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The Kratzer potential
$V\left(r\right)=2D\left(\frac{a}{r}\frac{{a}^{2}}{2{r}^{2}}\right)$
$\text{}$
was originally intended to approximate the interatomic interaction
in diatomic molecules.
This has long since been superseded by superior alternatives, such as the Morse potential.
This potential belongs to the small number of problems for which the
Schrödinger equation is exactly solvable.
We analyzed this potential mainly to verify the correctness of the eigenfunctions
data that our notebook finds exclusively numerically.
The agreement is excellent, at least up to the eighth decimal place.
We set the parameters:
$\hslash \text{}=1$;
$m=1$;
$a=1.25;$
$d=2.5.$ 

The Mathematica notebook, given a potential V(x), allows to numerically solve the equation, finding the energies (eigenvalues) and the relative functions (eigenfunctions).  
The problem can be solved analytically.
The exact eigenvalues are given by:
$E\left(n\right)=\frac{2{a}^{2}{d}^{2}}{{\left(n+\mu +\frac{1}{2}\right)}^{2}}$
in which
$\mu =\frac{1}{2}\sqrt{1+8{a}^{2}d}$. The first 8 exact eigenvalues obtained are the following: E8 = 0.080134912026 E7 = 0.157813998056 E6 = 0.220279456098 E5 = 0.280742493876 E4 = 0.362578678462 E3 = 0.485988851140 E2 = 0.685072149164 E1 = 1.037193591616 E0 = 1.751374661634 These values can be compared to those shown in the graph on the left, which were obtained numerically with our Wolfram notebook. 

The exact (analytic) eigenfunctions are obtained with the
Whittaker Function
as defined in Wolfram Mathematica:
WhittakerM. $\mu =\frac{1}{2}\sqrt{1+8{a}^{2}d}$ ${\kappa}_{n}=n+\frac{1}{2}+\mu $ ${\u03f5}_{n}=\frac{2{a}^{2}{d}^{2}}{{{\kappa}_{n}}^{2}}$ ${\phi}_{n}\left(r\right)=\mathrm{WhittakerM}\left[{\kappa}_{n},\mu ,\frac{4ad}{{\kappa}_{n}}r\right]$ 

In the graph on the left, the yellow line indicates the upper energy limit
set in the eigenvalue search. With this limit only 3 eigenvalues were found. The eigenvalues were calculated numerically with the Wolfram notebook. 