Schedule for: 18w5002 - Around Quantum Chaos

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk we relate concentration of Laplace eigenfunctions in position and momentum to sup-norms and submanifold averages. In particular, we present a unified picture for sup-norms and submanifold averages which characterizes the concentration of those eigenfunctions with maximal growth. We then exploit this characterization to derive geometric conditions under which maximal growth cannot occur. Moreover, we obtain quantitative gains in a variety of geometric settings.

Let $(M,g)$ be a compact Riemannian manifold and $\Omega\subset M$ a nonempty open set. Take an $L^2$ normalized eigenfunction $u$ of the Laplacian on $M$ with eigenvalue $\lambda^2$. What lower bounds can we get on the mass $m_\Omega(u)=\int_\Omega |u|^2$? There are two well-known bounds for general $M$: (a) $m_\Omega(u)\geq ce^{-C\lambda}$, following from unique continuation estimates, and (b) $m_\Omega(u)\geq c$, where $c>0$ is independent of $\lambda$, assuming that $\Omega$ intersects every sufficiently long geodesic (this is known as the geometric control condition). In general one cannot improve on the bound (a) for arbitrary $\Omega$, as illustrated by Gaussian beams on the round sphere. I will present a recent result which establishes the frequency-independent lower bound (b) for any choice of $\Omega$ when $M$ is a surface of constant negative curvature. This bound has numerous applications, such as control for the Schrödinger equation, exponential decay of damped waves, and the full support property of semiclassical measures. The proof uses the chaotic nature of the geodesic flow on $M$. The key new ingredient is a recently established fractal uncertainty principle, which states that no function can be localized close to a fractal set in both position and frequency. This talk is based on joint works with Jean Bourgain, Long Jin, and Joshua Zahl.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Let $(M,g)$ be a compact, Riemannian manifold and $V \in C^{\infty}(M; \R)$. Given a regular energy level $E > \min V$, we consider $L^2$-normalized eigenfunctions, $u_h,$ of the Schrodinger operator $P(h) = - h^2 \Delta_g + V - E(h)$ with $P(h) u_h = 0$ and $E(h) = E + o(1)$ as $h \to 0^+.$ The well-known Agmon-Lithner estimates \cite{Hel} are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region $\{ V>E \}.$ The decay rate is given in terms of the Agmon distance function $d_E$ associated with the degenerate Agmon metric $(V-E)_+ \, g$ with support in the forbidden region. Our main result is a partial converse to the Agmon estimates (ie. exponential {\em lower} bounds for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowable region $\{ V< E \}$ arbitrarily close to the caustic $\{ V = E \}.$ I will explain this result in my talk and then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates. This is joint work with Xianchao Wu.

Let $M$ be a compact hyperbolic manifold. The entropy bounds of Anantharaman et al. restrict the possible invariant measures on $T^1 M$ that can be quantum limits of sequences of eigenfunctions. Weaker versions of the entropy bounds also apply to approximate eigenfuctions ("log-scale quasimodes"), so it is interesting to construct such approximate eigenfunctions which converges to singular measures. Generalizing work of Brooks (hyperbolic surfaces) and Eswarathasan--Nonnenmacher (hyperbolic geodesics on Riemannian surfaces) we construct sequences of quasimodes on $M$ converging to totally geodesic submanifolds. A diagonal argument then realizes every invariant measure are a limit of quasimodes of fixed logarithmic width. Joint work with S. Eswarathasan

One of the fundamental problems in quantum chaos is to understand how high-frequency waves behave in chaotic environments. A famous but vague conjecture of Michael Berry predicts that they should look on small scales like Gaussian random waves. We will show how a notion of convergence for sequences of manifolds called Benjamini-Schramm convergence can give a satisfying formulation of this conjecture. The Benjamini-Schramm convergence includes the high-frequency limit as a special case but provides a more general framework. Based on this formulation, we will expand the scope and consider a case where the frequencies stay bounded and the size of the manifold increases instead. We will formulate the corresponding random wave conjecture and present some results to support it, including a quantum ergodicity theorem. Based on joint works with Tuomas Sahlsten, Miklos Abert and Nicolas Bergeron.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will present some recent results (joint with Tom Beck) about the behavior at infinity of nodal sets of eigenfunctions for small radial perturbations of the harmonic oscillator. For the unperturbed oscillator, separation of variables eigenfunctions are products of Laguerre functions and spherical harmonics. The angular momenta for a given \hbar that are present at fixed energy E = \hbar (n + d/2) are the spherical harmonics of frequency up to \hbar^{-1}. After a radial perturbation, the energy E eigenspace will break up into the different energies for different angular momenta. The radial rate of growth for the eigenfunctions is an increasing function of their energy E (namely r^{E-d/2}\exp{-r^2/2} in dimension d). Our results give precise information about which angular momenta give the biggest energies after perturbation and hence a good understanding of the size of the nodal set deep into the forbidden region.

