The Lambert-W step-potential affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potentials – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as
V
(
x
)
=
V
0
1
+
W
(
e
−
x
/
σ
)
.
where
W
is the Lambert function also known as the product logarithm. This is an implicitly elementary function that resolves the equation
W
e
W
=
z
.
The Lambert
W
-potential is an asymmetric step of height
V
0
whose steepness and asymmetry are controlled by parameter
σ
. If the space origin and the energy origin are also included, it presents a four-parametric specification of a more general five-parametric potential which is also solvable in terms of the confluent hypergeometric functions. This generalized potential, however, is a conditionally integrable one (that is, it involves a fixed parameter).
The general solution of the one-dimensional Schrödinger equation for a particle of mass
m
and energy
E
:
d
2
ψ
d
x
2
+
2
m
ℏ
2
(
E
−
V
(
x
)
)
ψ
=
0
,
for the Lambert
W
-barrier for arbitrary
V
0
and
σ
is written as
ψ
(
x
)
=
z
i
δ
/
2
e
−
i
s
z
/
2
(
d
u
(
z
)
d
z
−
i
δ
+
s
2
u
(
z
)
)
,
z
=
W
(
e
−
x
/
σ
)
,
where
u
(
z
)
is the general solution of the scaled confluent hypergeometric equation
u
″
(
z
)
+
(
i
δ
z
−
i
s
)
u
′
(
z
)
+
a
s
z
u
(
z
)
=
0
and the involved parameters are given as
a
=
δ
(
δ
+
s
)
2
s
+
σ
m
V
0
2
E
ℏ
,
δ
=
2
σ
2
m
(
E
−
V
0
)
ℏ
2
,
s
=
2
σ
2
m
E
ℏ
2
.
A peculiarity of the solution is that each of the two fundamental solutions composing the general solution involves a combination of two confluent hypergeometric functions.
If the quantum transmission above the Lambert
W
-potential is discussed, it is convenient to choose the general solution of the scaled confluent hypergeometric equation as
u
=
c
1
(
i
s
z
)
1
−
i
δ
1
F
1
(
1
+
i
(
a
−
δ
)
;
2
−
i
δ
;
i
s
z
)
+
c
2
U
(
i
a
;
i
δ
;
i
s
z
)
,
where
c
1
,
2
are arbitrary constants and
1
F
1
and
U
are the Kummer and Tricomi confluent hypergeometric functions, respectively. The two confluent hypergeometric functions are here chosen such that each of them stands for a separate wave moving in a certain direction. For a wave incident from the left, the reflection coefficient written in terms of the standard notations for the wave numbers
k
1
=
2
m
E
ℏ
2
,
k
2
=
2
m
(
E
−
V
0
)
ℏ
2
reads
R
=
e
−
2
π
σ
k
2
sinh
(
π
σ
2
k
1
(
k
1
−
k
2
)
2
)
sinh
(
π
σ
2
k
1
(
k
1
+
k
2
)
2
)
