General eigenvalue problems

CALCULUS OF VARIATIONS
Using (22.13) and the fact that y does not appear explicitly, we obtain
∂y
∂
∂y
∂
ρ
−
τ
= 0.
∂t
∂t
∂x
∂x
If, in addition, ρ and τ do not depend on x or t then
∂2 y
1 ∂2 y
= 2 2,
∂x2
c ∂t
where c2 = τ/ρ. This is the wave equation for small transverse oscillations of a taut
uniform string. 22.6 General eigenvalue problems
We have seen in this chapter that the problem of finding a curve that makes the
value of a given integral stationary when the integral is taken along the curve
results, in each case, in a differential equation for the curve. It is not a great
extension to ask whether this may be used to solve differential equations, by
setting up a suitable variational problem and then seeking ways other than the
Euler equation of finding or estimating stationary solutions.
We shall be concerned with differential equations of the form Ly = λρ(x)y,
where the differential operator L is self-adjoint, so that L = L† (with appropriate
boundary conditions on the solution y) and ρ(x) is some weight function, as
discussed in chapter 17. In particular, we will concentrate on the Sturm–Liouville
equation as an explicit example, but much of what follows can be applied to
other equations of this type.
We have already discussed the solution of equations of the Sturm–Liouville
type in chapter 17 and the same notation will be used here. In this section,
however, we will adopt a variational approach to estimating the eigenvalues of
such equations.
Suppose we search for stationary values of the integral
b
2
(22.22)
p(x)y (x) − q(x)y 2 (x) dx,
I=
a
with y(a) = y(b) = 0 and p and q any sufficiently smooth and differentiable
functions of x. However, in addition we impose a normalisation condition
b
ρ(x)y 2 (x) dx = constant.
(22.23)
J=
a
Here ρ(x) is a positive weight function defined in the interval a ≤ x ≤ b, but
which may in particular cases be a constant.
Then, as in section 22.4, we use undetermined Lagrange multipliers,§ and
§
We use −λ, rather than λ, so that the final equation (22.24) appears in the conventional Sturm–
Liouville form.
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22.6 GENERAL EIGENVALUE PROBLEMS
consider K = I − λJ given by
b
K=
2
py − (q + λρ)y 2 dx.
a
On application of the EL equation (22.5) this yields
d
dy
p
+ qy + λρy = 0,
dx
dx
(22.24)
which is exactly the Sturm–Liouville equation (17.34), with eigenvalue λ. Now,
since both I and J are quadratic in y and its derivative, finding stationary values
of K is equivalent to finding stationary values of I/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 extremising K. Thus we have the important
result that finding functions y that make I/J stationary is equivalent to finding
functions y that are solutions of the Sturm–Liouville equation; the resulting value
of I/J equals the corresponding eigenvalue of the equation.
Of course this does not tell us how to find such a function y and, 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 functions p(x), q(x) and ρ(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 eigenvalues λi of (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 eigenfunctions yi (x) satisfy the orthogonality relation (as
in (17.24))
b
yi yj ρ dx = δij .
(22.25)
a
Further, we take the boundary condition in the form
yi pyj
x=b
x=a
= 0;
(22.26)
this can be satisfied by y(a) = y(b) = 0, but also by many other sets of boundary
conditions.
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