Math 279, Semiclassical Analysis (Topics in PDE), Fall 2018

Instructor:Semyon Dyatlov
Class hours:TuTh 12:30–2PM, in 31 Evans
Office hours:Tu 2–3PM and by appointment, in 805 Evans
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, book in progress
[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
Description: click here
Grading:Based on homework. You can submit solutions in person or by email. Points will not be assigned but I will read your submissions and provide feedback. You are not required to solve every problem.

Schedule (in reverse time order)

Tue Dec 4 End of the proof of quantum ergodicity.
Thu Nov 29 Quantum ergodicity.
Suggested reading: these notes, [Zw, Chapter 15]
Some pictures of eigenfunctions (open with Adobe Reader to play the movie)
Tue Nov 27 Pseudodifferential operators on manifolds: functional calculus, Weyl law, Egorov's Theorem, semiclassical measures. Quantum ergodicity.
Suggested reading: [Zw, Chapter 14], [DZ, Appendix E.1–E.3]
Tue Nov 20 Class cancelled due to poor air quality
Thu Nov 15 Change of variables in pseudodifferential operators. Pseudodifferential operators on manifolds. Sobolev spaces.
Suggested reading: [Zw, §§9.2,9.3.3–9.3.4, 14.1–14.2]
or [DZ, Appendix E.1–E.2]
Tue Nov 13 Examples of Fourier integral operators: pseudodifferential operators, Fourier transform, pullbacks, propagators. Application to normal forms (very briefly). Application to trace, Duistermaat–Guillemin trace formula (very briefly). Symbol classes Sk with improvement in the ξ-derivatives.
Suggested reading: [Zw, §§12.2–12.3, 9.3.1–9.3.2]
Thu Nov 8 Fourier integral operators: general oscillatory integrals, multiplication by pseudodifferential operators, wavefront set mapping property, basic properties.
Suggested reading: this note, [Zw, §§10.1–10.2], [GS, Chapters 10–11], [H4, §§25.1–25.3]
Tue Nov 6 Generating functions of symplectomorphisms. Hamilton–Jacobi equation. Hyperbolic parametrix, transport equation. Application: dispersive estimate.
Suggested reading: [Zw, §§10.2–10.4]
Homework 9, due Thu Nov 29
Thu Nov 1 Applications of ellipticity and propagation of singularities to nonsemiclassical situations: elliptic regularity and the singular support of the fundamental solution to the wave equation. Hyperbolic parametrix: why the phase function has to be a generating function of the Hamiltonian flow.
Suggested reading: [Zw, §10.2]
Tue Oct 30 Egorov's Theorem up to Ehrenfest time (without proof). Propagation of singularities (proved using Egorov's Theorem, different from the book). Flow invariance of wavefront sets.
Suggested reading: [Zw, §§11.4, 12.3]
Homework 8, due Tue Nov 27
Thu Oct 25 Semiclassical wavefront sets: pseudolocality, wavefront sets of Schrödinger eigenfunctions. Egorov's Theorem.
Suggested reading: [Zw, §§8.4, 11.1]
Tue Oct 23 Semiclassical wavefront sets.
Suggested reading: [Zw, §8.4]
Thu Oct 18 Semiclassical measures associated to quasimodes of Schrödinger operators: flow invariance. Results on manifolds (without proof). Applications to control of eigenfunctions.
Suggested reading: [Zw, §5.2]
Students who want to see more recent results are invited to look at these slides or this paper
Tue Oct 16 Semiclassical defect measures: existence for subsequences and basic properties. Measures associated to quasimodes of Schrödinger operators: support property.
Suggested reading: [Zw, §§5.1–5.2]
Homework 7, due Tue Nov 13
