content_image/365315.jpg content_image/365313.jpg content_image/365312.jpg content_image/365314.jpg content_image/365310.jpg content_image/365311.jpg content_image/365309.jpg
## A. Negative region 

In the study of the discrete spectrum it is convenient to use a different variable. Now we choose y − 1 = 1 + e − a ( x + L $^{)}$, with d/dx = ay (1 − y ) d/dy , V ( y ) = − W y and m ( y ) = m$_{0}$ (1 − y ) and using the parametric transformation φ = y $^{σ}$(1 − y ) $^{ϵ}$h ( y ), we obtain from Eq. (7) 

<img src='content_image/365315.jpg'>

The most general condition that we can impose on the term multiplying 1 / [ y (1 − y )] in order that the equation be that of the hypergeometric function is that it be equal to a constant ζ . Therefore we get three equations: 

<img src='content_image/365313.jpg'>

<img src='content_image/365312.jpg'>

<img src='content_image/365314.jpg'>

From Eq. (46a) we can solve for σ , while summing Eq. (46b) and Eq. (46c) we obtain the equation for ϵ 

<img src='content_image/365310.jpg'>

<img src='content_image/365311.jpg'>

So σ = ν . A solution of the second one is ϵ = − i ( E + W ) /a , which is the same of µ in the scattering problem but with the replacement W → − W . In determining the parameters a and b ( c = 1 + 2 ν ) of the hypergeomtric equation we make use of the relation in Eq. (46c), where we define λ = i $^{√}$( W 2 − m 2 $_{0}$) /a 2 and finally obtain: 

<img src='content_image/365309.jpg'>

