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B1 [y] = Ī±11 y(a) + Ī±12 y (a) + Ī²11 y(b) + Ī²12 y (b) = A,

B2 [y] = Ī±21 y(a) + Ī±22 y (a) + Ī²21 y(b) + Ī²22 y (b) = B, (4.15)

with non-zero A, B ā” even though we will solve diļ¬erential equations with

such boundary conditions.

Also, for an n-th order operator, we will not constrain derivatives of order

higher than n ā’ 1. This is reasonable1 : If we seek solutions of Ly = f with L

1

There is a deeper reason which we will explain in chapter 9.

86 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS

a second-order operator, for example, then the values of y at the endpoints

are already determined in terms of y and y by the diļ¬erential equation. We

cannot choose to impose some other value. By diļ¬erentiating the equation

enough times, we can similarly determine all higher endpoint derivatives in

terms of y and y . These two derivatives, therefore, are all we can ļ¬x by ļ¬at.

The boundary and diļ¬erentiability conditions that we impose make D a

subset of the entire Hilbert space. This subset will always be dense: any

element of the Hilbert space can be obtained as a limit of functions in D. In

particular, there will never be a function in L2 [a, b] that is orthogonal to all

functions in D.

4.2 The Adjoint Operator

One of the important properties of matrices, established in the appendix,

is that a matrix that is self-adjoint, or Hermitian, may be diagonalized . In

other words, the matrix has suļ¬ciently many eigenvectors for them to form

a basis for the space on which it acts. A similar property holds for self-

adjoint diļ¬erential operators, but we must be careful in our deļ¬nition of

self-adjointness.

Before reading this section, We suggest you review the material on adjoint

operators on ļ¬nite-dimensional spaces that appears in the appendix.

4.2.1 The Formal Adjoint

Given a formal diļ¬erential operator

dn dnā’1

L = p0 (x) n + p1 (x) nā’1 + Ā· Ā· Ā· + pn (x), (4.16)

dx dx

and a weight function w(x), real and positive on the interval (a, b), we can

ļ¬nd another such operator Lā , such that, for any suļ¬ciently diļ¬erentiable

u(x) and v(x), we have

d

w uā— Lv ā’ v(Lā u)ā— = Q[u, v], (4.17)

dx

for some function Q, which depends bilinearly on u and v and their ļ¬rst nā’1

derivatives. We call Lā the formal adjoint of L with respect to the weight w.

4.2. THE ADJOINT OPERATOR 87

The equation (4.17) is called Lagrangeā™s identity. The reason for the name

āadjointā is that if we deļ¬ne an inner product

b

wuā—v dx,

u, v = (4.18)

w

a

and if the functions u and v have boundary conditions that make Q[u, v]|b =

a

0, then

u, Lv w = Lā u, v w , (4.19)

which is the deļ¬ning property of the adjoint operator on a vector space. The

word āformalā means, as before, that we are not yet specifying the domain

of the operator.

The method for ļ¬nding the formal adjoint is straightforward: integrate

by parts enough times to get all the derivatives oļ¬ v and on to u.

Example: If

d

L = ā’i (4.20)

dx

then let us ļ¬nd the adjoint Lā with respect to the weight w ā” 1. We have

ā—

d d dā—

ā—

ā’i v ā’ v ā’i u = ā’i

u (u v). (4.21)

dx dx dx

Thus

d

Lā = ā’i = L. (4.22)

dx

This operator (which you should recognize as the āmomentumā operator

from quantum mechanics) is, therefore, formally self-adjoint, or Hermitian.

Example: Let

d2 d

L = p0 2 + p1 + p2 , (4.23)

dx dx

with the pi all real. Again let us ļ¬nd the adjoint Lā with respect to the inner

product with w ā” 1. Now

ā—

uā— [p0 v + p1 v + p2 v] ā’ v [(p0 u) ā’ (p1 u) + p2 u]

d

p0 (uā— v ā’ v uā—) + (p1 ā’ p0 )uā— v , (4.24)

=

dx

so

d2 d

ā

L = p0 2 + (2p0 ā’ p1 ) + (p0 ā’ p1 + p2 ). (4.25)

dx dx

88 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS

What conditions do we need to impose on p0,1,2 for L to be formally self-

adjoint with respect to the inner product with w ā” 1? For L = Lā we

need

p0 = p 0

2p0 ā’ p1 = p1 ā’ p 0 = p1

p0 ā’ p1 + p2 = p2 ā’ p 0 = p1 . (4.26)

We therefore require that p1 = p0 , and so

d d

L= p0 + p2 , (4.27)

dx dx

which is a Sturm-Liouville operator.

Example: Reduction to Sturm-Liouville form. Another way to make the

operator

d2 d

L = p0 2 + p1 + p2 , (4.28)

dx dx

self-adjoint is by a suitable choice of weight function w. Suppose that p0 is

positive on the interval (a, b), and that p0 , p1 , p2 are all real. Then we may

deļ¬ne

1 xp

1

w= exp dx (4.29)

p0 p0

a

and observe that it is positive on (a, b), and that

1

Ly = (wp0y ) + p2 y. (4.30)

w

Now

= [wp0 (uā— v ā’ uā— v)]b ,

ā’ Lu, v

u, Lv (4.31)

a

w w

where

b

wuā— v dx.

u, v = (4.32)

w

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