| Instructors: | Semyon Dyatlov (Jul 29–Aug 6) and Kiril Datchev (Aug 8–16) |
| Class hours: | see schedule |
| Lecture notes: | see below. Click here for a classical/quantum correspondence chart which could be used as a partial roadmap for the course |
| Exercises: | see below. These can be used to further your understanding of the material and as possible conversation topics with your TAs/classmates. They will not be graded, you can freely choose which of them you would like to work on. Some of them ask to carefully go through something that was only presented briefly in lecture. Harder exercises are marked by an asterisk. |
| Textbook: | [Zw] Maciej Zworski, Semiclassical Analysis, AMS, 2012 |
| Additional reading: |
[DS] Mouez Dimassi and Johannes Sjöstrand,
Spectral Asymptotics in the Semi-Classical Limit,
Cambridge University Press, 1999
[DZ] Semyon Dyatlov and Maciej Zworski, Mathematical theory of scattering resonances, AMS Graduate Studies in Mathematics 200, 2019 [FJ] F. Gerard Friedlander and Mark Joshi, Introduction to the Theory of Distributions, 2nd edition, Cambridge University Press, 1998 [GS] Alain Grigis and Johannes Sjöstrand, Microlocal Analysis for Differential Operators: An Introduction, Cambridge University Press, 1994 [H1] Lars Hömander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, Springer, 2003 [H3] Lars Hömander, The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators, Springer, 2007 [H4] Lars Hömander, The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators, Springer, 2007 |
| Mon | Jul 29 |
§1: A motivating example
[ Movie ]
[ Exercises ] Introduction to classical/quantum correspondence, wavefront sets, propagation of singularities on a simple model of one-dimensional Schrödinger equation. Statement of the method of nonstationary phase. Suggested reading: [Zw, §§1.1–1.2] |
| Tue | Jul 30 |
§2: Method of stationary phase
[ Exercises ] Proof of the method of nonstationary phase. Nondegenerate critical points. Method of stationary phase and its proof: quadratic stationary phase, Morse Lemma. Suggested reading: [Zw, §§3.4–3.5] |
| Wed | Jul 31 |
§3: Semiclassical quantization
[ Exercises ] Definition of standard and Weyl quantization, basic mapping properties. Examples. Oscillatory testing. §4: Calculus for compactly supported symbols [ Exercises ] Asymptotic expansions, Borel's Theorem. Expansions for compositions and adjoints of quantizations of compactly supported symbols. Corollaries: Product Rule, Commutator Rule, Adjoint Rule. Suggested reading: [Zw, §§4.1–4.3] Note: Zworski's book uses primarily the Weyl quantization while in this course we use the standard quantization. The two quantizations are equivalent, and the proofs for Weyl quantization can be adapted to work for the standard quantization (or else one can use change of quantization, [Zw, Theorem 4.13]). |
| Thu | Aug 1 |
Continuing §4 §5: Calculus for general symbols [ Exercises ] Order functions and symbol classes corresponding to them. Mapping properties for quantizations of such symbols on Schwartz functions/tempered distributions. Calculus for general symbol classes. Suggested reading: [Zw, §4.4] |
| Fri | Aug 2 |
§6: L2 theory
[ Exercises ] Boundedness of operators with symbols in S(1), with proof only for Schwartz symbols. Compactness of operators with order function decaying to zero. Sharp Gårding inequality, with proof only in a simple case. §7: Ellipticity [ Exercises ] Elliptic parametrix. Elliptic estimate. Application: confinement to the classically allowed region for eigenfunctions of Schrödinger operators. Movies of Schrödinger eigenfunctions for an asymmetric and a symmetric double well potential. The symmetric case is related to Problem 4.8 in the quantum mechanics course. Suggested reading: [Zw, §§4.5–4.7] |
| Mon | Aug 5 |
§8: Change of variables
[ Exercises ] Change of variables for compactly supported symbols. Kohn–Nirenberg symbol classes and asymptotic expansions for these classes: compositions, adjoints, change of variables. Pseudolocality. §9: Calculus on manifolds [ Overview of calculus on manifolds ] [ Exercises ] Pseudodifferential operators on manifolds: definition, principal symbol. Product, commutator, and adjoint rules. Mapping properties on Sobolev spaces and sharp Gårding inequality. Quantization procedure. Suggested reading: [Zw, §§9.2–9.3, 14.1–14.2], [DZ, §§E.1.6–E.1.8] |
| Tue | Aug 6 |
§10: Wavefront sets
[ Exercises ] Semiclassical wavefront sets of families of distributions on Rn, equivalence of the Fourier transform definition and the pseudodifferential definition. Example: wavefront set of an oscillatory integral. Suggested reading: [Zw, §8.4], [DZ, §E.2.3] |
| Fri | Aug 16 | TBA |
| Thu | Aug 15 | TBA |
| Tue | Aug 13 | TBA |
| Mon | Aug 12 | TBA |
| Fri | Aug 9 | TBA |
| Thu | Aug 8 | TBA |