WEEK
13

HOMEWORK
SESSION 
WEEK
13

WARMUP
FOR
THE
FINAL
TEST 
 E49  The operator domain of Laplacian+V can
collapse to
{0} even if the form domain is dense.
 E50  A positive selfadjoint operator with a
gap from 0 is
invertible. The square root of positive
selfadjoint operator is
monotone. This is false for powers higher than 1.
Also the product of
two positive operators is not necessarily
positive.
 E51  Preparatory estimates for relative
compactness.
 E52  Estimating spectral eigenspaces.


WEEK
12

HOMEWORK
SESSION 
WEEK
12

TUTORIAL

 E41
 A
bounded operator on a Hspace is compact iff it
maps weakly convergent
into norm convergent sequences.
 E42
 Wave
operators: closure of domain and range, invariance
properties,
intertwining property.
 E43
 Cook's
criterion for the existence of the wave operator.
Application to
Laplacian+V with a square integrable V.
 E44
 The
oneparamenter strongly continuous unitary group
of dilations and the
computation of its generato

 f(x)g(\nabla) is defined via Ftransform as an
operator on
L^2(R^d). Its boundedness depends on the
integrability properties of f
and g.
 In particular if both f and g are
squareintegrable, then
f(x)g(\nabla) is HilbertSchmidt.
 If f and g are bounded and decay at infinity,
then
f(x)g(\nabla) is compact.

WEEK
11

WARMUP
FOR
THE
FINAL
TEST

WEEK
11

TUTORIAL

Selfadjointess,
spectral
theorem, functional calculus, and all that.

 Unitary transformations preserve
selfadjointness,
spectrum, point spectrum. (See solution to Exercise
40)
 Adjoint of a sum and of a product of operators
in the
unbounded case. (See Problem
15.)
 Stepbystep computation of the adjoint of the
momentum
operator on [0,2pi] with boundary conditions
f(0)=f(2pi)=0. (See Problem
14.) In fact,
if one knows already the definition and the
properties of H^1, then the
computation of the adjoint is immediate.
 You may want to practise with some of these problems
on
selfadjointness, spectral theorem, functional
calculus.

WEEK
10

HOMEWORK
SESSION 
WEEK
10

TUTORIAL

 E37  The Volterra integral operator on
L^2[0,1].
Boundedness,
compactness, absence of eigenvalues, spectrum
collapses to {0} only.
 E38  HilbertSchmidt operators on L^2 are
realised as
integral
operators with squareintegrable kernel.
 E39  Multiplication operator by a measurable
function.
Maximal
domain. Adjoint. The operator is selfadjoint iff
the function it
multiplies by is realvalued. Its spectrum
coincides with the essential
range of that function. Measure characterisation
of the eigenvalues.
 E40  Position and momentum operators:
selfadjointness,
absence
of eigenvalues, spectrum. In fact, P and Q are
unitarily equivalent,
via Ftransform.
Note: Vito
Volterra, the mathematical physicist, after
whom the integral
operator of E37 is named. Volterra,
the old
capital of Etruscans.

 Functional calculus from the Spectral Theorem.
The general
idea
is in Section
7.3
of
the
handout. The complete statement, i.e.,
construction of bounded Borel functions of a
(possibly unbounded)
selfadjoint operator, is Theorem VIII.5 of
ReedSimon.
 IMPORTANT
MESSAGE 1: it is part of (the proof of) the
Spectral Theorem that the
functional calculus produces a p.v.m. (when one
constructs functions of
A starting from characteristic functions of Borel
sets) that is
*precisely* the p.v.m. associated with A in the
statement of the
Spectral Theorem.
 IMPORTANT MESSAGE 2: in practice one just
constructs
polynomials of A in the usual way; the general
f(A) is given by a
strong limit (note part (d) of Theorem VIII.5 of
ReedSimon) using
the uniform density of polynomials in (bounded)
continuous functions
and the pointwise density of the latter in bounded
Borel functions.
 Mesaure types: decomposition of a Borel measure
in pp, sc,
ac
part. Reference: ReedSimon, Section I.4, pag
1922, and Teschl,
Section
A.7 of the appendix.

