MAQD Summer School

These talks provide a brief introduction to some aspects of semiclassical analysis.  These include semiclassical pseudodifferential operators and their properties, Sobolev spaces, and the semiclassical wavefront set.  The first lecture will also include some reminders about spaces of functions and distributions that we shall work with frequently.

TA: Jian Wang

Lecture 1: Monday 9:30-10:20am
Lecture 2: Tuesday 11-11:50am
Lecture 3: Wednesday 9:30-10:20am
Problem Session: Monday, 4:30-5:30pm
(problems)

I will speak about how to use abstract spectral theory to define eigenvalue problems associated with possibly singular settings. I will not assume anything known about unbounded operators and will focus on some of the following examples:

• Quantum graphs,
• Bessel equation and Schrödinger equation with an inverse square potential
• Dirac potentials and pseudo-laplacians
• Domains with cuts, and conical singularities.

TA: Mengxuan Yang

Lecture 1: Monday 11-11:50am
Lecture 2: Tuesday 2-2:50pm
Lecture 3: Thursday 9:30-10:20am
Problem Session: Thursday 11am-12pm (in Pancoe Auditorium)
(problems)

The guiding thread of this lecture will be a Schrödinger equation describing the dynamics of an electron in a crystal in the presence of an external potential.
Because the size of the cells of the crystal are supposed to be very small comparatively with the macroscopic scale, it is a multi-scale problem with periodic aspects. We shall use Bloch theory to deal with the periodicity, and semi-classical measures to take care of the multi-scale features. These notions will be explained and used for calculating the density of probability of presence of the electron in the limit where the size of the cells is much smaller than the macroscopic one.

The problem session consists in developments. A problem about super-adiabatic projectors and two other ones showing other applications of the use of two-scale semi-classical measures. It gives material for discussions.

TA: Lino Benedetto

Lecture 1: Monday 2-2:50pm
Lecture 2: Wednesday 11-11:50am
Lecture 3: Friday 11-11:50am
Problem Session: Thursday 2-3pm (in Pancoe Auditorium)
(problems)

This minicourse explores the relationship between the spectrum of certain quantum mechanical operators, such as the Laplacian and Schrödinger operators, and the dynamics of the underlying classical system. We will study this connection via trace formulas. The primary goal of the course is to outline the proof of the Gutzwiller trace formula. We will begin by presenting Weyl’s asymptotic formula for the eigenvalue counting function for bounded domains and compact Riemannian manifolds. Following this, we will discuss the Poisson summation formula, the Weyl remainder for flat tori, and fluctuations in the remainder term.

Next, we will introduce semiclassical Schrödinger operators and their spectra, followed by a discussion on the functional calculus of these operators and the Weyl formula. In the final lecture, we will cover the parametrix construction of the kernel of the propagator and establish the Gutzwiller trace formula.

Throughout the course, we will use several concepts from Tanya’s course, including the Fourier transform, tempered distributions, quantizations, and the stationary phase lemma. We will adhere to the notations used in Maciej Zworski’s book.

TA: Amir Vig

Lecture 1: Monday 3:30-4:20pm
Lecture 2: Tuesday 3:30-4:20pm
Lecture 3: Thursday 5-5:50pm
Problem Session: Tuesday 4:30-5:30pm
(problems)

The geodesic flow on a negatively-curved Riemannian manifold (e.g., a hyperbolic surface) is a prototype of an Anosov flow. These flows exhibit strong chaotic properties, such as sensitivity to initial conditions. The purpose of this mini-course is to describe the long-time asymptotic properties of Anosov flows (such as ergodicity, mixing, etc.) using microlocal analysis.

TA: Tristan Humbert

Lecture 1: Tuesday 9:30-10:20am
Lecture 2: Thursday 4-4:50pm
Lecture 3: Friday 9:30-10:20am
Problem Session: Friday 2-2:50pm
(problems)

Title: Random spherical harmonics
Abstract: I will discuss asymptotic properties of Laplace eigenfunctions in the
simple geometric model of the round 2-sphere. I will start by presenting
some of their concentration and non-concentration properties. Then,
building on ideas developped by Zelditch, I will show how probabilistic
methods allow to prove that typical eigenfunctions are in fact very well
behaved (boundedness of their L^p norms, equidistribution properties,
etc.).
Time: Friday 3:30-4:20pm