Wed 16.45-18.15 and Thu 13.15-14.45, starting on October 21, 2020

This lecture will take place online. Students who are planning to participate are advised to write an e-mail to k DOT merz AT tu-bs DOT de.

Analysis 1-3, and PDE. A nodding acquaintance with functional analysis, distribution theory and some familiarity with the Fourier transform are helpful.

- W. Amrein - Hilbert Space Methods in Quantum Mechanics
- L. Hörmander - The Analysis of Linear Partial Differential Operators
- V. Jakšić - Topics in spectral theory
- T. Kato - Perturbation Theory for Linear Operators
- M. Reed and B. Simon - Methods of Modern Mathematical Physics
- B. Simon - Schrödinger semigroups (available from the author here)
- B. Simon - Harmonic analysis. A Comprehensive Course in Analysis, Part 3.
- G. Teschl - Mathematical Methods in Quantum Mechanics (available from the author here)
- J. Weidmann - Linear Operators in Hilbert Spaces
- D. R. Yafaev - Mathematical scattering theory. Analytic theory

- Some crumbly notes on spectral theory for Schrödinger operators (German)
- See the article by David Damanik and Jake Fillman (pp. 3-28) for some examples of Schrödinger operators with thin spectra and further references.
- More or less everything on trace ideals is covered in this book by Barry Simon.
- New abstract and more general versions of the Birman-Schwinger principle by M. Hansmann and D. Krejcirik.
- More on localization of discrete spectrum by T. Kato (sections IV.3.4-5), S. D. Algazin, and H. Siedentop. See also S. Dyatlov and M. Zworski for a textbook reference.
- More on Hausdorff measures in Falconer (Ch. 1 and 8), Federer (Sect. 2.10), or Mattila (Sect. 4.3)
- Systematic survey of the Cantor function
- More information on decomposition of Borel measures with respect to Hausdorff measures and dynamics of their Fourier transform is discussed by Y. Last.
- Some notes on Hardy spaces and Poisson transforms of measures.

Lecture 1 (Unbounded operators in Hilbert spaces)

Lecture 2 (Some classes of linear operators and quadratic forms)

Lecture 3 (Stieltjes measures, Borel transform, and some dynamic aspects) (preliminary notes)

Lecture 4 (Spectral theorem and some dynamic aspects) (preliminary notes)

Wed 16.45-18.15 every second week. The first class is on Nov. 5.

Exercise sheets |
Remarks |

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Last modified: January 21, 2020.

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