Critical strength of attractive central potentials

Abstract

We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yield several sequences of upper and lower limits on the critical value, g ( ) c , of the coupling constant (strength), g, of the potential, V (r) = −gv(r), for which a first -wave bound state appears, which converges to the exact critical value.

An upper bound for asymmetrical spinless Salpeter equations

“…The total orbital angular momentum is r 1 p 0 + r 2 p 0 = r 0 p 0 . A semiclassical quantification gives thus r 0 p 0 = L + 1/2, and we obtain a system very similar to (16)- (18). Nevertheless, the AFM produces more general results: Equations are obtained not only for a circular motion; The quantum number Q is unambiguously determined by the choice of the power law auxiliary potential; A good choice of this potential can allow to obtain an upper bound; An eigenstate of H NR (7), with ν i = p 2 0 + m 2 i and ρ = K(r 0 ), is an approximation of the genuine eigenstate [29].…”

“…If the force acting on i comes from the potential V (r) generated by j, then F 1 = F 2 = V ′ (r 0 ). (20) and (21) can be recast onto the form (18), and it is obvious than (16) gives the mass of the system. The total orbital angular momentum is r 1 p 0 + r 2 p 0 = r 0 p 0 .…”

“…Sometimes, bounds yield analytical results which can give precious details about the solutions of the SSE. With some rare exceptions [9,10], most of the analytical results have been obtained for the symmetrical version of the SSE [11][12][13][14][15][16][17][18][19][20] σ p 2 + m 2 + V (r) |φ = M |φ .…”

Approximate solutions for N-body Hamiltonians with identical particles in D dimensions

“…The critical coupling constant κ c ({θ}), where {θ} stands for a set of quantum numbers, is such that the potential admits a bound state with the quantum numbers {θ} if κ > κ c ({θ}) (see for instance Refs. [28,29]).…”

Existence of mesons after deconfinement

“…We still refer the reader to Refs. [25][26][27][28][29] for methods to compute g 00 . Consequently, bound states will exist if…”