** TCC: Spectral and Functional Inequalities and their applications**

Ari Laptev

**Description of the course:
**

We shall start with some preliminary material describing basic concepts of self-adjoint operators in Hilbert Space such as compact, bounded and unbounded operators, semi-bounded operators and Friedrichs extensions, variational principle, Birman-Schwinger principle.

The course will include:

- Multi-dimensional Hardy and Sobolev inequalities.
- Inequalities for the lowest eigenvalues of Schrödinger operators with decaying potentials.
- Properties of the spectrum of Dirichlet and Neumann Laplacians.
- Polya conjecture, tiling domains. Berezin-Li-Yau inequalities.
- Weyl asymptotics.
- Lieb-Thirring inequalities for Schrödinger operators and their applications.
- Dual form of Lieb-Thirring inequalities.
- Spectrum of harmonic oscillators.
- Spectrum of Schrödinger operators with Coulomb potential.
- Schrödinger operators with reflection free potentials.
- Spectrum of Schrödinger operators with magnetic fields.

Lectures:

Lectures - week 2

Lectures - week 3

Lectures - week 4

Lectures - week 5

Lectures - week 6

Lectures - week 7

Lectures - week 8

**Basics information:**

Problems:

problem sheet 2

problem sheet 3

problem sheet 4

problem sheet 5

problem sheet 6

**Books:**

Mikhail S. Birman and Michael Z. Solomyak,
* Spectral Theory of self-adjoint operators in Hilbert Space*
D. Reidel Publishing House, 1987.

M. Reed and B. Simon,
* Methods of modern mathematical physics. IV: Analysis of Operators, *
Academic Press, 1978.

Elliott H. Lieb and Michael Loss,
* Analysis,*
American Mathematical Society; 2 edition, 2001.

Elliott H. Lieb and Robert Seiringer,
* The stability of matter in Quantum Mechanics,*
Cambridge University Press, 2009.

Vladimir Maz'ya,
* Sobolev Spaces,*
Springer, 2nd Edition., 2011.