$V\left(x\right)$ Domain (x) ${E}_{n}$ $\psi \left(x,{E}_{n}\right)=\sum _{m}{f}_{m}\left({E}_{n}\right){\varphi }_{m}\left(x\right)$ Constraints $\frac{A}{x}+\frac{B+l\left(l+1\right)/2}{{x}^{2}}$ $\left[0,\infty \left[$ $-\frac{1}{2}{\left(\frac{A}{n+\nu /2+1}\right)}^{2}$ ${\varphi }_{n}\left(x\right)={A}_{n}{x}^{1+\nu /2}{\text{e}}^{-x/2}{L}_{n}^{\nu }\left(x\right)$ $\nu =-1+2\sqrt{|{\left(l+1/2\right)}^{2}+2B|}$ $\nu >-1$ $\frac{1}{2}{\lambda }^{4}{x}^{2}+\frac{B+l\left(l+1\right)/2}{{x}^{2}}$ $\right]-\infty ,\infty \left[$ ${\lambda }^{2}\left(2n+\nu +2\right)$ ${\varphi }_{n}\left(x\right)={A}_{n}{x}^{\frac{\nu }{2}+\frac{3}{4}}{\text{e}}^{-x/2}{L}_{n}^{\nu }\left(x\right)$ $\nu =-1+2\sqrt{|{\left(l+1/2\right)}^{2}+2B|}$ $\nu >-1$ $\begin{array}{l}V\left(x\right)\\ =\frac{{\lambda }^{2}}{2}\left(A{\text{e}}^{-\lambda x}+{\left(\frac{\mu }{2}\right)}^{2}{\text{e}}^{-2\lambda x}\right)\end{array}$ $\right]-\infty ,\infty \left[$ $-\frac{{\lambda }^{2}}{2}{\left(\frac{A}{\mu }+n+\frac{1}{2}\right)}^{2}$ ${\psi }_{n}\left(x\right)={A}_{n}{\text{e}}^{-\lambda |\frac{A}{\mu }+n+\frac{1}{2}|x}{\text{e}}^{-\mu {\text{e}}^{-\lambda x}/2}{L}_{n}^{|\frac{A}{\mu }+n+\frac{1}{2}|}\left(\mu {\text{e}}^{-\lambda x}\right)$ $\mu ,\text{\hspace{0.17em}}\lambda >0$ $\frac{C}{{\left({\text{e}}^{\lambda x}-1\right)}^{2}}+\frac{A}{{\text{e}}^{\lambda x}-1}$ $\left[0,\text{\hspace{0.17em}}\infty \left[$ ${E}_{n}=-\frac{{\lambda }^{2}}{8}{\left[n+\frac{\nu +1}{2}+\frac{2\left(A-C\right)/{\lambda }^{2}}{n+\frac{\nu +1}{2}}\right]}^{2}$ $\begin{array}{c}{\psi }_{n}\left(x\right)={A}_{n}{\left(1-y\right)}^{\left(\nu +1\right)/2}{\left(1+y\right)}^{\left(\mu +1\right)/2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{P}_{n}^{\left(\mu ,\nu \right)}\left(y\right)\end{array}$ $C=\frac{{\lambda }^{2}\left({\nu }^{2}-1\right)}{8}$ $\mu ,\text{\hspace{0.17em}}\nu >-1$ $y=1-2{\text{e}}^{-\lambda x}$ $\begin{array}{c}V\left(x\right)=C\mathrm{tanh}\left(\lambda x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{A}{{\mathrm{cosh}}^{2}\left(\lambda x\right)}\end{array}$ $\right]-\infty ,\infty \left[$ ${E}_{n}=-\frac{{\lambda }^{2}}{2}\left[{\vartheta }_{n}^{2}+{\left(\frac{C}{{\lambda }^{2}}\right)}^{2}{\vartheta }_{n}^{-2}\right]$ ${\psi }_{n}\left(x\right)={A}_{n}{\left(1-y\right)}^{{\nu }_{n}/2}{\left(1+y\right)}^{{\mu }_{n}/2}{P}_{n}^{\left({\mu }_{n},{\nu }_{n}\right)}\left(y\right)$ $y=\mathrm{tanh}\left(\lambda x\right)$ , $\lambda >0$ $C={\left(\lambda \mu /2\right)}^{2}-{\left(\lambda \nu /2\right)}^{2}$ ${\vartheta }_{n}=\left(n+\frac{1}{2}-|D/\lambda |\right)$ ${E}_{n}=-{\left(\lambda \mu /2\right)}^{2}-{\left(\lambda \nu /2\right)}^{2}$ ${V}_{0}\mathrm{cos}\left(k\lambda x\right)$ $\left[0,\text{\hspace{0.17em}}L\right]$ See [28] ${\psi }_{n}\left(x\right)\propto {P}_{n}^{\left(±1/2,±1/2\right)}\left[\mathrm{cos}\left(k\lambda x\right)\right]$ $\lambda =\text{π}/L$ $k=0,1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}\cdots$