WEEK
9

HOMEWORK
SESSION 
WEEK
9

TUTORIAL

 E33  The bound states of a Schrödinger
Hamiltonian in
dimension 3 or higher with potential in
L^{d/2}+L^{\infty} are
exponentially localised wave functions.
 E34  System of two nuclei very much far apart
with two
electrons:
ground state energy.
 E35  Schur's test for kernel (integral)
operators.
 E36  Kernel (integral) operators with square
integrable
kernels.
The operator norm is dominated by the L^2norm of
the kernel.
Approximation in operator norm with finiterank
operators. The
HilbertSchmidt norm is the same irrespectively of
the orthonormal
basis considered.
Schur's test, discussed in E35, was introduced by
Schur in Bemerkungen
zur
Theorie
der
beschränkten
Bilinearformen
mit
unendlich
vielen
Veränderlichen (1911)

 Projectionvalued measures (equivalently,
spectral
resolution).
 Integration
w.r.t. a projectionvalued measure in complete
analogy to the
integration w.r.t. Lebesgue.
 Construction of a (possibly unbounded)
selfadjoint operator and characterisation of its
domain. Statement of
the Spectral Theorem in p.v.m. form (Theorem
7.3
of the handout,
Theorem VIII.6 of ReedSimon.)
 Example: the case of a symmetric matrix (working
out
explicitly the Example
in
pag. 29 of the handout).

WEEK
8

HOMEWORK
SESSION 
WEEK
8

TUTORIAL

 E25  If the solution f to (DV)f=0 in 3dim (V
nonzero,
nonnegative, and locally integrable) is smooth
and nonnegative, then
either f>0 or f=0.
 E26  The ground state of a single well
potential has only
one
single peak.
 E27  Helium Hamiltonian: variational upper
bounds to the
ground
state energy.
 E28  Antisymmetric wave functions: L2scalar
product of
wedge
products.

 Tensor product of Hilbert spaces. Rigorous
definitions and
properties. (References: Teschl, chapter
1.4, ReedSimon,
chapter
II.4.)
 Revision on bounded linear operators on a
Hilbert space,
first
part:
 Definition of the algebra B(H) of bounded
linear
operators on H. The operator norm makes it a
C*algebra. The abstract
definitions of resolvent set, spectrum,
resolvent operator, as well as
the properties discussed in abstract in the
tutorial of week 03, carry
over to this concrete case.
 Definition of the (Hilbert) adjoint of an
operator in
B(H).
 Definition of selfadjoint operator and of
orthogonal
projection.
(Reference: ReedSimon, chapter VI.)
Suggested problems
in
class:
 P11  Orthogonal projections on a Hilbert space.
Kernel and
range give an ortogonal decomposition of H.
Spectrum of the projection.
Resolvent operator of the projection.
 P12  Explicit computation of the norm and the
adjoint of
an operator on L^2(R).
 P13  Important formulas for a bounded operator
T on a
Hilbert space. KerT* is the orthogonal complement
to RanT. The special
case of normal operators.

WEEK
7

MIDTERM
EXAM

WEEK
7

TUTORIAL

midterm
exam
 Integral of sinc(x)^2: Ftransform, Parseval.
 The distributional solution to xu'+u=0 is a
linear
combination of the delta and the principal value.
 Weak convergence in H^1(Omega) implies weak
convergence in
L^2(Omega). This is always true (it's a general
Banach space fact),
here you were asked to prove it for Omega=R, thus
by means of Fourier
transform.
 Cyclic vectors for multiplication by x on
L^2[1,1]. The
function 1 is, the Heaviside function is not.
Multiplication by x^2 on
L^2[1.1] has no cyclic vectors.
 In 3 dimensions the H^2 norm controls the
L^infinity norm.
 D+tV in d>2 dimensions with a V that is
nonpositive,
vanishing at infinity, and d/2integrable, and
with negative ground
state energy for some t_0, has a ground state
energy E(t) that is
strictly decreasing in t for t>t_0.
Despite what claimed in some solution sheets, the
answer to the cyclic
vector problem is NOT Spongebob,
sorry.

problems
in class
 P07  In a Hspace convergence against elements
of an ONB + boundedness of norms is the same as
weak convergence.
 P08  Useful identities and inequalities
involving
resolvents.
 P09  Integral of sinc(x)^4: Ftransform,
Parseval.
 P10  Higher order Sobolev norms control the L^p
norm.

