Spectral Theory in Hilbert Space

2 0. INTRODUCTION
cos( √ λ t), it follows that
′′
−f = λf in I
(0.2)
f(a) = f(b) = 0,
and that g(t) = A sin( √ λ t) + B cos( √ λ t) for some constants A and
B. As is easily seen, (0.2) has non-trivial solutions only when λ is
an element of the sequence {λj} ∞ 1 , where λj = ( π
b−a j)2 . The numbers
λ1, λ2, . . . are the eigenvalues of (0.2), and the corresponding solutions
(non-trivial multiples of sin(j π (x − a))), are the eigenfunctions of
b−a
(0.2). The set of eigenvalues is called the spectrum of (0.2). In general,
a superposition of standing waves is therefore of the form u(x, t) =
(Aj sin( √ λj t) + Bj cos( √ λj t)) sin( λj (x − a)). If we assume that
we may differentiate the sum term by term, the initial conditions of
(0.1) therefore require that
Bj sin( π
b−aj(x − a)) and Aj π π j sin( j(x − a))
b−a b−a
are given functions. The question of whether (0.1) has a solution which
is a superposition of standing waves for arbitrary initial conditions, is
then clearly seen to amount to the question whether an ‘arbitrary’
function may be written as a series uj, where each term is an eigenfunction
of (0.2), i.e., a solution for λ equal to one of the eigenvalues.
We shall eventually show this to be possible for much more general
differential equations than (0.1).
The technique above was used systematically by Fourier in his Theorie
analytique de la Chaleur (1822) to solve problems of heat conduction,
which in the simplest cases (like our example) lead to what are
now called Fourier series expansions. Fourier was never able to give a
satisfactory proof of the completeness of the eigenfunctions, i.e., the
fact that essentially arbitrary functions can be expanded in Fourier
series. This problem was solved by Dirichlet somewhat later, and at
about the same time (1830) Sturm and Liouville independently but
simultaneously showed weaker completeness results for more general
ordinary differential equations of the form −(pu ′ ) ′ + qu = λu, with
boundary conditions of the form Au + Bpu ′ = 0, to be satisfied at the
endpoints of the given interval. Here p and q are given, sufficiently regular
functions, and A, B given real constants, not both 0 and possibly
different in the two interval endpoints. The Fourier cases correspond
to p ≡ 1, q ≡ 0 and A or B equal to 0.
For the Fourier equation, the distance between successive eigenvalues
decreases as the length of the base interval increases, and as the
base interval approaches the whole real line, the eigenvalues accumulate
everywhere on the positive real line. The Fourier series is then
replaced by a continuous superposition, i.e., an integral, and we get
the classical Fourier transform. Thus a continuous spectrum appears,