Problems Week 5
1. A sech2
potential. Verify that the solution to the associated Legendre
equation
ˆψ(x; k) = a(k)2ik
( sech x)−ik
F(c+, c−; 1 − ik;
1
2
(1 + tanh x)) ,
where
c± =
1
2
− ik ± U0 +
1
4
behaves asymptotically as
ˆψ =
e−ikx
+ beikx
x → +∞
ae−ikx
x → −∞
for
a(k) =
Γ(c+)Γ(c−)
Γ(1 − ik)Γ(ik)
, b(k) = a(k)
Γ(1 − ik)Γ(−ik)
Γ(c+ + ik)Γ(c− + ik)
.
The hypergeometric function is defined as
F(a, b; c; z) =
∞
n=0
(a)n(bn)
(c)n
zn
n!
, (q)n = q(q+1) · · · (q+n−1) (q)0 = 1 .
It satisfies the following identity due to Euler
F(a, b; c; z) =
Γ(c)Γ(c − a − b)
Γ(c − a)Γ(c − b)
F(a, b; a + b + 1 − c; 1 − z)
+ (1 − z)c−a−b Γ(c)Γ(a + b − c)
Γ(a)Γ(b)
F(c − a, c − b; 1 + c − a − b; 1 − z) .
2. The scattering coefficients. With u(x) = −U0 sech2
x show, by using the
properties of the gamma function, that a(k) and b(k) in the previous
problem satisfy the conditions
|a|2
+ |b|2
= 1
and a → 1 and b → 0 as k → ∞.
3. Inverse scattering about −∞. Find the equation for L(x, z) if
ψ−(x; k) = e−ikx
+
x
−∞
L(x, z)e−ikz
dz
is a solution of ψ + (λ − u(x))ψ = 0. What boundary conditions must
L(x, z) satisfy?
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