## Lectures for students

There were 2 courses for students (master or PhD), of 8h each, during the trimester at IHP. The courses were given in english.

*Click here to see the detailed schedule for the lectures in June**Click here to see the videos of the lectures*

**Lecture 1: Interpolation inequalities and applications to nonlinear PDE**

Enno Lenzmann (Basel, Switzerland)

**Duration:** 8h between June 3rd and June 13th, 2013

**Summary:** Interpolation inequalities and their optimizers play a central role in the analysis of many nonlinear evolution problems (e.g. nonlinear Schrödinger and wave equations, water wave problems etc.). The present course will be divided into three main sections as follows. In the first (introductory) part of this course, we will review some "classical" results and techniques to show existence, symmetry and uniqueness for optimizers for Sobolev and Hardy-Littlewood-Sobolev inequalities. In the second part of this course, we discuss some recent approaches to show uniqueness and symmetry of optimizers with particular emphasis on interpolation estimates involving the fractional Laplacian. In the final part of the course, we focus on various applications to nonlinear evolution PDE.

Prerequisites of the course: a good knowledge of advanced analysis (e.g. on the level of the Lieb & Loss textbook "Analysis"). Some further knowledge of PDE and variational calculus is desirable but not mandatory.

**Lecture 2: Operators and their perturbations**

Jan Derezinski (Warsaw, Poland)

**Duration:** 8h between June 3rd and June 13th, 2013

**Summary: **The main purpose of the course is to develop general theory of perturbations of linear operators on Hilbert space, with the emphasis on Schrödinger operators. Many concrete examples will be described in detail. List of subject that will be (partially) covered:

1) Reminder of basic spectral theory

- Unbounded operators

- Closed operators

- Spectrum

- Relative boundedness

- Pseudoresolvents

- Unbounded operators on Hilbert spaces

- (Essential) self-adjointness

- Scale of Hilbert spaces

- Closed and closable positive forms

- Friedrichs extensions

2) Reminder of basic harmonic analysis and its applications

- Young inequality

- Fourier transformation

- Sobolev inequalities

- Self-adjointness of Schrödinger operators

- Self-adjointness of many-body Schrödinger operators

3) Momentum and Laplacian on the line

- Momentum on half-line

- Momentum on an interval

- Laplacian on half-line

- Laplacian on an interval

4) Orthogonal polynomials

- Orthogonal polynomials in weighted *L*^{2} spaces

- Self-adjointness of Sturm-Liouville operators

- Classical orthogonal polynomials as eigenvectors of certain Sturn-Liouville operators

- Hermite polynomials

- Laguerre polynomials

- Jacobi polynomials

5) Finite rank perturbations and their renormalization

- Aronszajn-Donoghue Hamiltonians

- Delta potentials

- Friedrichs Hamiltonians

- Bound states and resonances of Friedrichs Hamiltonians

- Exponential decay from a unitary dynamics

6) Potential *1/|x|²*

- Hardy inequality

- Modified Bessel equation

- Bessel equation

- Operator *-∂ _{xx}+(m²-1/4)/x^{2}*