Reading Schrödinger group 
Time: 11:00–12:00 on Thursdays
of oddnumbered teaching weeks (12 & 26 Oct, 9 & 13 Nov, 7 Dec).
Place: MAGIC room.
Web: http://www.maths.leeds.ac.uk/~kisilv/Schrodinger18.html
email: kisilv@maths.leeds.ac.uk
1 Motivation
The most prominent examples of threedimensional Lie groups are: the
Heisenberg group ℍ and the group SL_{2}(ℝ).
The Heisenberg group initiates the class of nilpotent Lie groups and
SL_{2}(ℝ) is the smallest semisimple Lie group. In a sense, those two
classes are bearing many opposite properties.
For a wider community, ℍ and SL_{2}(ℝ) are significant
for their applications:

The Heisenberg group appears in harmonic analysis, quantum
mechanics, signal processing and many other areas.
 The affine group—a part of SL_{2}(ℝ)—is behind of numerous
techniques in real analysis, wavelet theory, etc. Furthermore, the
group SL_{2}(ℝ) itself

is the group of holomorphic automorphisms of the unit disk (or
the upper halfplane) in complex analysis;
 has the Lie algebra sl_{2} spanned by the
Hamiltonian of the quantum harmonic oscillator and the respective
ladder operators;
 has the discrete subgroup SL_{2}(ℤ)
consisting of the integer 2× 2 matrices, which is crucial
for number theory.
The above two lists are partially overlapping and partially
complementing. There are deep reasons for this. The mentioned
generators of the Lie algebra sl_{2}—the Hamiltonian of
the harmonic oscillator and the ladder operators—are quadratic
elements of the Heisenberg Lie algebra. In the opposite direction: the
group SL_{2}(ℝ) naturally acts by outer automorphisms of the
Heisenberg group, thus one can build the semidirect product
S=ℍ⋊ SL_{2}(ℝ), which is known as the
Schrödinger group.
The Schrödinger group naturally covers any of the above applications of
ℍ and SL_{2}(ℝ) and especially efficient in the areas
common to both: harmonic analysis, quantum mechanics and number
theory. It is enough to mention that:

S is the full group of symmetries of the wave, heat and
Schrödinger equations (hence its name).
 The discrete subgroup of S with integer components
defines the theta function.
2 Outline
Our aim is to learn fundamentals of the Schrödinger group in a
pedestrian way, which is suitable for young researchers. The initial
plan is:

12 Oct 2017 The general and special linear groups, the projective line and
Möbius transformations:

What are GL(2,R) and SL(2,R)?
 What are relations between SL(2,R) and Möbius transformations?
 What is the relation between eigenvectors and fixed points?
 Are Möbius maps 3transitive?
 What is the number of nonequivalent oneparameter subgroups
of SL(2,R)? (the classification is based on the number of fixed points).
 Can we represent the projective line as a homogeneous space SL(2,R)/H?
 What are the actions of SL(2,R) on the projective line, upper
half plane and unite disk?
 What are the similarities and differences, when we work in the
field of complex numbers or more complicated rings like, e.g.
dual or double numbers?
First PDF handouts.
 26 Oct 2017 Subgroups, homogeneous spaces, induced representations of the Heisenberg group, the Schrödinger
and Fock–Segal–Bargmann models:

The Heisenberg group and the corresponding Lie algebra.
 Classification of one and twodimensional continuous subgroups.
 The induction of representations from a subgroup.
 The induced representations of the Heisenberg group: the
Schrödinger representation and the FockSegalBargmann representation.
 The ladder operators and the vacuum state—the Gaussian.
 The Stonevon Neumann theorem, which classifies unitary irreducible
representations of the Heisenberg group.
Second PDF handouts.
 9 Nov 2017 Automorphisms of the Heisenberg group, semidirect
product, the Schrodinger group, discrete subgroups, subgroup
averaging and the Gaussian.
 23 Nov 2017 The classification of unitary irreducible representations of the
group SL_{2}(ℝ), realisations in the Hardy, Bergman and Dirichlet spaces.
 14 Feb 2018 Canonical commutation relations, the Hamiltonian of
harmonic oscillator, ladder (creation and annihilation) operators,
the vacuum vector and classification of unitary irreducible
representations of the Heisenberg group. Realisations in
Schrödinger, Fock and theta models.
 21 Feb 2018 Bargmann classification of unitary irreducible
representations of the group SL_{2}(ℝ) through raisinglowering
operators on the discrete spectrum of the maximal compact subgroup
K.
 7 Mar 2018 The realisations of unitary irreducible
representations of the group SL_{2}(ℝ), in the Hardy, Bergman and
Dirichlet spaces.
 9 Oct 2018 Ladder operators and the harmonic oscillator (notes).
 ???? 2018 The metaplectic representation, theta functions, the heat
equation.
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Last modified: October 19, 2018.
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