Institut for Matematiske Fag
Universitetsparken 5, DK2100
København Ø

Pseudodifferential operators and spectral theory (2011)
Lecture: 
Wednesday 1012 am  (Biocenter 4.0.10) 
 Wednesday 35 pm 
(Auditorium 4) 
  
Office hours Heiko Gimperlein:  by appointment  (04.2.19) 
Email:  gimperlein at math.ku.dk 
Formal matters: 
You can get 7.5 credit points for active participation.
The course intends to give an introduction to, for example,
pseudodifferential operators and semiclassical analysis on manifolds, the
corresponding resolvents and heat kernels/complex powers/zeta functions,
spectral theory and related topics. It should be a good preparation for thesis work in real analysis or mathematical physics. Parts of it might also be of interest to
noncommutative geometers interested in quantizations and index theory.
Especially at the beginning we are going to rely on some hard theorems from DifFun2. You can find the proofs here.

Some references:  E. B. Davies, Spectral Theory and Differential Operators, CUP,

 L. C. Evans and M. Zworski, Semiclassical Analysis, lecture notes,

 G. Grubb, Distributions and Operators, Springer (Ch. 7+8),

 M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer.

Further topics:  Perturbations of selfadjoint operators (Kim, slides, notes) 
 Fredholm operators and their index (Thomas, mostly Grubb 8.3, notes) 
 Ikehara's Tauberian theorem / The prime number theorem (Flemming, notes) 
 The Laplace operator on the sphere (Job, Shubin and Hörmander, notes) 
 The wave front set of a distribution (Sabrina, Shubin App. 1, notes) 
 The Cauchy problem for the wave equation on Rn (Isak, notes) 
 Rayleigh quotients and Weyl's law (Julie, notes) 
 Resolvents of pseudodifferential operators (Isak, notes) 
 The BirmanSchwinger principle for eigenvalues in gaps 
 Lowlying eigenvalues of Schrödinger operators (Dumassi/Sjöstrand) 
 Introduction to operator semigroups (e.g. Grubb 14.2 or Lax 34.2) 
 Lpbounds of eigenfunctions (Sogge or Hörmander) 
 Sharp Strichartz estimates (Keel/Tao) 
 
 to be continued 
