Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

Show that

∫b

a

(

y′jpy′i−yjqyi

)

dx=λiδij. (22.27)

Letyibe an eigenfunction of (22.24), corresponding to a particular eigenvalueλi,sothat
(
py′i

)′

+(q+λiρ)yi=0.

Multiplying this through byyjand integrating fromatob(the first term by parts) we
obtain
[
yj

(

py′i

)]b

a

−

∫b

a

yj′(pyi′)dx+

∫b

a

yj(q+λiρ)yidx=0. (22.28)

The first term vanishes by virtue of (22.26), and on rearranging the other terms and using
(22.25), we find the result (22.27).

We see at once that, if the functiony(x) minimisesI/J, i.e. satisfies the Sturm–

Liouville equation, then puttingyi=yj=yin (22.25) and (22.27) yieldsJand

Irespectively on the left-hand sides; thus, as mentioned above, the minimised

value ofI/Jis just the eigenvalueλ, introduced originally as the undetermined

multiplier.

For a functionysatisfying the Sturm–Liouville equation verify that, provided (22.26) is

satisfied,λ=I/J.

Firstly, we multiply (22.24) through byyto give

y(py′)′+qy^2 +λρy^2 =0.

Now integrating this expression by parts we have
[
ypy′

]b

a

−

∫b

a

(

py′^2 −qy^2

)

dx+λ

∫b

a

ρy^2 dx=0.

The first term on the LHS is zero, the second is simply−Iand the third isλJ. Thus
λ=I/J.

22.7 Estimation of eigenvalues and eigenfunctions

Since the eigenvaluesλiof the Sturm–Liouville equation are the stationary values

ofI/J(see above), it follows that any evaluation ofI/Jmust yield a value that lies

between the lowest and highest eigenvalues of the corresponding Sturm–Liouville

equation, i.e.

λmin≤

I

J

≤λmax,

where, depending on the equation under consideration, eitherλmin=−∞and