denotes the Fourier transform of f . This equality of sums is called the Poisson

summation formula.

Example: Since the Fourier transform of a Gaussian is another Gaussian, the

Poisson formula gives

∞

π ∞ ’m2 π2 /κ

’κm2

e = e . (2.115)

κ m=’∞

m=’∞

and, more usefully,

∞ ∞

2π 1 1

’ 2t (θ+2πn)2 2 t+inθ

e’ 2 n

e = . (2.116)

t n=’∞ n=’∞

The last identity is known as Jacobi™s imaginary transformation. It states the

equivalence of the eigenmode expansion and the method of images solution

of the heat equation

1 ‚2• ‚•

= (2.117)

2 ‚x2 ‚t

on the unit circle. Notice that when t is small the sum on the right-hand side

converges very slowly, while the sum on the left converges very rapidly. The

opposite is true for large t. The conversion of a slowly converging series into

a rapidly converging one is a standard application of the Poisson summation

formula.

68 CHAPTER 2. FUNCTION SPACES

Chapter 3

Linear Ordinary Di¬erential

Equations

In this chapter we will discuss linear ordinary di¬erential equations. We will

not describe tricks for solving any particular equation, but instead focus on

those aspects the general theory that we will need later.

We will consider either homogeneous equations, Ly = 0 with

Ly ≡ p0 (x)y (n) + p1 (x)y (n’1) + · · · + pn (x)y, (3.1)

or inhomogeneous equations Ly = f . In full,

p0 (x)y (n) + p1 (x)y (n’1) + · · · + pn (x)y = f (x). (3.2)

We will begin with homogeneous equations.

3.1 Existence and Uniqueness of Solutions

The fundamental result in the theory of di¬erential equations is the existence

and uniqueness theorem for systems of ¬rst order equations.

3.1.1 Flows for First-Order Equations

Consider a general ¬rst order non-linear di¬erential equation in Rn

dx1

= X 1 (x1 , x2 , . . . , xn , t),

dt

69

70 CHAPTER 3. LINEAR ORDINARY DIFFERENTIAL EQUATIONS

dx2

= X 2 (x1 , x2 , . . . , xn , t),

dt

.

.

.

n

dx

= X n (x1 , x2 , . . . , xn , t). (3.3)

dt

For a su¬ciently smooth vector ¬eld (X 1 , X 2 , . . . , X n ) there is a unique solu-

tion xi (t) for any initial condition xi (0) = xi . Rigorous proofs of this claim,

0

including a statement of exactly what “su¬ciently smooth” means, can be

found in any standard book on di¬erential equations. Here, we will simply

assume the result. It is of course “physically” plausible. Regard the X i as

being the components of the velocity ¬eld in a ¬‚uid ¬‚ow, and the solution

xi (t) as the trajectory of a particle carried by the ¬‚ow. An particle initially at

xi (0) = xi certainly goes somewhere, and unless something seriously patho-

0

logical is happening, that “somewhere” will be unique.

Now introduce a single function y(t), and set

x1 = y,

x2 = y,

™

x3 = y ,

¨

.

.

.

xn = y (n’1) , (3.4)

and, given smooth functions p0 , . . . , pn with p0 nowhere vanishing, look at

the particular system of equations

dx1

= x2 ,

dt

dx2

= x3 ,

dt

.

.

.

n’1

dx

= xn ,

dt

dxn 1

p1 xn + p2 xn’1 + · · · + pn x1 .

=’ (3.5)

dt p0 (t)

Clearly this is equivalent to

dn y dn’1 y dy

p0 (t) n + p1 (t) n’1 + · · · + pn’1 (t) + pn (t)y(t) = 0. (3.6)

dt dt dt

3.1. EXISTENCE AND UNIQUENESS OF SOLUTIONS 71

Thus an n-th order ordinary di¬erential equation (ODE) can be written as a

¬rst-order equation in n dimensions, and we can exploit the uniqueness result

cited above. We conclude, provided p0 never vanishes, that the di¬erential

equation Ly = 0 has a unique solution, y(t), for each set of initial data

(y(0), y(0), y(0), . . . , y (n’1) (0)). Thus,

™ ¨

i) If Ly = 0 and y(0) = 0, y(0) = 0, y (0) = 0, . . ., y (n’1) (0) = 0, we

™ ¨

deduce that y ≡ 0.

ii) If y1 (t) and y2 (t) obey the same equation Ly = 0, and have the same

initial data, then y1 (t) = y2 (t).

3.1.2 Linear Independence

Suppose we are given an n-th order equation

p0 (x)y (n) + p1 (x)y (n’1) + · · · + pn (x)y = 0. (3.7)

In this section we will assume that p0 does not vanish in the region of x we are

interested in, and that all the pi remain ¬nite and di¬erentiable su¬ciently

many times for our formul¦ to make sense.

Let y1 (x) be a solution with initial data

y1 (0) = 1,

y1 (0) = 0,

.

.

.

(n’1)

y1 = 0. (3.8)

Let y2 (x) be a solution with

y2 (0) = 0,