MA 342N
Assignment 1
Due 24 February 2016
Id:
342N-s2016-1.m4,v 1.1 2016/02/15 21:25:36 john Exp john
1. Suppose that q, in addition to satisfying the assumptions from lecture,
is an even function. Prove that η(λ) = 0, i.e. that there are no mixed
terms (θϕ or ϕθ terms) in the spectral decomposition.
Hint: Show that
ψ+ (λ, −x) = ψ− (λ, x),
ϕ(λ, −x) = −ϕ(λ, x),
ψ− (λ, −x) = ψ+ (λ, x),
θ(λ, −x) = θ(λ, x).
2. Consider the set of functions y which satisfy y ′′ + λy = 0 for x 6= 0, are
continuous at x = 0, and whose derivatives are continuous except for a
jump discontinuity of the form
lim y ′ (x) − lim− y ′ (x) = Ay(0).
x→0+
x→0
These are close enough to the situation considered in lecture that we
can still do scattering theory and spectral theory.1
1
If we take the limit of our special solutions to y ′′ + λy = qy with
h if |x| < w/2,
q(x) =
0 if |x| > w/2
with y and y ′ continuous at x = ±w/2 as w → 0 and h → ∞ in such a way that A = wh
is constant then we get solutions to the problem above. Formally this is often written as
y ′′ + λy = qy with q(x) = Aδ(x), where δ is the Dirac delta. Unfortunately that doesn’t
really make any sense, even if the equations are interpreted in the sense of distributions.
The simplest way of treating the problem is to ignore both delta functions and the h → 0
limit and just view the equation as y ′′ + λy = 0 with the jump condition given above.
1
Id:
342N-s2016-1.m4,v 1.1 2016/02/15 21:25:36 john Exp john 2
Find the scattering matrix for this problem, and verify that it is unitary.
Note: The basic existence and uniqueness theorem still applies here,
despite the strangeness of the equation at x = 0. That’s not hard
to prove, but it’s also not interesting, so don’t bother. You can just
assume there is a unique solution for any choice of initial data at any
point, except that if we are taking data at x = 0 then we need to specify
the average of the limits
1
2
lim+ y ′ (x) + lim− y ′(x)
x→0
x→0
rather than the (probably non-existent) y ′(0).
Hint: The functions θ and ϕ are still useful, but their definitions need
to be modified as described above:
θ(λ, 0) = 1,
ϕ(λ, 0) = 0,
1
2
1
2
x→0
lim+ θ′ (λ, x) + lim− θ′ (λ, x) = 0,
x→0
′
′
lim ϕ (λ, x) + lim− ϕ (λ, x) = −1.
x→0+
x→0
3. Find the spectral representation for the same problem.