’ 2 + V (|r|) ψ = Eψ

with V an attractive circular square well.

’»/πa2 , r<a

V (r) =

0, r > a.

The factor of πa2 has been inserted to make this a regulated version of

V (r) = ’»δ 3 (r). Let µ = »/πa2 .

i) By matching the functions

J0 (µr) , r<a

ψ(r) ∝

K0 (κr), r > a,

8.4. SINGULAR ENDPOINTS 251

at r = a, show that in the limit a ’ 0, we can scale » ’ ∞ in such a

way that there remains a single bound state with binding energy

4 ’2γ ’4π/»

E0 ≡ κ2 = ee .

a2

ii) Show that the associated wavefunction obeys

ψ(r) ’ 1 + ± ln r, r’0

where

1

±= .

γ + ln κ/2

Observe that this can be any real number, and so the entire range of

possible boundary conditions can be obtained by specifying the binding

energy of an attractive potential.

iii) Assume that we have ¬xed the boundary conditions by specifying κ,

and consider the scattering of unbound particles o¬ the origin. We

de¬ne phase shift δ(k) so that

ψk (r) = cos δJ0 (kr) ’ sin δN0 (kr)

2

∼ cos(kr ’ π/4 + δ), r ’ ∞.

πkr

Show that

2

cot δ = ln k/κ.

π

Exercise: Three-dimensional delta-function potential. Repeat the calculation

of the previous exercise for the case of a three-dimensional delta-function

potential

’»/(4πa3 /3), r<a

V (r) =

0, r > a.

i) Show that in the limit a ’ 0, the delta-function strength » can be

scaled to in¬nity so that the scattering length

’1

» 1

’

as =

4πa2 a

remains ¬nite.

252 CHAPTER 8. SPECIAL FUNCTIONS I

ii) Show that when this as is positive, the attractive potential supports a

single bound state with external wavefuction

1

ψ(r) ∝ e’κr

r

where κ = a’1 .

s

Exercise: The pseudo-potential. Consider a particle of mass µ con¬ned in

a large sphere of radius R. At the center of the sphere is a singular poten-

tial whose e¬ects can be parameterized by its scattering length as and the

resultant phase shift

δ(k) ≈ tan δ(k) = ’as k.

i) Show that the presence of the singular potential changes the energy En

of the l = 0, kn = nπ/R eigenstate by an amount

h2 2as kn

2

¯

∆En = .

2µ R

ii) Show that the unperturbed normalized wavefunction is

1 sin kn r

ψkn (r) = .

2πR r

iii) Show the energy shift can be written as if it were the result of applying

¬rst-order perturbation theory

d3 r|ψkn |2 Vps (r)

∆En ≈ n|Vps |n ≡

to a pseudo-potential

4πas h2 3

¯

Vps (r) = δ (r).

2µ

Although the energy shift is small, it is not a ¬rst order-e¬ect, and even the

sign of this “potential” may di¬er from the sign of the actual short distance

potential4 .

4

The pseudo-potential formula is often used to parameterize the pairwise interaction

of a dilute gas of particles of mass m, where it reads

4πas h2 3

¯

Vps (r) = δ (r).

m

The factor of two di¬erence in the denominator arises because the µ in the excercise must

be understood as the reduced mass µ = m2 /(m + m) = m/2 of the pair of interacting

particles.

8.4. SINGULAR ENDPOINTS 253

Example: The “l=0” part of the Laplace operator in n dimensions is

d2 (n ’ 1) d

+ .

dr 2 r dr

This is formally self adjoint with respect to the natural inner product

∞

r n’1 u— v dr.

u, v = (8.184)

n

0

The zero eigenvalue solutions are ψ1 (r) = 1 and ψ2 (r) = r 2’n . The second of

these ceases to be normalizable once n ≥ 4. In four dimensions and above,

therefore, we are in the limit-point case and no point interaction ” no matter

how strong ” can a¬ect the physics.

254 CHAPTER 8. SPECIAL FUNCTIONS I

Chapter 9

Integral Equations

A problem involving a di¬erential equation can often be recast as one involv-

ing an integral equation. Sometimes this new formulation suggests a method

of attack that would not have been apparent in the original language. It is

also sometimes easier to extract general properties of the solution when the

problem is expressed as an integral equation.

9.1 Illustrations

Here are some examples:

A boundary-value problem: Consider the di¬erential equation for the un-

known u(x)

’u + »V (x)u = 0 (9.1)

with the boundary conditions u(0) = u(L) = 0. To turn this into an integral

equation we introduce the Green function

1

’ L), 0 ¤ x ¤ y ¤ L,

x(y