Student
number
Semester 2 Assessment, 2024
School of Mathematics and Statistics
MAST30031 Methods of Mathematical Physics
Submission deadline: 23:59 on Monday 19 August 2024
This assignment consists of 5 pages (including this page) with 4 questions and 65 total marks
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MAST30031
Methods of Mathematical Physics
Page 2 of 5Semester 2, 2024
Question 1 (10 marks)
Write a summary of the first part of the lectures called “Hilbert spaces” which is covered in the
first three weeks.The summary should be between two and three A4 pages!To make
it simple, pickten out of the thirteen topics belowand briefly describe those in a concise
way.
•relation between inner products, norms, and metrics
•relation between inner product spaces, Hilbert spaces, Cauchy se-
quences and completeness
•relation between Riemann and Lebesgue integrals
•equivalence relations and classes
•orthogonality, orthonormality, bases and projections
•Hilbert space isomorphisms andL
2
andl
2
spaces
•bounded and unbounded operators and the operator norm
•adjoint operators and their domains
•Hermitian and self-adjoint operators and the Cayley transform
•matrix functions
•spectra of operators and the resolvent
•tempered distributions, Schwartz spaces and Gelfand triplets
•the spectral decomposition theorem for self-adjoint operators
Question 2 (10 marks)
LetPbe an orthogonal projector which is defined on a Hilbert spaceH.
(a) Show that the norm of this projector satisfies∥P∥≤1.
(b) Show that the norm of this operator satisfies∥P∥≥1 except ifP= 0.
(c) Combine the two results to specify the precise norm∥P∥of any orthogonal projectorP.
Question 3 (30 marks)
LetAbe a HermitianZ
N
-operator on a dense subspace D
A
of the Hilbert spaceH. This means
thatAis a Hermitian operator with range and domain being equal to each other andA
N
= id
the identity map on D
A
. Throughout this question we will assume thatω=e
2πi
N
, so that integer
powersω
k
ofωgenerate theNdistinctN
th
roots of unity.
(a) Letkbe an integer. Establish the orthogonality relation
1
N
N−1
X
n=0
ω
kn
=δ
(N)
k0
,
where the Kronecker deltaδ
(N)
kl
enforces equality ofkandlmoduloN.
MAST30031
Methods of Mathematical Physics
Page 3 of 5Semester 2, 2024
(b) Use the result of part (a) to show that the operatorsP
k
(k= 0,1,...,N−1) defined on
D
A
by
P
k
=
1
N
N−1
X
n=0
ω
−kn
A
n
(∗)
are projectors satisfyingP
k
P
l
=δ
kl
P
k
. Here we assume thatA
0
is the identity map.
Hint: You should think of the label of the operatorsP
k
as only being defined
moduloN, i.e.P
k
=P
k+N
(why?). Consequently, the Kronecker deltaδ
kl
is
also exhibiting this periodicity moduloN.
(c) ComputeP
†
k
, expressing the result in terms of projection operators. Under what condi-
tions onkisP
k
hermitian?
(d) Establish the completeness relation
N−1
X
k=0
P
k
= I.
(e) Use the result of (a) and (b) to express the operatorAin terms of the projectorsP
k
.
(f) Consider an arbitrary functionf:C−→C. Use your result of part (e) to explain why
one should expect the relation
f(A) =
N−1
X
n=0
1
N
N−1
X
k=0
f(ω
k
)ω
−kn
A
n
.(†)
(g) So far, all results concerned the dense subspace D
A
ofH. To extend these results to all
ofH, further work is required.
Prove thatAand the projectorsP
k
are bounded.
(If you can establish that the norms are∥A∥=∥P
k
∥= 1.)
(h) Extend the domain ofAto the whole Hilbert spaceH.
(i) Show that the spectrum ofAcan only contain the set{ω
k
|k= 0,1,...,N−1}, i.e. the
N
th
roots of unity. (Hint:Start with an educated guess for the explicit form of the
resolvent r
λ
(A) = (A−λI)
−1
with the help of (†) and show that it satisfies all three
properties for regular value, see Lecture Slides, whenλ̸=ω
k
.)
(j) Prove the identity (†) with the help of the spectral decomposition theorem. (Hint:show
thatHcan be decomposed intoNsubspaces namely the eigenspaces corresponding to
the eigenvaluesω
k
(k= 0,1,...,N−1), meaningAhas only a point spectrum.)
MAST30031
Methods of Mathematical Physics
Page 4 of 5Semester 2, 2024
Question 4 (15 marks)[This is a question from a previous exam]
Note: Some parts of this question can only be answered after the week 3 material of the lecture
has been covered.
Let{a
n
}
n∈N
0
⊂Rbe a convergent sequence with the limita
∞
= lim
n→∞
a
n
∈Rsatisfying
a
∞
̸=±a
n
for anyn∈N
0
. Consider the following Hilbert space of double-sided sequences
ℓ
2
(Z) =
(
v={v
n
}
n∈Z
⊂C:
∞
X
n=−∞
|v
n
|
2
<∞
)
with respect to the inner product
⟨v,w⟩=
∞
X
n=−∞
v
∗
n
w
n
withv={v
n
}
n∈Z
,w={w
n
}
n∈Z
∈ℓ
2
(Z)
and the operatorHonℓ
2
(Z) given by
Hv=wwithw
n
=a
|n|
v
−n
for alln∈Z.
(a) Show thatHis bounded and self-adjoint onℓ
2
(Z).
When answering this question state explicitly how self-adjointness is defined!
(b) Show that the resolventr
λ
(H) ofHis given by the operator
Kv=wwithw
n
=
λ
λ
2
−a
2
|n|
v
n
+
a
|n|
λ
2
−a
2
|n|
v
−n
for alln∈Z.
Comment: To get the result stated you will need to work with a resolvent defined by
r
λ
(H) = (λI−H)
−1
.
This is the definition you will encounter in many textbooks but it disagrees with the
definition we have seen in the lectures by an overall sign. You may use the definition
used in the lectures to do this problem but you should not be surprised to obtain an
answer that differs from the above formula by a sign.
(c) Show that the point spectrum ofHis equal to the sequence{±a
n
:n∈N
0
} ⊂Rand
the continuous spectrum is equal to the two points{±a
∞
}.
When answering this question do not forget to compute all normalised eigen-
vectors for the point spectrum!
Hint:Remember that every complex number that is not part of the set{±a
n
}
n∈N
0
∪
{±a
∞
}must have a non-vanishing distance to it. The equation for the eigenvectors can
be traced back to diagonalising 2×2 matrices.
(d) Letf:R→C. Write the explicit action off(H) on a vector inℓ
2
(Z) starting from the
spectral decomposition ofH.
End of Assignment — Total Available Marks = 65
MAST30031
Methods of Mathematical Physics
Page 5 of 5Semester 2, 2024
