In mathematical physics, a Pöschl–Teller potential, named after the physicists Herta Pöschl (credited as G. Pöschl) and Edward Teller, is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions.
In its symmetric form is explicitly given by
V ( x ) = − λ ( λ + 1 ) 2 s e c h 2 ( x ) and the solutions of the time-independent Schrödinger equation
− 1 2 ψ ″ ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) with this potential can be found by virtue of the substitution u = t a n h ( x ) , which yields
[ ( 1 − u 2 ) ψ ′ ( u ) ] ′ + λ ( λ + 1 ) ψ ( u ) + 2 E 1 − u 2 ψ ( u ) = 0 .
Thus the solutions ψ ( u ) are just the Legendre functions P λ μ ( tanh ( x ) ) with E = − μ 2 2 , and λ = 1 , 2 , 3 ⋯ , μ = 1 , 2 , ⋯ , λ − 1 , λ . Moreover, eigenvalues and scattering data can be explicitly computed. In the special case of integer λ , the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg-de Vries equation.
The more general form of the potential is given by
V ( x ) = − λ ( λ + 1 ) 2 s e c h 2 ( x ) − ν ( ν + 1 ) 2 c s c h 2 ( x ) . 
