## 2 A Necessary but Not Sufficient Condition  
We begin by stating the necessary condition for the existence of polynomial solutions to the general differential equation given by (3):  
**Theorem 2**. : _A necessary condition for the differential Equation (3) to have a polynomial solution of degree \(m\) is_  
\[\tau_{n-2}=\alpha_{n}\,m\,(m-1)+\beta_{n-1}\,m\,,\qquad n=2,\,3,\,\ldots\,. \tag{7}\]  
Proof. : Substitute the polynomial solution \(y=\sum_{i=0}^{m}\mathcal{C}_{i}\,r^{i}\) into the differential Equation (3). After collecting the common terms of the common \(r\) power, the necessary condition for the leading coefficient \(\mathcal{C}_{m}\) to be nonzero is \(\tau_{n-2}=\alpha_{n}\,m\,(m-1)+\beta_{n-1}\,m\) for all \(m\geq 0\) follows. For \(n=2\), Equation (7) represents both the necessary and sufficient conditions for the polynomial solutions of (3). Although the condition (7) is necessary, it is not sufficient for \(n\geq 3\) to guarantee the polynomial solutions of (3). There are precisely \(n-2\) additional conditions that are sufficient to ensure the existence of such polynomials. These additional conditions can be understood as constraints that relate the remaining parameters of \(R_{n-2}(r)\) with the coefficients \(\{\alpha_{k}\}_{k=0}^{n}\) and \(\{\beta_{k}\}_{k=0}^{n-1}\) in \(P_{n}(r)\) and \(Q_{n-1}(r)\), respectively. In later sections, we shall devise a procedure to find these sufficient conditions and provide a method of evaluating the corresponding polynomial solutions explicitly. The remainder of the section investigates the inverse square-root potential. We shall show that despite satisfying the necessary condition in Theorem 2 that the Schrodinger equation with this potential as mentioned in the literature has no polynomial solutions, thus demonstrating that the condition (7) is necessary but not sufficient. _The Inverse Square-Root Potential_  
The exact solutions of the radial Schrodinger equation  
\[-\frac{d^{2}\,\psi(r)}{dr^{2}}+\left(-\frac{v}{\sqrt{r}}+\frac{l\,(l+1)}{r^{2}} \right)\psi(r)=E\,\psi(r)\,,\qquad v>0\,,\quad r>0\,,\quad\psi(0)=\psi(\infty)=0\,, \tag{8}\]  
were recently studied by a number of authors [21; 22; 23; 24; 25]. The differential Equation (8) has two singular points, one regular at \(r=0\) and another irregular at \(r=\infty\) of rank 3 [50]. To analyze these singular points further, we employ a change of variables \(z=\sqrt{2\,\kappa\,r}\), for some constant \(\kappa\) to be determined shortly. Equation (8) may then be expressed as  
\[z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)+\left(\left(\frac{\sqrt{2}\,v }{\kappa^{3/2}}+\frac{E}{\kappa^{2}}z\right)z^{3}-4\,l\,(l+1)\right)\psi(z)=0 \,,\qquad\psi(0)=\psi(\infty)=0\,. \tag{9}\]  
As \(z\to 0\), the solution to the differential Equation (9) behaves asymptotically as the solution of the Euler equation, \(z^{2}\,\psi^{\prime\prime}(z)-z\,\psi^{\prime}(z)-4\,l\,(l+1)\,\psi(z)=0\), which has a physically acceptable solution \(\psi(z)\sim z^{2\,(l+1)}\). Meanwhile as \(z\to\infty\), the solution to the differential equation behaves asymptotically as the solution of  
\[\psi^{\prime\prime}(z)+\left(\frac{\sqrt{2}\,v}{\kappa^{3/2}}-z\right)\,z\, \psi(z)=0\,.\]  
To examine the solution of this equation we complete the square to  
\[-\psi^{\prime\prime}(z)+\left(\frac{v}{\sqrt{2\,\kappa^{3}}}-z\right)^{2}\, \psi(z)=\left(\frac{v}{\sqrt{2\,\kappa^{3}}}\right)^{2}\,\psi(z)\,. \tag{10}\]  
The substitution \(x=((v/\sqrt{2\,\kappa^{3}})-z)\) allows us to compare the solution of the equation to the solvable Schroinger equation with the harmonic oscillator potential, as \(z\to\infty\), via  
\[\psi(z)=\exp\left(-\int^{z}(u+\lambda)\,du\right)=\exp\left(-\left(\frac{z^{2 }}{2}+\lambda\,z\right)\right)\,,\quad\lambda=-\frac{v}{\sqrt{2\,\kappa^{3}} }\,. \tag{11}\]  
Hence, we may assume the solution of Equation (9) to be  
\[\psi(z)=N\,z^{2\,(l+1)}\,\exp\left(-\left(\frac{z^{2}}{2}+\lambda z\right) \right)\,f(z)\,,\qquad\lambda=-\frac{v}{\sqrt{2\,\kappa^{3}}}<0\,, \tag{12}\]  
up to a normalization constant \(N\).