Chapter 4 Linear Differential Operators

116 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS
and found that
L † = −i d
dx , Q[u, v] = −iu∗v. (4.23)
The operator −id/dx (which you should recognize as the “momentum” operator
from quantum mechanics) obeys L = L † , and is therefore, formally
self-adjoint, or Hermitian.
Example: Let
d
L = p0
2 d
+ p1
dx2 dx + p2, (4.24)
with the pi all real. Again let us find the adjoint L † with respect to the inner
product with w ≡ 1. Now, proceeding as above, but integrating by parts
twice, we find
u ∗ [p0v ′′ + p1v ′ + p2v] − v [(p0u) ′′ − (p1u) ′ + p2u] ∗
= d
p0(u
dx
∗ v ′ − vu ∗′ ) + (p1 − p ′ 0 )u∗v . (4.25)
From this we read off that
L † = d2
dx2 p0 − d
dx p1 + p2
= p0
d 2
dx 2 + (2p′ 0
d
− p1) + (p′′ 0
dx − p′ 1 + p2). (4.26)
What conditions do we need to impose on p0,1,2 for this L to be formally
self-adjoint with respect to the inner product with w ≡ 1? For L = L † we
need
We therefore require that p1 = p ′ 0
p0 = p0
2p ′ 0 − p1 = p1 ⇒ p ′ 0 = p1
p ′′
0 − p ′ 1 + p2 = p2 ⇒ p ′′
0 = p ′ 1. (4.27)
, and so
L = d
d
p0 + p2, (4.28)
dx dx
which we recognize as a Sturm-Liouville operator.
Example: Reduction to Sturm-Liouville form. Another way to make the
operator
d
L = p0
2 d
+ p1
dx2 dx + p2, (4.29)