Analytical results

Following Ref. [48], we express solutions of the radial Schrödinger equation with the PTG potential (1) as functions of the variable x:

$$x={1-(1+\Lambda^2)y^2\over 1-(1-\Lambda^2)y^2} ,$$ (20)

where y is a function of the radial coordinate R, given by Eqs. (4) and (5). In the variable x, the Schrödinger equation transforms into the Jacobi equation, and its general solution can be expressed by means of the hypergeometric function:

 

$$\Psi_{kLj}(R) = \chi_L(R) \left({ 1+x\over 2} \right)^{\beta\over2}$$ (21)

$$~~~~ \times F \left( {L\!+\!{3\over2}\!+\!\beta\!+\!\bar\nu_{Lj}\... ...over2}\!+\!\beta\!-\!\bar\nu_{Lj}\over 2}, L\!+\!{3\over2}; {1-x\over 2}\right)$$

where

$$\chi_L(R)={s^{1\over2}\over R} \bigl[ 1 + \Lambda^2-(1-\Lamb... ...)\bigl]^{1\over4} \biggl({1-x\over2}\biggr)^{ L+ 1 \over 2 } .$$ (22)

For a given complex momentum k, the wave function is specified by two dimensionless parameters:

$$\beta=-\frac{ik}{{s\Lambda}^{2}}$$ (23)

and

$$\bar\nu_{Lj} = \biggl[\bigl(\nu_{Lj}+{1\over2}\bigr)^2 +\beta^2\bigl(1-\Lambda^2\bigr) \biggr]^{1\over2} .$$ (24)

The bound states occur for parameters $\beta$ defined by:

$$\beta = \beta_{nLj},$$ (25)

where

$$\Lambda^2 \beta_{nLj} = \biggl[ \biggl(2n+L+{3\over2}\biggr)^2\bigl(1-\Lambda^2\bigr)$$

$$+\Lambda^2 \bigl(\nu_{Lj}+{1\over2}\bigr)^2 \biggr)\biggr]^{1\over2} - \biggl(2n+L+{3\over2}\biggr)$$ (26)

and $n\geq0$. In this case, the hypergeometric function reduces to the Jacobi polynomial, and the solution reads [48]:

 

$$\Psi_{nLj} = {\cal N}_{nLj} \bigl[1+\Lambda^2-(1-\Lambda^2)x\bigr]^{1\over4}$$

$$~~~ \times\biggl({1+x\over2}\biggr)^{\beta_{nLj}\over2} \biggl({1-x\over2}\biggr)^{L+1\over2} P_n^{(L+{1\over2},\beta_{nLj})}(x),$$ (27)

where ${\cal N}_{nLj}$ is a normalization factor, and the number of bound states is limited by condition $\beta_{nLj} > 0$, which ensures that the eigenfunctions vanish at the infinity.

The bound-state energies are given by

$${\epsilon}_{nLj}=\frac{\hbar^2 s^2}{2m}{\cal E}_{nLj},$$ (28)

where the dimensionless eigenenergies are

$${\cal E}_{nLj}=-{\Lambda}^4 \beta^2_{nLj} .$$ (29)

It can be seen from the above equation that the tail of a bound-state wave functions for $\eta \rightarrow \infty$,

$$\Psi_{nLj}(R) \propto e^{-\beta_{nLj}\Lambda^2(\eta-\eta_0)},$$ (30)

does not explicitly depend on L, i.e., there is no influence of centrifugal barrier on any partial wave.

If ${\Lambda}^2 > 2$, the resonances will occur for integer n such that:

$$n > \frac{1}{2}\bigl[ \Lambda ({\Lambda}^2-2)^{-1/2}(\nu_{Lj} + \frac{1}{2})-1 \bigr]$$ (31)

The dimensionless resonance energy is then:

$${\cal E}_{nLj} = \bigl(2n+L+{3\over2}\bigr)^2 \bigl(\Lambda^2-2)-\Lambda^2\bigl(\nu_{Lj}+ {1\over2}\bigr)^2 ,$$ (32)

and the dimensionless resonance width is:

 

$$\gamma_{nLj} = 4\left(2n+L+{3\over2}\right) \times \biggl[ \bigl(2n+L+{3\over2} \bigr)^2 \left(\Lambda^2-1\right)$$

$$-\Lambda^2\left(\nu_{Lj}+{1\over2}\right)^2 \biggr]^{1\over2} .$$ (33)

In the limit: ${\cal E}_{nLj} \rightarrow 0$, the resonance width is:

$$\gamma_{nLj} \rightarrow \gamma_{nLj}^{(0)} = 4(2n+L+\frac{3}{2})^2 ,$$ (34)

and, hence, the ratio: ${\gamma_{nLj}}/{{\cal E}_{nLj}} \rightarrow \infty$, for all values of parameter $\Lambda$. For small ${\cal E}_{nLj}$, we have:

$$\frac{\gamma_{nLj}}{{\cal E}_{nLj}} = \frac{\gamma_{nLj}^{(0... ... E}_{nLj}} + 2 - 4\frac{ {\cal E}_{nLj}}{\gamma_{nLj}^{(0)}} .$$ (35)

This ratio depends strongly on $\Lambda$. For large (n,L), the quantities ${\cal E}_{nLj}$ and $\gamma_{nLj}$ are proportional to (2n+L)² and their ratio is:

$$\frac{\gamma_{nLj}}{{\cal E}_{nLj}} = 4 \frac{({\Lambda}^2-1)^{1/2}} {{\Lambda}^2-2} ,$$ (36)

i.e. $\gamma_{nLj}/{\cal E}_{nLj} < 1$ for $\Lambda > (10+4{\sqrt 5})^{1/2}$. Therefore, in our numerical examples we present results for $\Lambda$=3 and 7, which are the values on two sides of the limiting case defined by the widths of resonances being equal to resonance energies.

Solutions in the continuum can be found by analytically continuing the eigenfunctions from a discrete negative energy to positive continuous energy (see Eq. (21)). The solutions obtained in this way are proportional to hypergeometric functions. Imposing the boundary condition that an incoming wave has the momentum k, one can determine the scattering function for each angular momentum [48]. Using the asymptotic behavior of the most general solution (21) one obtains the matrix elements of the S-matrix:

 

$$(-1)^{L+1} \exp \biggl( 2\beta [{\Lambda^2 \eta_0} + \ln \Lambda ] \biggr)$$ S (k) =

$$\times {\Gamma \bigl[-\beta\bigr] \Gamma \bigl[ (L+{3\over2}+\beta+\bar\... ...mma \bigl[ \beta\bigr] \Gamma\bigl[ (L+{3\over2}-\beta+\bar\nu_{Lj})/2 \bigr] }$$ (37)

$$\times {\Gamma \bigl[ (L+{3\over2}+\beta-\bar\nu_{Lj})/2 \bigr] \over \Gamma\bigl[ (L+{3\over2}-\beta-\bar\nu_{Lj})/2 \bigr] } .$$

Expression (37) yields the matrix elements of S-matrix for the Schrödinger problem with the PTG potential without any restrictions, i.e., (37) contains informations about all mathematical solutions, both physical and unphysical ones. The poles of the S-matrix in the variable $k=is\Lambda^2\beta$, correspond to the remarkable solutions [65] depending on the asymptotic behavior of solutions.