Let $M$ be a compact real analytic negatively curved manifold. It admits a complexification in which the metric induces a pluri-subharmonic function $\sqrt{\rho}$ whose sublevel sets are strictly pseudo-convex domains $M_\tau$, known as Grauert tubes. The Laplace eigenfunctions on $M$ analytically continue to the Grauert tubes, and their complex nodal sets are complex hypersurfaces in $M_\tau$. Zelditch proved that the normalized currents of integration over the complex nodal sets tend to a single weak limit $dd^c\sqrt{\rho}$ along a density one subsequence of eigenvalues. In this talk, we discuss a joint work with Steve Zelditch, in which we show that the weak convergence result holds `on small scale,' namely, on logarithmically shrinking Kaehler balls whose centers lie in $M_\tau \setminus M$. The main technique is a Poisson-FBI transform relating QE on Kaehler balls to QE on the real domain. Similar small-scale QE results were obtained in the Riemannian setting by Hezari-Riviere and Han, and in the ample line bundle setting by Chang-Zelditch.

Courant's nodal domain theorem provides a natural  generalization of Sturm–Liouville theory to higher dimensions; however,  the result is in general not sharp. It was recently shown that the nodal  deficiency of an eigenfunction is encoded in the spectrum of the  Dirichlet-to-Neumann operators for the eigenfunction's positive and  negative nodal domains. While originally derived using symplectic  methods, this result can also be understood through the spectral flow  for a family of boundary conditions imposed on the nodal set. In this  talk I will describe this flow for a Schrödinger operator with separable  potential on a rectangular domain, and describe a mechanism by which low  energy eigenfunctions do or do not contribute to the nodal deficiency.  Operators on non-rectangular domains and quantum graphs will also be  discussed. This talk represents joint work with Gregory Berkolaiko (Texas A&M) and  Jeremy Marzuola (UNC Chapel Hill).

We start by reviewing the notion of "quantum graph", its eigenfunctions and the problem of counting the number of their zeros. The nodal surplus of the n-th eigenfunction is defined as the number of its zeros minus (n-1), the latter being the "baseline" nodal count of Sturm-Liouville theory.  It appears from numerics that the distribution of the nodal surplus of large graphs has a universal form: it approaches Gaussian as the number of cycles grows.  We will discuss our recent progress towards proving this conjecture. When the graph is composed of two or more blocks separated by bridges, we propose a way to define a "local nodal surplus" of a given block. Since the eigenfunction index n has no local meaning, the local nodal surplus has to be defined in an indirect way via the nodal-magnetic theorem of Berkolaiko, Colin de Verdiere and Weyand.  By studying the symmetry properties of the distribution of the local nodal surpluses we show that for a graphs with disjoint cycles the distribution of (total) nodal surplus is binomial. Based on joint work with Lior Alon and Ram Band.

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. We present the main results concerning Neumann domains on manifolds and on graphs. We compare manifolds to graphs and relate the Neumann domain results on each of them to the nodal domain study. The talk is based on joint works with Lior Alon, Michael Bersudsky, Sebastian Egger, David Fajman and Alexander Taylor.

The asymptotic of the number of nodal domains of eigenfunctions on a manifold is closely related to the dynamics of the geodesic flow on the manifold. For instance, if a surface with non-empty boundary has an ergodic geodesic flow, then for any given Dirichlet eigenbasis, one can find a subsequence of density one where the number of nodal domains tends to +\infty. In this talk, I'm going to discuss what happens to the unit circle bundle over a manifold. When equipped with a metric which makes the Laplacian to commute with the circular action on each fiber, the geodesic flow never is ergodic. Recently I and Steve Zelditch proved that among such metrics the following property is generic: for any given orthonormal eigenbasis one can find a subsequence of density 1 where the number of nodal domains is identically 2. This highlights how underlying dynamics can impact the nodal counting. I will sketch proof when we are considering a unit tangent bundle of a compact surface with the genus \neq 1. I also will present an explicit orthonormal eigenbasis on the 3 torus where all of them have only two nodal domains.

Given a smooth Morse function and a Riemannian metric on a compact manifold, Witten defined a semiclassical operator which is now referred as the Witten Laplacian. In light of the recent developpement towards the spectral analysis of hyperbolic dynamical systems, I will discuss some well-known properties and some new ones of these Witten Laplacians. Namely, I will explain that the spectrum of these operators converges in the semiclassical limit to the so-called Pollicott-Ruelle spectrum. This is a joint work with N.V. Dang (Univ. Lyon 1).

