straightforwardly applicable to the Woods-Saxon potential [6] or the kink-like potential [10] but not to the Hellmann potential (2) since…”

“…Fourier transformation immediately provides the momentum-space representation of these basis states; explicit expressions of these functions can be found in, e.g., Refs. [6,8,9,20,23]. Table 1 illustrates the application of the variational technique recalled above to spinless relativistic Hellmann problems by presenting the set of upper limits on the binding energies…”

“…At present, we are aware of just two attempts [28,29] to exploit for that purpose a longstanding but very approximate way of dealing with spinless Salpeter equations. Unfortunately, from our point of view this latter approach is very problematic in (at least) two well-known respects: On the one hand, as a first step towards treatability, the Table 1: Variational upper bounds, in units of the identical mass, m, of the two bound-state constituents, on the binding energies (8) of the low-lying bound states of generic relativistic Hellmann problems (identifying any bound state by radial quantum number n r and orbital angular momentum quantum number ℓ), derived by relying on the trial-space basis (7) and employing, as numerical input values, for the two variational parameters β = 1 and µ = m, for the Yukawa range parameter b = m, and three illustrative combinations of the coupling strengths: κ = υ = The pseudo-spinless-Salpeter Hamiltonian emerging from such mistreatment, however, can be shown [6] to be an operator unbounded from below for all interaction potentials that are not too singular at the spatial origin, including all generalized Hellmann potentials. On the other hand, as a second step, the formulation of an equation of Schrödinger type is enforced by reinsertion of that eigenvalue equation of the pseudo-spinless-Salpeter Hamiltonian that holds for its lower-order truncation.…”

“…Needless to say, upon having a set of solutions at one's disposal, its significance can be easily examined by various really powerful tools, for instance, the generalization of the virial theorem of nonrelativistic quantum theory to the incorporation of relativistic kinematics [4,5]. Utilizing techniques of this kind has proven extremely efficient in the separation of the wheat from the chaff [6][7][8][9][10].…”