|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|
[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
|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.|
Suggested reading: these notes, [Zw, Chapter 15]
Some pictures of eigenfunctions (open with Adobe Reader to play the movie)
Pseudodifferential operators on manifolds: functional calculus, Weyl law, Egorov's Theorem, semiclassical measures.
Suggested reading: [Zw, Chapter 14], [DZ, Appendix E.1–E.3]
|Tue||Nov 20||Class cancelled due to poor air quality|
Change of variables in pseudodifferential operators.
Pseudodifferential operators on manifolds.
Suggested reading: [Zw, §§9.2,9.3.3–9.3.4, 14.1–14.2]
or [DZ, Appendix E.1–E.2]
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]
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]
Generating functions of symplectomorphisms.
Hyperbolic parametrix, transport equation.
Application: dispersive estimate.
Suggested reading: [Zw, §§10.2–10.4]
Homework 9, due Thu Nov 29
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]
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
Semiclassical wavefront sets: pseudolocality,
wavefront sets of Schrödinger eigenfunctions.
Suggested reading: [Zw, §§8.4, 11.1]
Semiclassical wavefront sets.
Suggested reading: [Zw, §8.4]
Semiclassical measures associated to quasimodes of Schrödinger operators:
flow invariance. Results on manifolds (without proof).
Applications to control
Suggested reading: [Zw, §5.2]
Students who want to see more recent results are invited to look at these slides or this paper
Semiclassical defect measures: existence for subsequences and basic properties.
Measures associated to quasimodes of Schrödinger operators:
Suggested reading: [Zw, §§5.1–5.2]
Homework 7, due Tue Nov 13
Functional calculus for pseudodifferential operators. Semiclassical defect measures.
Suggested reading: [Zw, §§14.3.2, 5.1], [DS, §8]
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
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]
Compactness. Sobolev spaces. Eigenvalues of Schrödinger operators.
Suggested reading: [Zw, §§4.6,6.3]
Homework 5, due Tue Oct 30
Cotlar–Stein Theorem. Inverting globally elliptic operators.
Suggested reading: [Zw, Theorem C.5, §4.7]
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
Product Rule for Weyl quantization, semiclassical expansions.
Suggested reading: [Zw, §4.3–4.4]
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]
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]
Semiclassical Fourier transform.
Suggested reading: [Zw, §§3.6, 4.1]
or [GS, §1] (for an alternative development of oscillatory integrals)
Homework 3, due Tue Oct 9
Method of stationary phase. Quantization formulas. Oscillatory integrals.
Suggested reading: [Zw, §§3.5–3.6, 4.1.1]
Homework 2, due Thu Sep 27
The spaces of distributions D'(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]
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)
Differential operators on Rn
Suggested reading: lecture notes, [Zw, Chapter 1]
Homework 1, due Thu Sep 20: exercises 1–5 from the lecture notes
Overview of semiclassical analysis using three examples: Schrödinger propagation, quantum harmonic oscillator, and quantum ergodicity
Slides (use Adobe Reader to play movies)