WEEK
6

HOMEWORK
SESSION

WEEK
6

TUTORIAL

assignment
06
 E21  Construction of cutoff functions. IMS
localisation
formula.
 E22  Potentials vanishing at infinity in dim=3
(or higher)
give
rise to nonpositive ground states.
 E23  Variational bound from above and
localisation
estimate from
below of the ground state of a system made of an
electron and two fixed
nuclei.
 E24  Potentials that scale under dilations give
rise to a
virial
theorem for the ground state energy E, and E turns
out to be non
positive.

problems
in class
 P01  Variational characterisation of f as
sup
of the duality products against elements of the
unit ball of the dual.
Convenient fact: it suffices to use only a
normdense subspace of
the dual in order to compute f with this
variational
characterisation
 P02  L2norm of gradient of f is controlled by
the L2norm
of f
and the L2norm of Laplacian of f. L2norm of
Laplacian of f controls
the L2 norm of any second derivative of f. L2norm
of f and of some
higher derivative of f controls Lpnorm of f.
 P03  In 3 dim or more, if the potential V is
sufficiently
small
in the L^{d/2} sense then the ground state of
Laplace+V is
nonnegative.
 P04  Practise with Young's inequality. The
convolution of
1/x
times a density rho.
 P05  Examples of spaces that are / are not
C*algebras.
 P06  Simple properties on the spectrum of A^n
and aA. The
spectra of A and of A* coincide

WEEK
5

HOMEWORK
SESSION

WEEK
5

TUTORIAL

assignment
05
 E17  The free Schrödinger evolution is a
strongly
continuous
(not norm continuous) unitary group on L^2. It
converges weakly to zero
as time goes to infinity.
 E18  The spectrum of an element of a C*algebra
is a
compact
subset of complex number contained in the disk of
radius A. The
C*condition implies that for a normal element A
one has
A^m=A^m for any integer m. In a
noncommutative C*algebra the
only scalar commutator possible is 0.
(Consequence: observables P and Q
cannot be simultaneously bounded.)
 E19  If the commutant of the range of a
representation of
a
C*algebra consists only of multiples of the
identity, then the
representation is irreducible. (This is in fact
<=>.) If the
representation is irreducible, then any vector of
the representation
space is cyclic. (This is in fact <=>.) Pure
states can be
interpreted as (positive) bounded linear
functionals on the C*algebra
of observables.
 E20  Polarisation observable for an EPR pair of
transverse
photons flying apart in opposite directions.
Angles for which the
corresponding Bell's inequality is
quantummechanically violated.
Problem 20 is easy
because there
are two entangled photons only... Look
at this October
2012
PRL
article where they entangled more then 10000
photons!

 Retrospective and concluding remarks on the
C*algebraic
formulation of Quantum Mechanics.
(Informal
notes of the tutorial.)

WEEK
4

HOMEWORK
SESSION

WEEK
4

TUTORIAL

assignment 04
 E13  For a sequence to converge weakly in L^p
it is not
enough to
converge when tested on a dense only of L^q. Weak
convergence plus
convergence of the norms imply norm convergence in
L^2 (in fact, in
L^p).
 E14  Weak convergence of a sequence in L^2 and
of the
sequence of
the weak derivatives implies that the limit is in
H^1. Differences
between norm/weak convergence in L^2 and in H^1.
 E15  Examples of functionals on L^p that are or
are not
continuous
in norm/weakly.
 E16  The dispersive estimate for the free
Schroedinger
evolution.

