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. |
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 S^{k} 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 S_{δ}(m).
L^{2} boundedness for Schwartz symbols and for
symbols in S_{1/2}(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'(R^{n}),
E'(R^{n}).
Fourier transforms of compactly supported distributions.
Integral kernels and the Schwartz kernel theorem.
Fourier transforms of imaginary exponentials e.g. exp(ix^{2}).
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(R^{n}).
The space of tempered distributions
S'(R^{n}) 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 R^{n}
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) |