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
)
.
