ASSIGNMENT 2, 18.155
DUE MIDNIGHT MONDAY 30 NOVEMBER, 2011
Date: Mon Nov 28 11:11:55 2011
This assignment replaces Problem set 10 and should represent a little
more work than for one of the Problem sets. You can submit work at
any time (electronically as usual) for me to review and return – the
mark will be based on the final submission only. You can also ask for
help etc, conditions as for a Problem set – I do not mind what or who
you consult as long as the final product is your own.
In particular I do not mind if two, or three, people collaborate on
one project subject to the conditions above.
Here are the projects, with a list of who has asked to work on what
to date. If someone really wants to work on something else I am willing
to negtotiate.
(1) If P (D) is an elliptic operator with constant coefficients and
Ω ⊂ Rn show that P (D) : C ∞ (Ω) −→ C ∞ (Ω) is surjective.
Involves estimates!
(2) Give a proof of Kuiper’s theorem ,that the group of unitary operators on a separable but infinite-dimensional Hilbert space is
‘weakly contractible’ (meaning all its homotopy groups vanish)
in the norm topology. Has topological consequences (e.g. that
the projective unitary group is a K(Z, 2)). Standard proofs are
a bit of a slog!
P
(3) Show that the harmonic oscillator, (Dj2 +x2j ) is an unbounded
j
self-adjoint operator on L2 (Rn ) with domain H 2 (Rn ) ∩ {u ∈
L2 ; |x|2 u ∈ L2 }. Show that its eigenfunctions are of the form
pα (x) exp(− 21 |x|2 ) with the pα polynomials and that these can
also be chosen to be eigenfunctions for the Fourier transform.
(4) Construct the forward fundamental solution for the wave opern
P
ator P = Dt2 −
Dj2 on R × Rn (the one with support in t ≥ 0)
j=1
and show that it is unique. Use this to prove that ‘forcing problem’ P u = f where f ∈ C −∞ (R1+n ) has support in t ≥ 0 has a
unique solution u ∈ C −∞ (R1+n ) with support in t ≥ 0 and that
this solution is smooth if f is smooth.
(5) Describe Wagner’s proof of the Ehrenpreis-Malgrange theorem
– that any non-trivial P (D) has a fundamental solution. MR2510844.
1
2
ASSIGNMENT 2, 18.155 DUE MIDNIGHT MONDAY 30 NOVEMBER, 2011
(6) Fredholm properties of the Laplacian on Euclidean space.
(7) Discuss Lidskii’s theorem – that the trace of a compact operator
is the sum of its eigenvalues. Harder than you might think but
not impossible.
(8) Radon transform. Discuss some of the properties of the Radon
transform as a map S(Rn ) −→ C ∞ (R × Sn−1 ) given by
Z
f (x)dH(s, ω), H(s, ω) = {x ∈ Rn ; x · ω = s}.
(1) Rf (s, ω) =
H(s,w)
(9) A little Fréchet analysis. Show that the operators on L2 (Rn )
with Schwartz kernels in S(R2n ) form a topological algebra with
the product giving a (metrically) continuous map
Z
n
2n
(2) S(R ) × S(R ) 3 (a, b) 7−→ a ◦ b(x, y) =
a(x, z)b(z, y)dz.
Rn
(3)
Prove that the group of those elements a ∈ S(R2n ) such that
Id +a (as an operator) has inverse Id +b for b ∈ S(R2n ) is an
open and dense subset of S(R2n ). If you want, you can go on to
show that the (unnormalized) Chern forms
Tr (g −1 dg)2k+1
are well-defined closed forms on this group.
(10) Dirichlet problem in a (smoothly) bounded domain – existence
of a complete orthonormal eigenbasis of L2 . (Jonathan)
(11) Poisson summation formula.
(12) Eigenvalues for Hill’s operator – including some asymptotics
(Alex)
Elliptic operators1
No one has asked about this. I think you can find a proof in Hörmander’s
four volume treatise, probably in Volume 2. It is a special case of ‘P pseudoconvexity’ in the sense that any open set is pseudoconvex with
respect to an elliptic operator.
Kuiper’s Theorem 2
No one has asked about this. The theorem is that a norm continuous map from a sphere of any dimension into GL(H) or U(H) can be
extended to a continuous map from the ball – with the original map as
the restriction to the sphere. This is ‘weak contractibility’ which here
means that all the homotopy groups vanish.
