Time: 11:00–12:00 on Thursdays of odd-numbered teaching weeks (12 & 26 Oct, 9 & 13 Nov, 7 Dec).
Place: MAGIC room.

Web: http://www.maths.leeds.ac.uk/~kisilv/Schrodinger-18.html
email: kisilv@maths.leeds.ac.uk

## 1  Motivation

The most prominent examples of three-dimensional Lie groups are: the Heisenberg group ℍ and the group SL2(ℝ). The Heisenberg group initiates the class of nilpotent Lie groups and SL2(ℝ) is the smallest semisimple Lie group. In a sense, those two classes are bearing many opposite properties.

For a wider community, ℍ and SL2(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 SL2(ℝ)—is behind of numerous techniques in real analysis, wavelet theory, etc. Furthermore, the group SL2(ℝ) itself
• is the group of holomorphic automorphisms of the unit disk (or the upper half-plane) in complex analysis;
• has the Lie algebra sl2 spanned by the Hamiltonian of the quantum harmonic oscillator and the respective ladder operators;
• has the discrete subgroup SL2(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 sl2—the Hamiltonian of the harmonic oscillator and the ladder operators—are quadratic elements of the Heisenberg Lie algebra. In the opposite direction: the group SL2(ℝ) naturally acts by outer automorphisms of the Heisenberg group, thus one can build the semidirect product S=ℍ⋊ SL2(ℝ), which is known as the Schrödinger group.

The Schrödinger group naturally covers any of the above applications of ℍ and SL2(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 3-transitive?
• What is the number of non-equivalent one-parameter 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 two-dimensional continuous subgroups.
• The induction of representations from a subgroup.
• The induced representations of the Heisenberg group: the Schrödinger representation and the Fock-Segal-Bargmann representation.
• The ladder operators and the vacuum state—the Gaussian.
• The Stone-von 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 SL2(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 SL2(ℝ) through raising-lowering operators on the discrete spectrum of the maximal compact subgroup K.
• 7 Mar 2018 The realisations of unitary irreducible representations of the group SL2(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.