Pseudodifferential operators and spectral theory (2011)

Weeklies

Week 1: We surveyed the definitions and basic properties of locally estimated pseudodifferential operators up to the composition formula following Ch. 7, Sections 1-3, in Gerd Grubb's book. Also some further results like kernel estimates and mapping properties in Lp-Sobolev spaces were recalled from DifFun2.

Week 2: Kim talked about perturbations of selfadjoint operators, in particular the Kato-Rellich theorem and selfadjointness of the many-body Schrödinger operator (Reed/Simon, Section X.2 / slides). After lunch, we continued the discussion of pseudodifferential operators as in the notes, first discussing ellipticity, boundedness and compactness on L2, and then discussing the behavior of functions and distributions under coordinate transformations. You also should have studied page 11 of the notes about the classical expansion of the parametrix symbol. A very general proof of L2-boundedness, which is closely related to the techniques from DifFun2, can be found here.

Week 3: Thomas discussed Fredholm operators, their index and its topological invariance (mostly Section 8.3 in Grubb / notes). Then we considered the analytic Fredholm theorem. See here how it can be used to derive the spectral theory of compact operators. After lunch we studied pseudodifferential operators and Sobolev spaces on manifolds (as in Grubb, Ch. 8, or Shubin).

Week 4: We finished Chapter 1, discussing in particular the parametrix construction and basic properties of the spectrum for elliptic operators on a compact manifold. Then we studied the resolvent of parameter-elliptic differential operators and defined their complex powers (partially following Shubin).

Week 5: Flemming talked about Ikehara's Tauberian theorem and how to use it to count prime numbers. After lunch, we continued to study the complex powers for a parameter-elliptic differential operator and defined the associated zeta functions.

Week 6: Job talked about the spectrum of the Laplace operator on the sphere, based on Shubin and (very concise!) Hörmander. After lunch we finished Chapter 2.

Week 7: Before lunch Morten Risager gave a guest lecture about the spectral theory of the Laplace operator on hyperbolic surfaces. In the afternoon we reviewed the spectral theorem for selfadjoint operators and studied their essential spectrum mostly following Davies' book.

Week 8: Sabrina talked about the wave front set of a distribution, following the first few pages in Shubin's Appendix 1 and some supplementing notes. We then discussed the relation with pseudodifferential operators and applications to products and restrictions of distributions. We also continued our discussion of the essential spectrum.

Week 9: Before lunch, Jan Philip Solovej gave a guest lecture about Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities, concerning the negative eigenvalues of Schrödinger operators. After lunch, we discussed the essential spectrum of Schrödinger operators, the measure-theoretic decomposition of the spectrum as well as variational characterizations of eigenvalues below the essential spectrum.

Week 10: Before lunch, Isak talked about the Cauchy problem for the wave equation on Rn. We postponed the microlocal propagation of singularities to January.