Chapter 4 Linear Differential Operators

130 CHAPTER 4. LINEAR DIFFERENTIAL OPERATORS
60
40
20
0.2 0.4 0.6 0.8 1
Figure 4.2: The sum 70
n=1 2 sin(nπx) sin(nπx′ ) for x ′ = 0.4. Take note of
the very disparate scales on the horizontal and vertical axes.
Warning: The convergence of the series
n φn(x)φ ∗ n (x′ ) to δ(x − x ′ ) is
neither pointwise nor in the L 2 sense. The sum tends to a limit only in the
sense of a distribution — meaning that we must multiply the partial sums by
a smooth test function and integrate over x before we have something that
actually converges in any meaningful manner. As an illustration consider our
favourite orthonormal set: φn(x) = √ 2 sin(nπx) on the interval [0, 1]. A plot
of the first 70 terms in the sum
∞ √ √
′ ′
2 sin(nπx) 2 sin(nπx ) = δ(x − x )
n=1
is shown in figure 4.2. The “wiggles” on both sides of the spike at x =
x ′ do not decrease in amplitude as the number of terms grows. They do,
however, become of higher and higher frequency. When multiplied by a
smooth function and integrated, the contributions from adjacent positive and
negative wiggle regions tend to cancel, and it is only after this integration
that the sum tends to zero away from the spike at x = x ′ .
Rayleigh-Ritz and completeness
For the Schrödinger eigenvalue problem
Ly = −y ′′ + q(x)y = λy, x ∈ [a, b], (4.86)