In hyperbolic dynamics (Anosov dynamics) each trajectory is strongly unstable and its behavior is unpredictable. A smooth probability distribution evolves also in a complicated way since it acquires higher and higher oscillations. Nevertheless using micro-local analysis, this evolution is predictable in the sense of distributions. It is similar to a quantum scattering problem in cotangent space as treated by Helffer and Sjöstrand using escape functions in (86'). In this talk we will use wave-packet transform (or FBI transform) and explain how to derive some spectral properties of the dynamics, as the existence of the intrinsic discrete spectrum of Ruelle resonances, a fractal Weyl law, estimates on the wave front set of the resonances, and band structure in the case of geodesic flow. Collaboration with Masato Tsujii.

There have been many works studying resonances generated by compactly supported potentials and by potentials which decay sufficiently quickly near infinity. However, in the case of random potentials there are only very few results. In this talk I will give an overview over some recent results in this direction by J. Sjöstrand, A. Drouot and F. Klopp. In particular I will discuss results obtained in collaboration with F. Klopp on the distribution of resonances close to the real axis and their link to the eigenstates of a full random Schrödinger operator in the localized regime.

Beginning with the predictions of Bogomolny-Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo-Seymour-Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations. The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a `typical' real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert's 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara-Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus. This is a joint with with D. Beliaev and S. Muirhead

In this talk we will discuss the asymptotic behavior of zeros and critical points for monochromatic random waves on compact, smooth, Riemannian manifolds, as the energy of the waves grow to infinity. This is joint work with Boris Hanin

If a real smooth function is given at random on the plane,  what is the probability that its vanishing locus has a large connected component ? I will explain some recent answers we obtained with Vincent Beffara  to this question, for some natural models coming from algebraic geometry and spectral analysis.

The behaviour of quantum chaotic states of billiard systems is believed to be well described by Berry's random plane wave model $$u=\sum_{j}c_{j}e^{i\lambda x\cdot \xi_{j}}$$ where the $c_{j}$ are Gaussian random variables. However, in $\R^{n}$ there are many other candidate waves over which we could randomise. Some are easier to adapt to manifolds than others. In this talk I will discuss when  (in $\R^{n}$) we can replace the $e^{i\lambda x\cdot \xi_{j}}$ with other waves and how those can be adapted to manifolds.

In this talk I will present some results on the exact controllability of Schrödinger equation on the torus. In a general setting, these questions are well understood for wave equations with continuous localisation functions, while for Schrödinger, we only have partial results. For rough localisation functions, I will first present some partial results for waves. Then I will show how one can take benefit from the particular simplicity of the geodesic flow on the torus to get (for continuous localisation functions) strong results (works by Haraux, Jaffard, Burq-Zworski, Anantharaman-Macia). Finally, for general localisation functions (typically characteristic functions of measurable sets) I will show how one can go further, by taking benefit from dispersive properties (on the 2 dimensional torus), to show that in this setting the Schrödinger equation is exactly controllable by any $L^2$ (non trivial) localisation function (and in particular by the characteristic function of any set (with positive measure).

We study the contribution to the wave trace of diffractive periodic orbits on  Euclidean surfaces with conical singularities. Using a new description of the propagator near  the so-called geometrically diffractive rays, we are able to compute the leading term of any kind of  diffractive periodic orbit. Joint work with A. Ford and A. Hassell.

Consider a semiclassical Schr\"odinger operator $P=h^2\Delta+V-E,$ where $V$ has a conormal singularity along a hypersurface.  The singular structure of $V$ affects the propagation of semiclassical singularities for solutions to $Pu=0,$ and in particular there is a `diffraction' of wavefront set by the interface: singularities are reflected as well as transmitted as they cross the interface transversely.  The reflected wave, however, is more regular, with the improvement depending on the regularity of the interface; moreover singularities cannot (for high enough regularity) glide along the interface.  (Joint work with Oran Gannot.)

Laughlin states are N-particle wave functions, successfully describing the fractional quantum Hall effect (QHE) for plateaux with simple fractions. It was understood early on, that much can be learned about QHE when Laughlin states are considered on a Riemann surface. Mathematically, it is interesting to know how do the Laughlin states depend on the Riemannian metric, magnetic potential function, complex structure moduli, singularities -- for the large number of particles N. I will review the results, conjectures and further questions in this area, and relation to topics such as Coulomb gases/beta-ensembles, Bergman kernels for holomorphic line bundles, Quillen metric, zeta determinants.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.