$V (x)$	Domain (x)	$E_{n}$	$ψ (x, E_{n}) = \sum_{m} f_{m} (E_{n}) ϕ_{m} (x)$	Constraints
$\frac{A}{x} + \frac{B + l (l + 1) / 2}{x^{2}}$	$[0, \infty [$	$- \frac{1}{2} {(\frac{A}{n + ν / 2 + 1})}^{2}$	$ϕ_{n} (x) = A_{n} x^{1 + ν / 2} e^{- x / 2} L_{n}^{ν} (x)$	$ν = - 1 + 2 \sqrt{\| {(l + 1 / 2)}^{2} + 2 B \|}$ $ν > - 1$
$\frac{1}{2} λ^{4} x^{2} + \frac{B + l (l + 1) / 2}{x^{2}}$	$] - \infty, \infty [$	$λ^{2} (2 n + ν + 2)$	$ϕ_{n} (x) = A_{n} x^{\frac{ν}{2} + \frac{3}{4}} e^{- x / 2} L_{n}^{ν} (x)$	$ν = - 1 + 2 \sqrt{\| {(l + 1 / 2)}^{2} + 2 B \|}$ $ν > - 1$
$\begin{array}{l} V (x) \\ = \frac{λ^{2}}{2} (A e^{- λ x} + {(\frac{μ}{2})}^{2} e^{- 2 λ x}) \end{array}$	$] - \infty, \infty [$	$- \frac{λ^{2}}{2} {(\frac{A}{μ} + n + \frac{1}{2})}^{2}$	$ψ_{n} (x) = A_{n} e^{- λ \| \frac{A}{μ} + n + \frac{1}{2} \| x} e^{- μ e^{- λ x} / 2} L_{n}^{\| \frac{A}{μ} + n + \frac{1}{2} \|} (μ e^{- λ x})$	$μ, λ > 0$
$\frac{C}{{(e^{λ x} - 1)}^{2}} + \frac{A}{e^{λ x} - 1}$	$[0, \infty [$	$E_{n} = - \frac{λ^{2}}{8} {[n + \frac{ν + 1}{2} + \frac{2 (A - C) / λ^{2}}{n + \frac{ν + 1}{2}}]}^{2}$	$\begin{matrix} ψ_{n} (x) = A_{n} {(1 - y)}^{(ν + 1) / 2} {(1 + y)}^{(μ + 1) / 2} \\ \times P_{n}^{(μ, ν)} (y) \end{matrix}$	$C = \frac{λ^{2} (ν^{2} - 1)}{8}$ $μ, ν > - 1$ $y = 1 - 2 e^{- λ x}$
$\begin{matrix} V (x) = C \tanh (λ x) \\ + \frac{A}{\cosh^{2} (λ x)} \end{matrix}$	$] - \infty, \infty [$	$E_{n} = - \frac{λ^{2}}{2} [ϑ_{n}^{2} + {(\frac{C}{λ^{2}})}^{2} ϑ_{n}^{- 2}]$	$ψ_{n} (x) = A_{n} {(1 - y)}^{ν_{n} / 2} {(1 + y)}^{μ_{n} / 2} P_{n}^{(μ_{n}, ν_{n})} (y)$ $y = \tanh (λ x)$ , $λ > 0$	$C = {(λ μ / 2)}^{2} - {(λ ν / 2)}^{2}$ $ϑ_{n} = (n + \frac{1}{2} - \| D / λ \|)$ $E_{n} = - {(λ μ / 2)}^{2} - {(λ ν / 2)}^{2}$
$V_{0} \cos (k λ x)$	$[0, L]$	See [28]	$ψ_{n} (x) \propto P_{n}^{(\pm 1 / 2, \pm 1 / 2)} [\cos (k λ x)]$	$λ = π / L$ $k = 0, 1, 2, 3, \dots$