Lecture: Selected Topics in Partial Differential Equations (WS 2020/2021)

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.

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

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


Course material
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
Sheet 1
Corrected a typo in exercise 1.3.3.
Sheet 2
Sheet 3
Sheet 4
Sheet 5
Some more details
Sheet 6
More hints. Corrected statement of Ex 6.3.


Last modified: January 21, 2020.

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