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Institute for Analysis
Workgroup Nonlinear Partial Differential Equations
JProf. Dr. Xian Liao

Announcement

Announcement for Seminar "Dirac Operators", Summer Semester 2025 (2SWS, 3LP)

Instructor: JProf. Dr. Xian Liao (Xian.liao@kit.edu)

Assistant: M.Sc. Rebekka Zimmermann (Rebekka.Zimmermann@kit.edu)

Time & Place: 11:30 - 13:00, Monday, SR2.067

Preliminary meeting: '''Wednesday 13:15pm, February 5th, 2025''', in SR 3.068 (Mathematical Building).

Description:
The Dirac operators play an important role in relativistic quantum mechanics, serving as the counterpart to the Schrödinger operator in the nonrelativistic case.
One particularly intriguing feature of the one-dimensional Dirac operator is that the linear Dirac operator naturally emerges as the Lax operator in the Lax-pair formulation of the nonlinear completely integrable modified Korteweg-de Vries (mKdV) equation. Specifically, if $\phi(t,x)$ is a solution of the mKdV equation, the spectrum of the time-dependent Dirac operator with the scalar potential $\phi(t,x)$ remains time-independent.

In this seminar, we aim to explore:

  • The nonrelativistic limit from the Dirac operator to Schrödinger operators (in the first three presentations).
  • The soliton solutions of the mKdV equation, constructed from KdV-solitons via the Miura transformation (in the fourth and fifth presentations).

This will be achieved by studying the mathematical theory of Dirac operators, following B. Thaller's book (see reference below).

Presentation topics are as follows:

  1. (19.05.25) Dirac Operators with Supersymmetry. (Thaller, Section 5.1, 5.2, 5.4, 5.5.1)
  2. (26.05.25) c-Dependence of Dirac Operators. (Thaller, Section 6.1)
  3. (02.06.25) c-Dependence of Eigenvalues. (Thaller, Section 6.2)
  4. (23.06.25) Inversion of the Miura Transformation. (Thaller, Section 9.1, 9.2, 9.3)
  5. (30.06.25) mKdV-solitons. (Thaller, Section 9.4, 9.5)

Requirement:
Each participant is supposed to present one topic (around 75min. $+$ 15min. questions/discussions), and to participate actively in other presentations.
Please contact the instructor four weeks before your presentation.
It is recommended to prepare handouts and distribute them before your presentation.

Prerequisites:
Analysis 1-3 (or Höhere Mathematik 1-3), Basics of Spectral Theory.

Reference:
B. Thaller, The Dirac Equation, 1992, Springer.