Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.6 GENERAL EIGENVALUE PROBLEMS

considerK=I−λJgiven by

K=

∫b

a

[

py′

2

−(q+λρ)y^2

]

dx.

On application of the EL equation (22.5) this yields

d

dx

(

p

dy

dx

)

+qy+λρy=0, (22.24)

which is exactly the Sturm–Liouville equation (17.34), with eigenvalueλ. Now,

since bothIandJare quadratic inyand its derivative, finding stationary values

ofKis equivalent to finding stationary values ofI/J. This may also be shown

by considering the functional Λ =I/J, for which

δΛ=(δI/J)−(I/J^2 )δJ

=(δI−ΛδJ)/J

=δK/J.

Hence, extremising Λ is equivalent to extremisingK. Thus we have the important

result thatfinding functionsythat makeI/Jstationary is equivalent to finding

functionsythat are solutions of the Sturm–Liouville equation; the resulting value

ofI/Jequals the corresponding eigenvalue of the equation.

Of course this does not tell us how to find such a functionyand, naturally, to

have to do this by solving (22.24) directly defeats the purpose of the exercise. We

will see in the next section how some progress can be made. It is worth recalling

that the functionsp(x),q(x)andρ(x) can have many different forms, and so

(22.24) represents quite a wide variety of equations.

We now recall some properties of the solutions of the Sturm–Liouville equation.

The eigenvaluesλiof (22.24) are real and will be assumed non-degenerate (for

simplicity). We also assume that the corresponding eigenfunctions have been made

real, so that normalised eigenfunctionsyi(x) satisfy the orthogonality relation (as

in (17.24))

∫b

a

yiyjρdx=δij. (22.25)

Further, we take the boundary condition in the form

[

yipy′j

]x=b

x=a

= 0; (22.26)

this can be satisfied byy(a)=y(b) = 0, but also by many other sets of boundary

conditions.