New classes of potentials for which the radial Schrödinger equation can be solved at zero energy

Abstract

In a previous paper 1 , submitted to Journal of Physics A -we presented an infinite class of potentials for which the radial Schrödinger equation at zero energy can be solved explicitely. For part of them, the angular momentum must be zero, but for the other part (also infinite), one can have any angular momentum. In the present paper, we study a simple subclass (also infinite) of the whole class for which the solution of the Schrödinger equation is simpler than in the general case. This subclass is obtained by combining another approach together with the general approach of the previous paper. Once this is achieved, one can then see that one can in fact combine the two approaches in full generality, and obtain a much larger class of potentials than the class found in ref. 1 We mention here that our results are explicit, and when exhibited, one can check in a straightforward manner their validity.In a recent paper, 1 we showed how, starting from two regular potentials for which the radial Schrödinger equation can be solved explicitly at zero energy -there are many of them 1,2 -one can construct explicitely an infinite number of potentials for which one can solve again, explicitely, the radial equation at zero energy. The solutions for these new potentials are given very simply in terms of the solutions of the two initial potentials. By construction, it is then seen that the method can be applied without any modifications to potentials which are singular (repulsive) at the origin, or are long range (Coulomb, etc), or are even confining, like λr 2 , λ > 0, etc.According to the case, one can include, as well, the angular momentum potential ℓ(ℓ + 1)/r 2 . Many examples, both for regular and singular potentials, covering all cases, except the confining potentials, are given in ref. 1 , with explicit solutions of the radial equation.

Some remarks on effective range formulae in potential scattering

“…Because of (7), ϕ 0 obviously satisfies (8). If we have r 2 V (r) ∈ L 1 (1, ∞), then ϕ 0 = ψ 0 + αχ 0 , and because of (9), ϕ 0 satisfies also (10). The solution ψ 0 being unique, the same is true for ϕ 0 .…”

“…Again, if V (r) keeps a constant sign, the extra condition r 2 V (r) ∈ L 1 (1, ∞) is both necessary as well as sufficient for the existence of such a solution ψ 0 . Obviously, φ 0 and χ 0 , or ψ 0 and χ 0 , are two independent solutions of ( 6) since the Wronskians at r = ∞, are, according to (7), ( 8) and (10),…”

“…In short, one gets theorem 2 from theorem 2 by replacing χ 0 by r χ 0 , and φ 0 and ψ 0 by r − φ 0 and r − ψ 0 , and changing the right-hand side of ( 11) to (11 ). From these formulae ( 7)- (10 ), one can immediately obtain the asymptotic properties of χ 0 , φ 0 and ψ 0 themselves, to be used in (31).…”

A Coulomb-related family of potentials having zero-energy bound states

“…where V (ρ) is chosen to be V (ρ) = V 1/18 (ρ) = − 64 81 1 ρ 10/9 − 2 361ρ 2/9 (25) and it is obtained for n = 10, λ = 9, l = 8 and k = 1/18. Eq.…”