Thu Oct 11 Functional calculus for pseudodifferential operators. Semiclassical defect measures.
Suggested reading: [Zw, §§14.3.2, 5.1], [DS, §8]
Tue Oct 9 Hilbert–Schmidt and trace class operators. Trace and eigenvalues of a self-adjoint operator. Trace and integral kernel. Trace of a pseudodifferential operator. Functional calculus for pseudodifferential operators; almost analytic extensions, Helffer–Sjöstrand formula. Proof of the Weyl law using the functional calculus and trace.
Suggested reading: [H3, §19.1, pp.185–187], [Zw, §§C.3,14.3.2–14.3.4, Theorem 3.6]
Homework 6, due Tue Nov 6
Thu Oct 4 Discreteness of spectrum for Schrödinger operators. Quantum harmonic oscillator. Hilbert–Schmidt and trace class operators.
Suggested reading: [Zw, §§6.3,6.1,C.3], [H3, §19.1, pp.185–187]
Tue Oct 2 Compactness. Sobolev spaces. Eigenvalues of Schrödinger operators.
Suggested reading: [Zw, §§4.6,6.3]
Homework 5, due Tue Oct 30
Thu Sep 27 Cotlar–Stein Theorem. Inverting globally elliptic operators. Gårding inequalities.
Suggested reading: [Zw, Theorem C.5, §4.7]
Tue Sep 25 Change of quantization. Symbol classes Sδ(m). L2 boundedness for Schwartz symbols and for symbols in S1/2(1).
Suggested reading: [Zw, §§4.3.3,4.5.1]
Homework 4, due Tue Oct 23
Thu Sep 20 Product Rule for Weyl quantization, semiclassical expansions.
Suggested reading: [Zw, §4.3–4.4]
Tue Sep 18 Asymptotic series and Borel's Theorem. Mapping properties of quantization of symbols in S(m), for Weyl quantization. Product Rule for Weyl quantization.
Suggested reading: [Zw, §4.2.3–4.2.5, 4.3.1, 4.4]
Thu Sep 13 Quantization of symbols which are Schwartz functions or tempered distributions; basic mapping properties. Oscillatory testing (for standard quantization). Product Rule for standard quantization and Schwartz class symbols. Order functions. Symbol classes S(m).
Suggested reading: [Zw, §§4.1, 4.2.1–4.2.2, 4.3.4, 4.4.5, 4.4.1]
Tue Sep 11 Semiclassical Fourier transform. Oscillatory integrals. Semiclassical quantization.
Suggested reading: [Zw, §§3.6, 4.1]
or [GS, §1] (for an alternative development of oscillatory integrals)
Homework 3, due Tue Oct 9
Thu Sep 6 Method of stationary phase. Quantization formulas. Oscillatory integrals.
Suggested reading: [Zw, §§3.5–3.6, 4.1.1]
Homework 2, due Thu Sep 27
Tue Sep 4 The spaces of distributions D'(Rn), E'(Rn). Fourier transforms of compactly supported distributions. Integral kernels and the Schwartz kernel theorem. Fourier transforms of imaginary exponentials e.g. exp(ix2). Method of nonstationary phase.
Suggested reading: [FJ, §6.1], [Zw, §§3.2, 3.4–3.5]
Thu Aug 30 Fourier transform on the Schwartz space S(Rn). The space of tempered distributions S'(Rn) and Fourier transform on it. Basic operations with distributions: differentiation and multiplication by smooth functions. Support of a distribution.
Suggested reading: [Zw, §§3.1–3.2] (the minimal knowledge that we need)
or [FJ, up to Chapter 8] (a comprehensive introduction for those of you who want more)
or [H1, up to Chapter 7] (a very comprehensive introduction for those of you with plenty of time)
Tue Aug 28 Differential operators on Rn
Lecture notes
Suggested reading: lecture notes, [Zw, Chapter 1]
Homework 1, due Thu Sep 20: exercises 1–5 from the lecture notes
Thu Aug 23 Overview of semiclassical analysis using three examples: Schrödinger propagation, quantum harmonic oscillator, and quantum ergodicity
Slides (use Adobe Reader to play movies)

Last updated: Dec 4, 2018