Upper limit on the number of bound states of the spinless Salpeter equation

Abstract

We obtain, using the Birman-Schwinger method, upper limits on the total number of bound states and on the number of -wave bound states of the semirelativistic spinless Salpeter equation. We also obtain a simple condition, in the ultrarelativistic case (m = 0), for the existence of at least one -wave bound states: C( , p/(p − 1))

Upper limit on the critical strength of central potentials in relativistic quantum mechanics

“…18-27͒, only two results are known for the spinless Salpeter equation. 28,29 The first result, obtained in Ref. 28, is an upper bound on the total number of bound states yielding a lower limit on the critical value, g c ͑0͒ , of the coupling constant ͑strength͒, g, for which a first S-wave ͑ᐉ =0͒ bound state appears ͑ᐉ being obviously the angular momentum͒ in the potential V͑r͒ =−gv͑r͒.…”

“…This has been done in Ref. 29 but since we need some modifications in the development, we recall the main line here. We must calculate the Green function of the kinetic energy operator.…”

The spinless relativistic Woods–Saxon problem

“…For the spinless Salpeter equation, results on the number of bound states are sparse [15,16]; an upper limit on this quantity as easy to handle as the Bargmann limit has been proved by I. Daubechies [15]: for technical reasons, let the Hamiltonian operator H = K+V acting on L 2 (R 3 ) be composed of a kinetic term K that is a positive, strictly increasing, differentiable function of |p| only that vanishes at |p| = 0 (which in the relativistic case can be effected by subtracting appropriate multiples of m) and rises beyond bounds with increasing |p| and of some potential V that is a negative smooth function of compact support, V ∈ C ∞ 0 (R 3 ), i.e., H ≡ K(|p|)+V (x) , K(|p|) ≥ 0 ,…”

Analytical solution of two-body spinless Salpeter equation for Hellmann potential

“…This equation is used when relativistic kinetic term effect is not negligible. So, it is suitable for bosons as well as the spin averaged spectra of bound states of fermions [3,4,7]. In addition, the two-body spinless Salpeter equation gives acceptable results, especially, for the bound states of deuteron, exciton and mesons [6,8].…”