Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.7 WKB METHODS

one function and the same function may need different expansions for

different values of argz.

Finally in this subsection we note that, although the form of equation (25.42) may

appear rather restrictive, in that it contains no term iny′, the results obtained so

far can be applied to an equation such as

d^2 y

dz^2

+P(z)

dy

dz

+Q(z)y=0. (25.56)

To make this possible, a change of either the dependent or the independent

variable is made. For the former we write

Y(z)=y(z)exp

(

1

2

∫z

P(u)du

)

⇒

d^2 Y

dz^2

+

(

Q−

1

4

P^2 −

1

2

dP

dz

)

Y=0,

whilst for the latter we introduce a new independent variableζdefined by

dζ

dz

=exp

(

−

∫z

P(u)du

)

⇒

d^2 y

dζ^2

+Q

(

dz

dζ

) 2

y=0.

In either case, equation (25.56) is reduced to the form of (25.42), though it

will be clear that the two sets of WKB solutions (which are, of course, only

approximations) will not be the same.

25.7.4 The Stokes phenomenon

As we saw in subsection 25.7.2, the combination of WKB solutions of a differential

equation required to reproduce the asymptotic form of the accurate solutiony(z)

of the same equation, varies according to the region of thez-plane in whichzlies.

We now consider this behaviour, known as the Stokes phenomenon, in a little

more detail.

Lety 1 (z)andy 2 (z) be the two WKB solutions of a second-order differential

equation. Then any solutionY(z) of the same equation can be written asymptot-

ically as

Y(z)∼A 1 y 1 (z)+A 2 y 2 (z), (25.57)

where, although we will be considering (abrupt) changes in them, we will continue

to refer toA 1 andA 2 as constants, as they are within any one region. In order to

produce the required change in the linear combination, as we pass over a Stokes

line from one region of thez-plane to another, one of the constants must change

(relative to the other) as the border between the regions is crossed.

At first sight, this may seem impossible without causing a discernible discon-

tinuity in the representation ofY(z). However, we must recall that the WKB

solutions are approximations, and that, as they contain a phase integral, for

certain values of argzthe phaseφ(z) will be purely imaginary and the factors