 Revision on weak convergence in infinitedim
Hspaces and
in L^p
spaces, 1<p<inf. Definition: it's the
convergence when tested
against all dual elements. Convergence against a
dense only is not
enough. (See Exericse
13.) Nor
it
suffices to use an ONB of the Hspace. But
convergence against elements
of an ONB + boundedness of norms is the same as
weak convergence. (See Problem
7
here.)
 The weak limit is unique. (The
underlying topology is Hausdorff,
it separates points. A basis of
neighbourhoods at 0 are cylinders in all but
finitely many dimensions.)
Warning: we shall only consider sequential
convergence.
 Norm conv. => Weak conv. Opposite is false.
But weak
convergence + convergence of norms imply norm
convergence (Exericse
13).
 The norm closure of a subset is contained in its
weak
closure. If a vector
suspace is norm closed, then it is also weakly
closed.
 Every normseparable Hspace is also weakly
separable.
Every Hspace is also (sequentially!) weakly
complete, i.e.,
every weakly Cauchy sequence converges weakly to
some
element of H. Proof of last fact involves Uniform
Boundedness
Principle + Riesz Representation Thm, that is,
completeness is
heavily used.
 Pointwise convergence does not imply weak conv.
Weak
conv. does not imply pointwise conv. Weak conv.
=> distributional
conv. but not vice versa. Examples.
 The norm is normcontinuous, not weakly cont.,
but weakly
lower semicont. In general, nonlinearity destroys
weak
convergence (examples).
 Scalar products (or L^pL^q products) of two
weakly
convergences does not
necessarily
converge. They do if one sequence converges in
norm.
 General mechanisms for weak convergence in
L^p(Omega): (1)
rapid
oscillation, (2) concentration, (3) wandering off
to infinity.
 Relevant consequence of duality L^p/L^q (or of
selfduality
of a Hspace): variational characterisation of
f as sup
of the duality products against elements of the
unit ball of the dual.
Convenient fact: it suffices to use only a
normdense subspace of
the dual in order to compute f with this
variational
characterisation. (See Problem
1
here.)
 Our main tool based upon weak convergence: the
BanachAlaoglu
theorem in L^p(Omega), 1<p<inf. It is a
compactness theorem:
extracting weakly convergent subsequences is
crucial for us to prove
the existence of minimisers of energy functionals.
Proof of
BanachAlaoglu is constructive in this case
because L^p is separable:
it boils down to a Cantor diagonal trick (see
LiebLoss, Section 2.18).
Remark: being a
weaklycompactness fact, BanachAlaouglu is NOT
quantitative in the
rate of convergence of the extracted subsequence.

WEEK
3

HOMEWORK
SESSION

WEEK
3

TUTORIAL

assignment 03
 E09  General expression of a smooth function
times the
delta
distribution. General distributional solution to
(x^k T)=0. A first
order
distributional O.D.E.
 E10  The free evolution of a coherent state is
again a
coherent
state whose position and momentum expectations
evolve classically.
 E11  Three dimensional Green function of
Laplacian+m^2.
 E12  The principal value (PV) distribution.

 C*algebra theory: abstraction of the structure
of bdd
operators
on Hspace. Provides conceptually clean language
for Q.M. of large
systems (infinite particles) and natural language
to axiomatize Q.M.
when emphasis is put on observables instead of
states.
 Def of: commutative, non commutative, *,
normed, Banach,
Banach* algebra. The C* condition. Examples of
C*algebras.
 At most one identity in a C*alg. If not
present, natural
embedding in a larger algebra with identity.
 Left/right/twosided(bilateral) ideals. Closed,
twosided
ideals
have special role: they make the quotient algebra
again a C*algebra.
Notion
of simple C*algebras. Examples.
 Normal, selfadjoint, unitary, projection,
positive
elements of a
C*alg. Inverse.
 Resolvent set, spectrum, resolvent. Spectrum of
A+z, A*,
A^{1}.
Spectrum is nonempty and compact in C. Typical
technique: series
expansion and analytic continuation. Neuman series
for the resolvent.
 Spectral radius, spectral radius formula for
selfadj
elements.
It's the key technical ingredient towards the
Spectral Theorem.
 *homomorphisms between C*algebras. They turn
out to map
positive elements into positive elements (easy)
and to be continuous
with norm at most 1
(less evident). Examples.
 Representation of a C*alg on a Hilbert space.
Faithful
representations. Irreducible representations. Many
examples.
 Structure theorems: any C*alg is *isomorphic
to a
normclosed,
selfadjoint subalgebra of bdd operators on
Hspace. Any commutative
C*alg is *isomorphic to continuous functions
vanishing at infinity
over a locally compact Hausdorff space. (Existence
of nontrivial
repr., as well as structure thms, follows from
HahnBanach thm.)
 Characters of commutative C*algebras. Examples.
 References: BratteliRobinson,
sections
2.1.1,
2.2.1,
2.2.2.,
2.2.3,
2.3.1,
2.3.4,
and
2.3.5, Thirring,
sections
I.2.2,
I.2.3,
and Strocchi,
sections
1.4,
1.5,
and
2.6.
 Informal notes of the
tutorial.