ASSIGNMENT 2, 18.155
DUE MIDNIGHT MONDAY 30 NOVEMBER, 2011 3
Harmonic oscillator 3
Here is a slightly modified version of what I wrote to Yasha.
One of the significant features of the harmonic oscillator is that one
can get elliptic regularity directly, using what we already know and
some commutation. The important point is that although the coefficients are not constant, they are pretty nice in that for instance
[∂j , xj ] = 1.
Here is one way to proceed.PSuppose you know that u ∈ L2 (Rn ) and
Hu ∈ L2 (bbRn ) where H = (−∂j2 + x2j ) is the harmonic oscillator.
j
This for instance is where you would start if you wanted to show that
H is an unbounded self-adjoint operator, of course Hu ∈ S 0 (Rn ) so
the assumption does make sense. Now, by local elliptic regularity, we
can see that u ∈ H 2 (Rn ) locally – just write the equation Hu = f as
∆u = f −|x|2 u. Now, consider a cutoff ψn = ψ(x/n) where ψ ∈ Cc∞ (Rn )
is equal to 1 near 0. The idea is to start from the integral
Z
(ψn2 u)Hu = hφ2n u, f uiL2
and integrate by parts. In fact, take the real part of this – which cancels
a term. Working on the (real part of) the left side, the piece |x|2 of H
contributes
Z
|x|2 φ2n |u|2
which is positive and useful. Integration by parts from the derivative
terms gives
Z
|∇(φn u)|2
which is also good, plus some commutator terms which you should
work out. These are of the form (I am being very casual, look at it
carefully!)
φn φ00n |u|2 + φn φ0n u∂j u.
Because we take the real part we can integrate again and get a cancelation from the second terms so that the whole commutator looks, after
integration like the first term above. Really then what one gets is
Z
|∇φn u|2 + |x|2 |φn u|2 = Bn (u) + Rehφ2n u, f i.
where Bn is quadratic in u but involves no derivatives and in fact you
can easily see that since u ∈ L2 , Bn (u) → 0 as n → ∞.
It follows easily from this that φn u converges to u in H 1 (Rn ) and
also that |x|u ∈ L2 (Rn ). Going back to the beginning you can now see
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ASSIGNMENT 2, 18.155 DUE MIDNIGHT MONDAY 30 NOVEMBER, 2011
that Hφn u → Hu in L2 because the commutator terms vanish in the
limit.
Now do a similar thing again, looking at kHφn uk2L2 . There are two
big positive terms and some cross terms. If you integrate by parts
in the cross term you can get another positive term, basically the sum
kxj ∂j φn uk2L2 plus a term which is now controlled by the fact that φn u →
u in H 1 etc. This now shows that u ∈ H 2 and |x|2 u ∈ L2 .
For an eigenfunction you can easily iterate the argument to get u ∈
∞
H , or you can use this to prove that H is self-adjoint with domain
consisting of those u ∈ H 2 such that |x|2 u ∈ L2 . In that case you
can see easily that H is invertible with inverse a compact self-adjoint
operator.
This still leaves the problem of showing that all eigenfunctions are
as stated.
Wave equation 4
No one asked about this so far – I will probably do some version of
it in the last few lectures.
Ehrenpreis Malgrange theorem 5
No one has asked about this, but probably the reference above is
enough.
Euclidean Laplacian6
Find Sobolev (type) spaces between which the standard Laplacian
on Euclidean space is Fredholm.
No one has asked about this, but I will outline some version of it in
the last few lectures.
Lidskii’s Theorem 7
Discuss the properties of theresolvent (z − A)−1 for A ∈ K(H)
(we almost did this) far enough to define the eigenvalues (the poles
of the resolvent) and their multiplicity, the dimension of the associated generalized eigenspaces – which is to say the integer m(z) =
limk→∞ dim(Nul(A − z)k ) (the finiteness of which is the main issue).
The Lidskii’s theorem says that for A of trace class
X
(4)
Tr(A) =
m(z).
z∈C
ASSIGNMENT 2, 18.155
DUE MIDNIGHT MONDAY 30 NOVEMBER, 2011 5
Radon tranform 8
You can find a treatment of this in the book ‘Scattering theory’ by
Lax and Phillips and other places too. The main thing to do is to get
boundedness and invertibility properties on Sobolev spaces using the
Fourier transform. More information on request.
