$V\left(x\right)=S\left(x\right)$ $W\left(x\right)$ ${\epsilon }_{n}$ ${\psi }_{n}^{+}\left(x\right)$ $\frac{{\lambda }^{2}}{2}\left[A{\text{e}}^{-\lambda x}+{B}^{2}{\text{e}}^{-2\lambda x}\right]$ 0 ${\epsilon }_{n}^{2}-{m}^{2}=-{\lambda }^{2}{\left[\frac{A}{B}\sqrt{\left({\epsilon }_{n}+m\right)}+n+\frac{1}{2}\right]}^{2}$ $\begin{array}{c}{\psi }_{n}^{+}\left(x\right)={A}_{n}{\mathrm{e}}^{-\lambda |\frac{A}{B}\sqrt{\left({\epsilon }_{n}+m\right)}+n+\frac{1}{2}|-\sqrt{B}{\text{e}}^{-\lambda x}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{L}_{n}^{|\frac{2A}{B}\sqrt{\left({\epsilon }_{n}+m\right)}+2n+1|}\left(2B\sqrt{\left(\epsilon +m\right)}\text{ }{\text{e}}^{-\lambda x}\right)\end{array}$ $|\epsilon | , $B>0$ , ${A}_{n}=\sqrt{\lambda \Gamma \left(n+1\right)/\Gamma \left(n+\nu +1\right)}$ ${V}_{0}{\text{e}}^{-\lambda x}$ ${W}_{0}{\text{e}}^{-\lambda x}$ ${\epsilon }_{n}^{2}-{m}^{2}=-{\lambda }^{2}{\left[\frac{2{V}_{0}}{{W}_{0}\lambda }\left({\epsilon }_{n}+m\right)+n+\frac{3}{2}\right]}^{2}$ $\begin{array}{c}{\psi }_{n}^{+}\left(x\right)={A}_{n}{\mathrm{e}}^{-\lambda |\frac{2{V}_{0}}{{W}_{0}\lambda }\left(\epsilon +m\right)+n+\frac{3}{2}|-{W}_{0}{\text{e}}^{-\lambda x}/\lambda }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{L}_{n}^{|\frac{4{V}_{0}}{{W}_{0}\lambda }\left(\epsilon +m\right)+2n+3|}\left(2{W}_{0}{\text{e}}^{-\lambda x}/\lambda \right)\end{array}$ $|\epsilon | , ${W}_{0}>0$ , ${A}_{n}=\sqrt{\lambda \Gamma \left(n+1\right)/\Gamma \left(n+\nu +1\right)}$ $V={V}_{0}\mathrm{tanh}\left(\lambda x\right)$ $W={W}_{0}\mathrm{tanh}\left(\lambda x\right)$ ${\epsilon }_{n}^{2}-{m}^{2}={W}_{0}^{2}-{\lambda }^{2}\left[{\vartheta }_{n}^{2}+{V}_{0}^{2}{\left(\frac{{\epsilon }_{n}+m}{{\lambda }^{2}}\right)}^{2}{\vartheta }_{n}^{-2}\right]$ ${\vartheta }_{n}=\left(n+\frac{1}{2}-|D/\lambda |\right)$ ${D}^{\text{\hspace{0.17em}}2}={W}_{0}^{\text{\hspace{0.17em}}2}+\alpha {W}_{0}+{\lambda }^{2}/4$ ${W}_{0}\left({W}_{0}+\alpha \right)>-{\lambda }^{2}/4$ $\begin{array}{c}{\psi }_{n}^{+}\left(x\right)={A}_{n}{\left(1+\mathrm{tanh}\left(\lambda x\right)\right)}^{{\nu }_{n}/2}{\left(1-\mathrm{tanh}\left(\lambda x\right)\right)}^{{\mu }_{n}/2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{P}_{n}^{\left({\mu }_{n},\text{\hspace{0.17em}}{\nu }_{n}\right)}\left(\mathrm{tanh}\left(\lambda x\right)\right)\end{array}$ ${\mu }_{n}=\sqrt{-\left[{\epsilon }_{n}^{2}-{m}^{2}-2{V}_{0}\left({\epsilon }_{n}+m\right)\right]}/\lambda$ ${\nu }_{n}=\sqrt{-\left[{\epsilon }_{n}^{2}-{m}^{2}+2{V}_{0}\left({\epsilon }_{n}+m\right)\right]}/\lambda$ ${\mu }_{n},\text{\hspace{0.17em}}{\nu }_{n}>-1$ $\frac{C}{{\left({\text{e}}^{-\lambda x}-1\right)}^{2}}+\frac{A}{{\text{e}}^{-\lambda x}-1}$ 0 ${\epsilon }_{n}^{2}={m}^{2}-\frac{{\lambda }^{2}}{4}{\left[n+\frac{\nu +1}{2}+\frac{2\left({\epsilon }_{n}+m\right)\left(A-C\right)/{\lambda }^{2}}{n+\frac{\nu +1}{2}}\right]}^{2}$ ${\epsilon }_{n}^{2}-{m}^{2}=-{\lambda }^{2}{\left(\mu +1\right)}^{2}/4$ ${\psi }_{n}\left(x\right)={A}_{n}{\left(1-y\right)}^{\left(\nu +1\right)/2}{\left(1+y\right)}^{\left(\mu +1\right)/2}{P}_{n}^{\left(\mu ,\nu \right)}\left(y\right)$ $y=1-2{\text{e}}^{-\lambda x}$ $\frac{{\lambda }^{2}\left({\nu }^{2}-1\right)}{8}=C\left({\epsilon }_{n}+m\right)$ $\begin{array}{l}\frac{{V}_{+}}{{\mathrm{sinh}}^{2}\left(\lambda r\right)}\\ +2\frac{{V}_{0}+{V}_{1}\left(2{\mathrm{tanh}}^{2}\left(\lambda r\right)-1\right)}{{\mathrm{cosh}}^{2}\left(\lambda r\right)}\end{array}$ 0 see [22] see [22] $\frac{{V}_{0}+{V}_{1}\mathrm{tanh}\left(\lambda x\right)}{{\mathrm{cosh}}^{2}\left(\lambda x\right)}$ 0 see [22] see [22] $\begin{array}{l}V\left(r\right)=\frac{1}{{\text{e}}^{\lambda r}-1}\\ ×\left[{V}_{0}+\frac{{V}_{+}/2}{1-{\text{e}}^{-\lambda r}}+{V}_{1}\left(1-2{\text{e}}^{-\lambda r}\right)\right]\end{array}$ 0 see [22] see [22] $\begin{array}{l}V\left(x\right)=\frac{1/4}{1-{\left(x/L\right)}^{2}}\left[2{V}_{0}+\frac{{V}_{+}}{{\left(x/L\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{V}_{-}}{1-{\left(x/L\right)}^{2}}\right]-{V}_{1}\frac{{\left(x/L\right)}^{2}-\frac{1}{2}}{{\left(x/L\right)}^{2}-1}\end{array}$ 0 see [22] see [22]