WEEK
2

HOMEWORK
SESSION

WEEK
2

TUTORIAL

assignment
02
 E05  Distributional Ftransform of 1/x and
1/x^2.
Variational upper bound on the ground state of
hydrogenic atoms using
coherent states.
 E06  Sequences of L1_loc functions that
converge or not in
the sense of distributions.
 E07  Link between the decay of a function and
the
differentiability of its Ftransform. Also, a
function and its
Ftransform cannot be both compactly supported.
 E08  Proof of Hardy inequality (with the right
constant)
through the vector field method.

 The Sobolev space H^1(Omega), our natural
"energy space":
defined
as the space of L^2 functions with weak derivative
in L^2 (recall the
def. of weak derivative from class).
 H^1 is complete and has a natural structure of
Hilbert
space.
 Smooth (C^inf) functions with L^2derivative are
dense in
H^1(Omega). In particular compactly supported
smooth functions are
dense in H^1(R^d).
 A bounded smooth function times an H^1 function
is still
H^1 and
chain rule applies. Partial integration between
H^1 functions applies
in the form int(fg')=inf(f'g).
 For H^1 functions, gradient of f has a smaller
L^2 norm
than
gradient of f.
 Fourier characterisation of H^1.
 Useful facts to know (try to prove them as an
exercise for
your
own preparation at home, possibly future
homework):
(1) An L^2 function is in H^1 if and only if its
difference quotient
converges in L^2 sense to the weak derivative of
the function.
(2) Functions in H^1(R) (or of an interval) are
continuous, and Hoelder
continuous almost everywhere.
(3) The characteristic function of a set in R^d
with positive measure
cannot belong to H^1(R^d).
 References: LiebLoss, sections 7.2 to 7.9.

WEEK
1

HOMEWORK
SESSION

WEEK
1

TUTORIAL

assignment
01
 E01

Indeterminacy
relation
in
Ftransform
language
in
d
dimension.
Indeterminacy is minimised by coherent states.
 E02
 Fourier
transform
of
a
positive
definite
quadratic
form
in d dmensions.
 E03
 Kernel
of
the
free
Schroedinger
evolution.
 E04
 Bessel
kernel,
the
kernel
of
(1Laplacian)^{1}.

 Topology to have a structure in which we can say
what
convergence means, even without the need of a
distance.
 From metric to topological spaces. Definition of
topology:
a family of subsets (the "opens") with given
consistency conditions.
Subspace topology.
 Hierarchy
topological/metric/normed/Banach/Hilbert spaces.
 Open and closed. Neighbourhood. Interior.
Boundary. Limit
point.
 Base for a topology. Basis of neighbourhoods.
 1st countable / 2nd countable / separable
topological
spaces.
 Convergence and continuity in a topological
space.
 Compact/bounded topological spaces.
 References: standard textbooks in Topology; and
this
handout.

WEEK
0

TUTORIAL

WEEK
0

TUTORIAL

 Lebesgue integral on R^n, visual picture of
Riemann vs
Lebesgue, why Lebesgue
measure/integral is
superior to Riemann.
 Dominated and monotone convergence,
Fatou's Lemma, Fubini. Examples and
counterexamples.
 References: ReedSimon, volume 1, section I.3;
LiebLoss,
sectons
1.6, 1.7, 1.8., 1.12, 2.1; plus section 1 of this
handout.

 Fourier transform from physical point of view:
convenient
unitary
transformation of wavefunctions: implements
indeterminacy principle,
diagonalises energy, monitors short x with large p
and vice versa.
 Ftransform in L^1: well defined, linear; has
symmetry
under
translation/scaling/rotation; is bounded and
injective from L^1 to
C_inf, not surjective though; maps Gaussians in
Gaussians, convolutions
in pointwise products.
 We like/need to invert F, natural space for
inversion is
the Schwartz
class. Here F becomes a linear bijection, with
explicit inversion
formula.
Up to factors 2pi, interchanges differentiation
and multiplication by
monomials. Preserves L^2 norm
(Plancherel/Parseval) => it extends
uniquely to a unitary operator on L^2.
 Computationally: Ftransf of f in L^1
intersected L^2 is
computed
with the explicit formula, if f is in L^2\L^1 then
first compute F(f)
on L^1 (or Schwartz) approximants. Typical
approximations: truncation
out of a big ball, smoothing, suppressing the
tale, etc.
 References: ReedSimon, volume 2, sections IX.1,
IX.2;
LiebLoss, sections 5.1 to 5.5; Teschl, section
7.1; plus section 2 of this
handout.
