Chapter 4 Linear Differential Operators

128 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS
Since −1 a b
=
c d
1 d −b
, (4.78)
ad − bc −c a
we see that this requires
a
c
b
= e
d
iφ
mL A
mR C
B
,
D
(4.79)
where φ, A, B, C, D are real, and AD−BC = 1. Demanding self-adjointness
has therefore cut the original eight real parameters down to four. These
can be determined either by experiment or by performing the microscopic
calculation. 3 Note that 4 = 2 2 , a perfect square, as required by the Weyl-
Von Neumann theory.
Exercise 4.5: Consider the Schrödinger operator ˆ H = −∂ 2 x on the interval
[0, 1]. Show that the most general self-adjoint boundary condition applicable
to ˆ H can be written as
ϕ(0)
ϕ ′
= e
(0)
iφ
a b ϕ(1)
c d ϕ ′
,
(1)
where φ, a, b, c, d are real and ac − bd = 1. Consider ˆ H as the quantum
Hamiltonian of a particle on a ring constructed by attaching x = 0 to x = 1.
Show that the self-adjoint boundary condition found above leads to unitary
scattering at the point of join. Does the most general unitary point-scattering
matrix correspond to the most general self-adjoint boundary condition?
4.3 Completeness of eigenfunctions
Now that we have a clear understanding of what it means to be self-adjoint,
we can reiterate the basic claim: an operator T that is self-adjoint with
respect to an L 2 [a, b] inner product possesses a complete set of mutually orthogonal
eigenfunctions. The proof that the eigenfunctions are orthogonal
is identical to that for finite matrices. We will sketch a proof of the completeness
of the eigenfunctions of the Sturm-Liouville operator in the next
section.
The set of eigenvalues is, with some mathematical cavils, called the spectrum
of T . It is usually denoted by σ(T ). An eigenvalue is said to belong to
3 For example, see: T. Ando, S. Mori, Surface Science 113 (1982) 124.