Frechet analysis 9
To show that these ‘Schwartz’ operators form a topological group
you only need to show that the composition and inversion maps are
continuous. For the composition this is straightforward since the kernel
of the composite of two operators Id +A and Id +B is Id +C where
C = A + B + AB and the kernel of AB is
Z
0
(5)
e(x, x ) = a(x, x00 )b(x00 , x0 )dx00 .
It is easy to see that this gives a continuous bilinear map with respect
to the metric. Continuity of the inverse follows by use of Neumann
series and a little care about the estimates above. Density is not so
hard either, I can offer help with this if anyone asks.
Dirichlet problem10
Suppose that Ω ⊂ Rn is a bounded ‘domain’ meaning it is a bounded
open set and is the interior of its closure. Now, define H01 (Ω) to be the
closure in H 1 (bbRn ) of Cc∞ (Ω). On this space we can define a bilinear
form, the Dirichlet form, by
Z
∇u · ∇vdx
(6)
BD (φ, ψ) =
Ω
where ∇ : H01 (Ω) −→ L2 (Ω) is the continuous extension of the gradient. The dual of H01 (Ω) can be identified (in terms of the extension of
the usual L2 pairing) with H −1 (Ω) which is the image in C −∞ (Ω) of
H −1 (Rn ). The Poincaré inequality here then shows that ∆ : H01 (Ω) −→
H −1 (Ω), defined distributionally, is a topological isomorphism. Thus,
for instance for every f ∈ L2 (Ω) ,→ H −1 (Ω)
unique
R there exists a ∞
1
u ∈ H0 (Ω) satisfying ∆u = f (or BD (u, φ) = f φ for all φ ∈ Cc (Ω).
Now show that the inverse of ∆ in this sense is compact as an operator on L2 (Ω) and hence there is a complete orthonomal basis of
eigenfunctions for the Dirichlet problem.
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ASSIGNMENT 2, 18.155 DUE MIDNIGHT MONDAY 30 NOVEMBER, 2011
Poisson summation formula 11
Consider the periodic functions on Rn with respect to the standard
lattice, i.e. those satisfying
(7)
f (x + 2πkei ) = f (x)
for all k ∈ Z and i = 1, . . . , n. If f ∈ L2loc is periodic in this sense then
it is certainly a tempered distribution (show this of course) and the
main thing to do is to relate the Fourier transform – another tempered
distribution – and the Fourier coefficients
Z
(8)
cl =
exp(−il · x)f (x), l ∈ Zn
C
where C is the product [0, 2π]n ⊂ Rn . [You can change the normalization I have chosen here if you want.] In particular, express fˆ in terms
of the cl .
The Poisson summation formula takes this a little further. Suppose
g ∈ S(Rn ) then one can get an f as above by summing the translates
of g
X
g(x + 2πl).
(9)
f (x) =
l∈Zn
Now, express the Fourier coefficients of f in terms of the Fourier transform of g. You should also answer the question:P
What is the Fourier transform of
δ(x + 2πl)?
l∈Zn
Hill’s operator12
One of the standard applications of spectral theory of compact selfadjoint operators is to unbounded self-adjoint operators and a particular case of this is Hill’s equation. Suppose that V is a real-valued and
continuous 2π-periodic function on R. The basic question is to ask for
eigenfunctions of the differential operator
d2
+ V (x), so P u = λu, u 2π-periodic.
dx2
Of course the choice of period is a matter of taste. For a classical
solution one would ask for u to be twice contiuously differentiable, we
2
could just ask for u to be in Hloc
(R) (and 2π-periodic).
So, the theorem is that there is a sequence of real eigenvalues λj → ∞
and a corresponding set of eigenfunctions which form an orthonormal
basis of the Hilbert space of 2π-periodic functions on the line. I would
also like some estimate showing that the λj are close to j 2 as j → ∞.
The latter can be proved by using the trace ideal.
(10)
P =−
ASSIGNMENT 2, 18.155
DUE MIDNIGHT MONDAY 30 NOVEMBER, 2011 7
I said some more privately to Alex and if anyone else wants to do
this I will do the same for them (or write it out).