Since Q† = ’e 2 x ‚x e’ 2 x , we can write

1 2

ψn (x) = Hn (x)e’ 2 x , (4.106)

where

dn ’x2

n x2

Hn (x) = (’1) e e (4.107)

dxn

are the Hermite Polynomials.

Exercise: Show that these are the only eigenfunctions and eigenvalues. Hint:

Show that Q lowers the eigenvalue by 2 and use the fact that Q† Q cannot

have negative eigenvalues.

This is a useful technique for any second-order operator that can be fac-

torized ” and a surprising number of the equations for “special functions”

can be. You will see it later, both in the exercises and in connection with

Bessel functions.

4.3.2 Continuous spectrum

Rather than a give formal discussion, we will illustrate this subject with some

examples drawn from quantum mechanics.

The simplest example is the free particle on the real line. We have

2

H = ’‚x . (4.108)

4.3. COMPLETENESS OF EIGENFUNCTIONS 105

We eventually want to apply this to functions on the entire real line, but we

will begin with the interval [’L/2, L/2], and then take the limit L ’ ∞

The operator H has formal eigenfunctions

•k (x) = eikx , (4.109)

corresponding to eigenvalues » = k2 . Suppose we impose periodic boundary

conditions at x = ±L/2:

•k (’L/2) = •k (+L/2). (4.110)

This selects kn = 2πn/L, where n is any positive, negative or zero integer,

and allows us to ¬nd the normalized eigenfunctions

1

χn (x) = √ eikn x . (4.111)

L

The completeness relation is

∞

1 ikn x ’ikn x

= δ(x ’ x ), x, x ∈ [’L/2, L/2].

ee (4.112)

L

n=’∞

As L becomes large, the eigenvalues become so close that they can hardly be

distinguished; hence the name continuous spectrum5 , and the spectrum σ(H)

becomes the entire positive real line. In this limit, the sum on n becomes an

integral

∞

dn

. . . ’ dn . . . = dk ... , (4.113)

dk

n=’∞

where

dn L

= (4.114)

dk 2π

is called the (momentum) density of states. If we divide this by L to get a

density of states per unit length, we get an L independent “¬nite” quantity,

the local density of states. We will often write

dn

= ρ(k). (4.115)

dk

5

When L is strictly in¬nite, •k (x) is no longer normalizable. Mathematicians do not

allow such un-normalizable functions to be considered as true eigenfunctions, and so a

point in the continuous spectrum is not, to them, actually an eigenvalue. Instead, mathe-

maticians say that a point » lies in the continuous spectrum if for any > 0 there exists

an approximate eigenfunction • such that • = 1, but L• ’ »• < . This is not a

pro¬table de¬nition for us.

106 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS

If we express the density of states in terms of the eigenvalue » then, by

an abuse of notation, we have

dn L

= √.

ρ(») ≡ (4.116)

d» 2π »

Note that

dn dn dk

=2 , (4.117)

d» dk d»

which looks a bit weird, but remember that two states, ±kn , correspond to

the same » and that the symbols

dn dn

, (4.118)

dk d»

are ratios of measures, i.e. Radon-Nykodym derivatives, not ordinary deriva-

tives.

In the L ’ ∞ limit, the completeness relation becomes

dk ik(x’x )

∞

= δ(x ’ x ),

e (4.119)

2π

’∞

and the length L has disappeared.

Suppose that we now apply boundary conditions y = 0 on x = ±L. The

normalized eigenfunctions are then

2

χn = sin kn (x + L/2), (4.120)

L

where kn = nπ/L. We see that the allowed k™s are twice as close together as

they were with periodic boundary conditions, but now n is restricted to being

a positive non-zero integer. The momentum density of states is therefore

dn L

ρ(k) = =, (4.121)

dk π

which is twice as large as in the periodic case, but the eigenvalue density of

states is

L

ρ(») = √ , (4.122)

2π »

which is exactly the same as before.

4.3. COMPLETENESS OF EIGENFUNCTIONS 107

That the number of states per unit energy per unit volume does not

depend on the boundary conditions at in¬nity makes physical sense: no

local property of the sublunary realm should depend on what happens in

the sphere of ¬xed stars. This point was not fully grasped by physicists,

however, until Rudolph Peierls6 explained that the quantum particle had to

actually travel to the distant boundary and back before the precise nature

of the boundary could be felt. This journey takes time T (depending on

the particle™s energy) and from the energy-time uncertainty principle, we

can distinguish one boundary condition from another only by examining the

spectrum with an energy resolution ¬ner than h/T . Neither the distance nor

¯

the nature of the boundary can a¬ect the coarse details, such as the local

density of states.

The dependence of the spectrum of a general di¬erential operator on

boundary conditions was investigated by Hermann Weyl. Weyl distinguished

two classes of singular boundary points: limit-circle, where the sepctrum

depends on the choice of boundary conditions, and limit-point, where it does

not. For the Schr¨dinger operator, the point at in¬nity, which is “singular”

o

simply because it is at in¬nity, is in the limit-point class. We will discuss the

Weyl™s theory of singular endpoints in chapter 8.

Phase-shifts

Consider the eigenvalue problem

d2