E. Match of solution at x = 0 So far we have derived asymptotic expressions at x → ±∞ for the wave function of the scattering problem in the negative region, x < 0, ( φ$_{L}$ ), and in the positive region, x > 0, ( φ$_{R}$ ) which depend respectively on two, D$_{1}$ and D$_{2}$ , and one, d$_{1}$ , unknown constants. In order to have a physical solution of the scattering problem one needs to match the two solutions φ$_{L}$ and φ$_{R}$ at x = 0. This is done by imposing the continuity of the wave function and of its derivative at x = 0 which gives two conditions and two of the three unknown constants can be expressed in terms of the one left out as the ordinary normalization constant. We need to find the behavior of the function φ$_{L}$ ( x ) and φ$_{R}$ ( x ) in the vicinity of x = 0. As x → 0 we have | y | ≈ e − aL ≪ 1 as our only assumption throughout the paper is that aL ≫ 1. Thus (1 − y ) − λ ≈ 1 and from Eq. 26 we obtain for φ$_{L}$ ( y ): having evaluated the two hypergeometric functions to unity as their argument vanishes, the above equation can be put as: In order to extract the behavior of φ$_{R}$ ( x ) as x → 0 we use the continuation identity of the hyper- geometric function [26] Proceeding similarly to the case x < 0 we find (for x > 0), that as x → 0 $^{+}$, z → 1, and 1 − z ≈ e a ( x − L ) ≪ 1 (we assume aL ≫ 1) and recalling that µ = ρ , from Eq. 32 with d$_{2}$ = 0 we find:, where: 12 M = Γ(1 − 2 ν )Γ(1 + 2 µ ) Γ(1 + µ − ν + λ )Γ(1 + µ − ν − λ ) (38a) N = Γ(1 − 2 ν )Γ( − 1 − 2 µ ) Γ( − µ − ν − λ )Γ( − µ − ν + λ ) (38b) φ ( x )$_{R}$ ∼ d$_{1}$ M e − aµ ( x − L ) + d$_{1}$ N e ( µ +1) a ( x − L ) (37) $_{2}$F$_{1}$ ( a, b ; c ; z ) = (36) Γ( c )Γ( c − a − b ) Γ( c − a )Γ( c − b ) $_{2}$F$_{1}$ ( a, b ; a + b − c + 1; 1 − z ) +(1 − z ) c − ( a + b $_{)}$Γ( c )Γ( a + b − c ) Γ( a )Γ( b ) $_{2}$F$_{1}$ ( c − a, c − b ; c − a − b + 1; 1 − z ) . φ$_{L}$ ( x ) ∼ D$_{1}$ e $^{iπµ}$e − aµ ( x + L ) − D$_{2}$ e − $^{iπµ}$e − a ( x + L $^{)}$e aµ ( x + L ) (35) φ$_{L}$ ( x ) ∼ D$_{1}$ ( − e − a ( x + L $^{)}$) µ + D$_{2}$ ( − e − a ( x + L $^{)}$)1 − µ