Abstract
Observing renewed interest in long-standing (semi-) relativistic descriptions of bound states, we would like to make a few comments on the eigenvalue problem posed by the spinless Salpeter equation and, illustrated by the examples of the nonsingular Woods-Saxon potential and the singular Hulth\'en potential, recall elementary tools that practitioners looking for analytic albeit approximate solutions might find useful in their quest.
Highlights
Recent years have witnessed a rise of attempts to study bound states by relativistic equations of motion, such as the Klein–Gordon equation, the Dirac equation, or the spinless Salpeter equation, with all its merits and drawbacks, derived by nonrelativistic reduction of the Bethe–Salpeter equation [6, 7]
For two particles of equal masses, m, and relative momentum p, interacting via a potential V(x) depending on their relative coordinate, x, the spinless Salpeter equation may be regarded as the eigenvalue equation of the nonlocal Hamiltonian
In contrast to the Coulomb potential VC(r) ≡ −κ/r, κ > 0, lots of rather popular potentials admit only a finite number, N, of bound states: this number is a crucial characteristic of bound-state problems
Summary
Recent years have witnessed a rise of attempts to study bound states by (semi-) relativistic equations of motion, such as the Klein–Gordon equation, the Dirac equation, or (as a straightforward generalization of the Schrödinger equation) the spinless Salpeter equation, with all its merits and drawbacks (consult, for instance, Refs. [1, 2] for details), derived by nonrelativistic reduction (cf., for instance, Refs. [3,4,5]) of the Bethe–Salpeter equation [6, 7]. Recent years have witnessed a rise of attempts to study bound states by (semi-) relativistic equations of motion, such as the Klein–Gordon equation, the Dirac equation, or (as a straightforward generalization of the Schrödinger equation) the spinless Salpeter equation, with all its merits and drawbacks For two particles of (just for notational simplicity) equal masses, m, and relative momentum p, interacting via a potential V(x) depending on their relative coordinate, x, the spinless Salpeter equation may be regarded as the eigenvalue equation of the nonlocal Hamiltonian. In view of the interest noted, we revisit this equation for central potentials V(x) = V(r), r ≡ |x|, by recalling (and exploiting) a couple of well-known results.
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