|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 Rn, scattering in the exterior of an obstacle in Rn, 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.|
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.
Hyperbolic surfaces; classification of infinite ends.
Scattering on surfaces with cusps:
meromorphic continuation of the resolvent,
Wavefront sets of resonant states. Propagation of
singularities with boundary (brief overview).
Incoming and outgoing tails and the trapped set (§6.1.1).
Spectral gaps for nontrapping manifolds without boundary (§6.2).
Scattering on manifolds with Euclidean ends: meromorphic continuation
of the resolvent (§4.2). Resonant states.
Scattering on manifolds with Euclidean ends: resolvent in the upper half-plane.
Proof of propagation of singularities (§E.5).
Sharp Gårding inequality.
Propagation of singularities (Theorem E.49).
Problem set 7, due May 11 | Solutions
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)
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.
Problem set 6, due Apr 27 | Solutions
Semiclassical quantization: oscillatory testing, products, adjoints, pseudolocality, mapping properties
on Sobolev spaces.
Microlocal analysis: semiclassical pseudodifferential operators,
Special guest lecture
Office hours cancelled
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).
Problem set 5, due Apr 13 | Solutions
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).
Resonance expansion of waves (Theorems 2.7, 3.9).
Outgoing asymptotics (Theorem 3.5).
Starting Rellich's Uniqueness Theorem (Theorem 3.30).
Class and office hours cancelled (snow).|
Problem set 3 due Thursday, Mar 16.
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).
Problem set 4, due Apr 6 | Solutions
Potential scattering in odd dimensions: meromorphic continuation of
the resolvent (Theorems 2.2, 3.6).
Grushin problems and Analytic Fredholm Theory (Theorem C.5).
Free resolvent in odd dimensions (§3.1).
Office hours cancelled
1D potential scattering: semiclassical asymptotics (mentioned).
Starting potential scattering in higher dimensions.
Problem set 3, due Mar 16 | Solutions
MATLAB demonstration of resonances in the semiclassical limit:
Numerical pictures: here and here
1D potential scattering: complex scaling (§2.7) and
Movies of semiclassical wave propagation: high frequency behavior of solutions to the semiclassical wave equation with a potential
Plane waves and scattering matrix in 1D (§2.4).
Complex scaling in 1D (§2.7).
Problem set 2, due Mar 7 | Solutions
Meromorphic continuation of the resolvent in 1D (§2.2). Resonances
in the closed upper half-plane. Scattering matrix (§2.4).
|Thu||Feb 9||Class and office hours cancelled (snow).|
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.
Problem set 1, due Feb 23 | Solutions