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we have a point x = a such that

p0 (x) = (x ā’ a)2 P (x), p1 (x) = (x ā’ a)Q(x), p2 (x) = R(x), (3.61)

where P and Q and R are analytic1 and P and Q non-zero in a neighbourhood

of a then the point x = a is called a regular singular point of the equation.

All other singular points are said to be irregular . Close to a regular singular

point a the equation looks like

P (a)(x ā’ a)2 y + Q(a)(x ā’ a)y + R(a)y = 0. (3.62)

The solutions of this reduced equation are

y1 = (x ā’ a)Ī»1 , y2 = (x ā’ a)Ī»2 , (3.63)

where Ī»1,2 are the roots of the indicial equation

Ī»(Ī» ā’ 1)P (a) + Ī»Q(a) + R(a) = 0. (3.64)

The solutions of the full equation are then

y1 = (x ā’ a)Ī»1 f1 (x), y2 = (x ā’ a)Ī»2 f2 (x), (3.65)

where f1,2 have power series solutions convergent in a neighbourhood of a.

An exception is when Ī»1 and Ī»2 coincide or diļ¬er by an integer, in which

case the second solution is of the form

y2 = (x ā’ a)Ī»1 ln(x ā’ a)f1 (x) + f2 (x) , (3.66)

1

A function is analytic at a point iļ¬ it has a power-series expansion that is convergent

to the function in a neighbourhood of the point.

3.4. SINGULAR POINTS 81

where f1 is the same power series that occurs in the ļ¬rst solution, and f2 is

a new power series. You will probably have seen these statements proved by

the tedious procedure of setting

f1 (x) = b0 + b1 (x ā’ a) + b2 (x ā’ a)2 + Ā· Ā· Ā· , (3.67)

and obtaining a recurrence relation determining the bi . Far more insight is

obtained, however, by extending the equation and its solution to the com-

plex plane, where the structure of the solution is related to its monodromy

properties. If you are familiar with complex analytic methods, you might like

to look at the discussion of monodromy in 9.2.1 of the MMB lecture notes.

82 CHAPTER 3. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

Chapter 4

Linear Diļ¬erential Operators

In this chapter we will begin to take a more sophisticated approach to dif-

ferential equations. We will deļ¬ne, with some care, the notion of a linear

diļ¬erential operator, and explore the analogy between such operators and

matrices. In particular, we will investigate what is required for a diļ¬erential

operator to have a complete set of eigenfunctions.

Formal vs. Concrete Operators

4.1

We will call the object

dn dnā’1

L = p0 (x) n + p1 (x) nā’1 + Ā· Ā· Ā· + pn (x), (4.1)

dx dx

which we also write as

n nā’1

p0 (x)ā‚x + p1 (x)ā‚x + Ā· Ā· Ā· + pn (x), (4.2)

a formal linear diļ¬erential operator . The word āformalā refers to the fact

that we are not yet worrying about what sort of functions the operator is

applied to.

4.1.1 The Algebra of Formal Operators

Even though they are not acting on anything in particular, we can still form

products of operators. For example if v and w are smooth functions of x we

can deļ¬ne the operators ā‚x + v(x) and ā‚x + w(x) and ļ¬nd

2

(ā‚x + v)(ā‚x + w) = ā‚x + w + (w + v)ā‚x + vw, (4.3)

83

84 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS

or

2

(ā‚x + w)(ā‚x + v) = ā‚x + v + (w + v)ā‚x + vw, (4.4)

We see from this example that the operator algebra is not usually commuta-

tive.

The algebra of formal operators has some deep applications. Consider,

for example, the operators

2

L = ā’ā‚x + q(x) (4.5)

and

3

P = ā‚x + a(x)ā‚x + ā‚x a(x). (4.6)

In the last expression, the combination ā‚x a(x) means āļ¬rst multiply by a(x),

and then diļ¬erentiate the result,ā so we could also write

ā‚x a = aā‚x + a . (4.7)

We can now form the commutator [P, L] ā” P L ā’ LP . After a little eļ¬ort,

we ļ¬nd

2

[P, L] = (3q + 4a )ā‚x + (3q + 4a )ā‚x + q + 2aq + a . (4.8)

3

If we choose a = ā’ 4 q, the commutator becomes a pure multiplication oper-

ator, with no diļ¬erential part:

1 3

[P, L] = q ā’ qq . (4.9)

4 2

The equation

dL

= [P, L], (4.10)

dt

or, equivalently,

1 3

q = q ā’ qq ,

Ė™ (4.11)

4 2

has solution

L(t) = etP L(0)eā’tP , (4.12)

showing that the time evolution of L is given by a similarity transformation,

which (at least formally) does not change its eigenvalues. The partial dif-

ferential equation (4.11) is the famous Korteweg de Vries (KdV) equation,

which has āsolitonā solutions whose existence is intimately connected with

the fact that it can be written as (4.10). The operators P and L are called

a Lax pair , after Peter Lax who uncovered much of the structure.

4.1. FORMAL VS. CONCRETE OPERATORS 85

4.1.2 Concrete Operators

We want to explore the analogies between linear diļ¬erential operators and

matrices acting on a ļ¬nite-dimensional vector space. Now the theory of

matrix operators makes much use of inner products and orthogonality. Con-

sequently the analogy is closest if we work with a function space equipped

with these same notions. We therefore let our diļ¬erential operators act on

L2 [a, b], the Hilbert space of square integrable functions on [a, b]. A diļ¬er-

ential operator cannot act on all functions in the Hilbert space, however,

because not all of them are diļ¬erentiable. We must at least demand that the

domain D, the subset of functions on which we allow the operator to act,

contain only functions that are suļ¬ciently diļ¬erentiable that the function

resulting from applying the operator is itself an element of L2 [a, b]. We will

usually restrict the set of functions even further, by imposing boundary con-

ditions at the endpoints of the interval. A linear diļ¬erential operator is now

deļ¬ned as a formal linear diļ¬erential operator, together with a speciļ¬cation

of its domain D.

The boundary conditions that we will impose will always be linear and

homogeneous. We require this so that the domain of deļ¬nition is a linear

space. In other words we demand that if y1 and y2 obey the boundary

conditions then so does Ī»y1 + Āµy2. Thus, for a second-order operator

2

L = p0 ā‚x + p1 ā‚x + p2 (4.13)

on the interval [a, b], we might impose

B1 [y] = Ī±11 y(a) + Ī±12 y (a) + Ī²11 y(b) + Ī²12 y (b) = 0,

B2 [y] = Ī±21 y(a) + Ī±22 y (a) + Ī²21 y(b) + Ī²22 y (b) = 0 (4.14)

but we will not, in deļ¬ning the diļ¬erential operator , impose inhomogeneous

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