There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as

$$V(r) = - \frac{\kappa}{r} + \frac{r}{a^2}$$

The mathematical problem is reduced to solving the radial part of the Schrodinger equation

$$-\frac{\hbar^2}{2m} \frac{d^2 u}{dr^2} + \left[ V(r) + \frac{l(l+1)}{2mr^2}\right]u = Eu$$

for the above potential, hence obtaining the reduced wavefunction $u(r)$.

  • The reduced wavefunction $u(r)$ is subject to the boundary conditions $u(r=0) = 0$, and $u'(r=0) = R(0) =$ some number $C$.

  • Each of the constants $\hbar$, $m$, $\kappa$, and $a$ are positive numbers.

  • $l$ serves the role of azimuthal quantum number, and can take non-negative integer values, i.e. $l = 0, 1, 2, 3 \ldots$ However, this is generally fed in as a constant input, and choice of $l$ segregates solutions into various categories, e.g. $l=0$ states are the $s-$wave states, $l=1$ are the $p-$wave states, and so on.

  • The energy eigenvalue $E$ can, in general, be either positive (called a Scattering State solution), or negative (called a Bound State solution); however, in this case, we are concerned with the latter variety ($E<0$) since the model caters precisely to bound heavy-quark systems.

Scanning across the literature, one only finds (abundantly) numerical/iterative approaches applied to this mathematical problem, and Wolfram Alpha also succumbs before it.

My questions are:

  • Is this physical model exactly solvable, i.e. is there any general solution to this differential equation for $u(r)$, for any $E<0$?

  • If not, what is the exact problem with this potential which hinders reaching an exact solution. (I say that because the singularity at $r=0$ is due to inverse $r$ part, but if the second term was missing from the potential, then the pure inverse $r$ potential is known to be exactly solvable, it is a textbook problem taught in all undergraduate Physics courses.)

  • Aside from numerical/iterative approaches, if there is any alternative method which can be used for this purpose, or can be used in circumventing this difficulty?

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    $\begingroup$ I'm not familiar with either quantum mechanics or differential equations, but the given differential equation can be rewritten to $$\frac{d^2 u}{dr^2}=\frac{2m}{(a\hbar)^2}\left(\frac{r^3-a^2Er^2-a^2\kappa r+\tfrac{a^2l(l+1)}{2}}{r^2}\right)u$$ so setting $$C:=\frac{2m}{(a\hbar)^2},\qquad c_0:=a^2\binom{l}{2},\qquad c_1:=-a^2\kappa,\qquad c_2:=-a^2E,$$ this becomes $$\frac{d^2y}{dx^2}=C\frac{x^3+c_2x^2+c_1x+c}{x^2}y.$$ And this looks like something mathematicians have thought a lot about. $\endgroup$ – Servaes Feb 17 at 12:31
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    $\begingroup$ Well, by setting some constants equal to a other constant we can rewrite your problem in the following sense: $$\alpha\cdot x''(t)+x(t)\cdot\left(\eta\cdot t-\rho\cdot\frac{1}{t}+\beta\cdot\frac{1}{t^2}\right)=\gamma\cdot x(t)\tag1$$ We can rewrite equation $(1)$ as follows: $$\frac{x''(t)}{x(t)}=\frac{\gamma-\eta\cdot t+\rho\cdot\frac{1}{t}-\beta\cdot\frac{1}{t^2}}{\alpha}\tag2$$ $\endgroup$ – Jan Feb 17 at 13:28
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    $\begingroup$ Here's a start: For $f:=e^{\sqrt{c_2}x}$ and $g:=\left(\frac{\sqrt{2}x}{a}\right)^{l+1}$ we have $$\frac{d^2f}{dx^2}=c_2f \qquad\text{ and }\qquad \frac{d^2g}{dx^2}=\frac{c_0}{x^2}g,$$ so for $z:=y-(f+g)$ we have $$\frac{d^2z}{dx^2}=C\left(x+\frac{c_1}{x}\right)z.$$ I haven't done differential equations since my first year in uni (over a decade ago), but I'm sure people have come up with a trick to solve this. $\endgroup$ – Servaes Feb 17 at 14:04
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    $\begingroup$ @Servaes: Your insights and comments are interesting. I suggest you summarize them and give a partial answer. It may lead someone to a more complete answer. $\endgroup$ – Tito Piezas III Feb 17 at 15:34
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    $\begingroup$ No, it is, of course, not solvable. Asking for a reason is opinion-based: you want speculation on why nobody succeeded. I gather you have exhausted this one. $\endgroup$ – Cosmas Zachos Feb 22 at 20:55

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