Instructor: | Semyon Dyatlov |
Textbook: | Semyon Dyatlov and Maciej Zworski, Mathematical theory of scattering resonances The book is still in progress though most of the sections are already written up. The book will be updated regularly and I encourage you to email me with any mistakes you find! Scattering theory describes long time evolution of waves in a noncompact setting, where energy can escape to infinity. Examples include potential scattering in R^{n}, scattering in the exterior of an obstacle in R^{n}, and scattering on asymptotically hyperbolic manifolds. We will focus in particular on scattering resonances, which are complex frequencies featured in expansions of linear waves; they are the analogs in the noncompact setting of the eigenvalues of the Laplace operator. The mathematical study of resonances is a very active field combining tools from microlocal analysis and dynamical systems, and resonances have important applications in many fields from nuclear science and general relativity to airfoil design and climate change; this course will focus on the fundamentals of the mathematical theory. We will start with the more elementary case of potential scattering in odd dimensions (Chapters 2–3). We will then move on to the more advanced topics in the second and third parts of the book. As the course progresses, I will also explain the necessary tools from functional analysis, complex analysis, and microlocal analysis. For the latter we will use parts of the book `Semiclassical Analysis' by Maciej Zworski, which is available for free to the MIT community. The last two lectures are a brief introduction to scattering on hyperbolic surfaces. An excellent source on the latter is the book `Spectral theory of infinite-area hyperbolic surfaces' by David Borthwick, available for free to the MIT community. |
Class hours: | TR 1–2:30 in 2-139 |
Office hours: | TR 2:30–3:30 in 2-273 |
Grading: | Based on problem sets, which will be released weekly (on average). You can submit solutions in person or by email. Points will not be assigned but I will read your submissions and provide feedback. Some homeworks may be long, however you are not required to solve every problem. |
Thu | May 18 |
Hyperbolic plane and its isometries.
Algebraic approach to scattering on hyperbolic surfaces.
Scattering coefficient for the modular surface.
How not to prove the Riemann hypothesis.
Lecture notes |
Tue | May 16 |
Hyperbolic surfaces; classification of infinite ends.
Scattering on surfaces with cusps:
meromorphic continuation of the resolvent,
scattering coefficient.
Lecture notes |
Thu | May 11 |
Wavefront sets of resonant states. Propagation of
singularities with boundary (brief overview).
Lecture notes |
Tue | May 9 |
Incoming and outgoing tails and the trapped set (§6.1.1).
Spectral gaps for nontrapping manifolds without boundary (§6.2).
Lecture notes |
Thu | May 4 |
Scattering on manifolds with Euclidean ends: meromorphic continuation
of the resolvent (§4.2). Resonant states.
Lecture notes |
Tue | May 2 |
Scattering on manifolds with Euclidean ends: resolvent in the upper half-plane.
Lecture notes |
Thu | Apr 27 |
Proof of propagation of singularities (§E.5).
Sharp Gårding inequality.
Lecture notes |
Tue | Apr 25 |
Hamiltonian flows.
Propagation of singularities (Theorem E.49).
Lecture notes Problem set 7, due May 11 | Solutions |
Thu | Apr 20 |
Semiclassical elliptic estimate and elliptic regularity.
Semiclassical wavefront set of a distribution (§E.2).
Lecture notes (ignore the sharp Gårding inequality for now, it will be covered next week; you can read about the Hamiltonian flow though) |
Thu | Apr 13 |
Principal symbol of a pseudodifferential operator.
Semiclassical quantization on manifolds (briefly).
Fiber-radially compactified cotangent bundle.
Wavefront set and elliptic set of pseudodifferential operators.
Elliptic parametrix and semiclassical elliptic estimate.
Lecture notes Problem set 6, due Apr 27 | Solutions |
Tue | Apr 11 |
Semiclassical quantization: oscillatory testing, products, adjoints, pseudolocality, mapping properties
on Sobolev spaces.
Lecture notes |
Thu | Apr 6 |
Microlocal analysis: semiclassical pseudodifferential operators,
basic properties.
Lecture notes |
Tue | Apr 4 |
Special guest lecture
Office hours cancelled |
Thu | Mar 23 |
Solutions with prescribed incoming part (Theorem 3.39).
Scattering operator (Theorem 3.38) and its unitarity
(Theorem 3.40). An overview of more advanced results
in potential scattering (see pages 7–8 in lecture notes).
Lecture notes Problem set 5, due Apr 13 | Solutions |
Tue | Mar 21 |
End of the proof of Rellich's Uniqueness Theorem,
including a Carleman estimate (Lemma 3.31).
Method of stationary phase (Zworski's book, Theorem 3.16).
Incoming/outgoing decomposition for plane waves
in free space (Theorem 3.35).
Lecture notes |
Thu | Mar 16 |
Resonance expansion of waves (Theorems 2.7, 3.9).
Outgoing asymptotics (Theorem 3.5).
Starting Rellich's Uniqueness Theorem (Theorem 3.30).
Lecture notes |
Tue | Mar 14 |
Class and office hours cancelled (snow). Problem set 3 due Thursday, Mar 16. |
Thu | Mar 9 |
Basic properties of the scattering resolvent;
structure of the Laurent expansion (Theorems 2.4, 3.7).
Resonance free region (Theorems 2.8, 3.8).
Resonance expansions of waves (Theorems 2.7, 3.9).
Lecture notes Problem set 4, due Apr 6 | Solutions |
Tue | Mar 7 |
Potential scattering in odd dimensions: meromorphic continuation of
the resolvent (Theorems 2.2, 3.6).
Grushin problems and Analytic Fredholm Theory (Theorem C.5).
Lecture notes |
Thu | Mar 2 |
Free resolvent in odd dimensions (§3.1).
Office hours cancelled |
Tue | Feb 28 |
1D potential scattering: semiclassical asymptotics (mentioned).
Starting potential scattering in higher dimensions.
Lecture notes Problem set 3, due Mar 16 | Solutions MATLAB demonstration of resonances in the semiclassical limit: demo156.m
and demo156_2.m ,
to be used with David Bindel's code
Numerical pictures: here and here |
Thu | Feb 23 |
1D potential scattering: complex scaling (§2.7) and
semiclassical asymptotics.
Lecture notes Movies of semiclassical wave propagation: high frequency behavior of solutions to the semiclassical wave equation with a potential |
Thu | Feb 16 |
Plane waves and scattering matrix in 1D (§2.4).
Complex scaling in 1D (§2.7).
Lecture notes Problem set 2, due Mar 7 | Solutions |
Tue | Feb 14 |
Meromorphic continuation of the resolvent in 1D (§2.2). Resonances
in the closed upper half-plane. Scattering matrix (§2.4).
Lecture notes |
Thu | Feb 9 | Class and office hours cancelled (snow). |
Tue | Feb 7 |
Overview of scattering theory: waves on closed vs. open systems (§§2.1,2.3).
A basic example: 1D potential scattering. Meromorphic continuation
of the scattering resolvent. Resonances. Resonance expansions for the wave equation.
Lecture notes Problem set 1, due Feb 23 | Solutions |