Pseudodifferential operators and spectral theory (2011)

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.

Weeklies

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 material:	Basic properties of oscillatory integrals
	Almost orthogonality and L2-boundedness: Calderon-Vaillancourt's theorem
	Applications of the analytic Fredholm theorem to compact operators
	Simple analytical formulas for the index
	Classes of compact operators (Birman/Solomjak, Ch. 11)
	Local solvability of differential operators
	Propagation of singularities

Notes:	Chapter 1: Pseudodifferential operators on manifolds
(beware of typos!)	Appendix 1: Fredholm operators and index theory
	Appendix 2: The wave front set of a distribution and applications
	Chapter 2: Complex powers, zeta functions and spectral asymptotics
	Appendix 3: Transition formulas for heat and resolvent expansions
	Appendix 4: Ikehara's Tauberian theorem and the prime number theorem
	Appendix 5: The Laplace operator on the sphere
	Appendix 6: Resolvents of pseudodifferential operators
	Chapter 3: Elements of spectral theory for differential operators
	Appendix 7: Perturbations of selfadjoint operators
	Appendix 8: Rayleigh quotients and Weyl's law
	Chapter 4: Propagation of singularities
	Appendix 9: The Cauchy problem for the wave equation on Rn
	Appendix 10: Existence theorems for ODEs and applications

Guest lectures:	Morten Risager (No. 30, 10 am): Spectral theory of hyperbolic surfaces
	Jan Philip Solovej (Dec. 14, 10 am): Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities (notes)

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 Birman-Schwinger principle for eigenvalues in gaps
	Low-lying eigenvalues of Schrödinger operators (Dumassi/Sjöstrand)
	Introduction to operator semigroups (e.g. Grubb 14.2 or Lax 34.2)
	Lp-bounds of eigenfunctions (Sogge or Hörmander)
	Sharp Strichartz estimates (Keel/Tao)
	to be continued