Conference schedule

Monday 9th September Tuesday 10th September Wednesday 11th September Thursday 12th September Friday 13th Septemer
9.00am--10.00am
Registration
10.00am--11.00am
Bolte
9.30am--10.30am
Nonnenmacher
9.30am--10.30am
Polterovich
9.30am--10.30am
Helffer
9.30am--10.30am
Macià
11.30am--12.30pm
Pollicott
11.00am--12.00m
Hillairet
11.00am--12.00m
Zelditch
11.00am--12.00m
Safarov
11.00am--12.00m
Schubert
Lunch Lunch Lunch Lunch Lunch
2.30pm--3.30pm
Sharp
2.00pm--3.00pm
Jakobson
2.00pm--3.00pm
Rueckriemen
Future directions
4.00pm--5.00pm
Pushnitski
3.30pm--4.30pm
Guillarmou
3.30pm--4.30pm
Kurlberg
Problem session Problem session Posters Problem seesion
Conference Dinner
For titles and abstracts, click onto the speakers' names (or scroll down).

Abstracts

Jens Bolte (Royal Holloway)

"Many-particle quantum graphs: spectra and Bose-Einstein condensation"

Quantum graphs are popular models in quantum chaos and in spectral geometry. We extend existing models by introducing many-particle systems with singular contact interactions. Self-adjoint operators are constructed via associated quadratic forms and spectral properties are established. We then classify non-interacting many-particle systems according to the presence or absence of Bose-Einstein condensation (BEC). This classification involves negative Laplace eigenvalues. We also prove that BEC is absent when repulsive hardcore interactions are switched on.

Mark Pollicott (Warwick)

"Zeta functions for infinite area surfaces"

Abstract to follow.

Richard Sharp (Warwick)

"Pair correlations for length spectra on negatively curved manifolds"

Naturally associated to a (compact) negatively curved Riemannian manifold is its length spectrum, i.e. the set of lengths of closed geodesics. This has been the subject considerable study using techniques from both spectral theory (particularly in the case of constant curvature) and ergodic theory. We will discuss "pair correlations" within this spectrum: asymptotics for pairs of closed geodesics, the difference of whose length lies in some (possibly shrinking) interval. In our approach, the closed geodesics are counted according to a discrete length which, in certain cases, can be chosen to be the word length of the corresponding conjugacy class in the fundamental group. (This is joint work with Mark Pollicott.)

Alexander Pushnitski (King's College)

"The spectral density of the scattering matrix of the magnetic Schrodinger operator for high energies"

Let S(k) be the scattering matrix of the Schrodinger operator with smooth short-range electric and magnetic potentials; k>0 is the energy parameter. The eigenvalues of S(k) are located on the unit circle. I will discuss two recent results (joint with my PhD student Daniel Bulger) on the asymptotic density of these eigenvalues as the energy k goes to infinity. It turns out that this asymptotic density can be described by explicit formulas, involving the electric potential and the magnetic vector-potentials. These formulas have a semiclassical nature, although this is not obvious.

Stéphane Nonnenmacher (CEA-Saclay)

"Quantum resonances in a normally hyperbolic setting"

(Joint work with Maciej Zworski)

A quantum (or wave) scattering system consists of waves coming from infinity, interacting with a "scatterer" (which may consist in a complicated geometry), and then escaping to infinity. The (selfadjoint) Hamiltonian operator ruling the system does not admit stationary modes, but rather "metastable" modes associated with complex valued resonances.

We are interested in the high-frequency distribution of these resonances, in particular the presence of a "resonance gap" (namely a uniform quantum decay rate) in terms of the corresponding classical Hamiltonian dynamics. The dynamics on the classical "trapped set" appears as the "classical backbone" for such questions.

After reviewing a few examples, we will concentrate on the case where the trapped set is a normally hyperbolic symplectic submanifold. This situation is relevant for instance in quantum chemistry or general relativity, but it also helps to understand a purely classical dynamical problem, namely the decay of correlations for contact Anosov flows.

Luc Hillairet (Orléans)

"Variational formulas for the spectrum of translation surfaces"

Translation surfaces are special flat surfaces with conical singularities whose variations are well understood from the geometric viewpoint. We study how different spectral quantities may change when the geometry does. We relate these variations to finite rank operators that account for the presence of the singularities. (Joint with A. Kokotov).

Dmitry Jakobson (Montréal)

"Averaging over manifold of metrics with the fixed volume form"

This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor.

We study the manifold of all metrics with the fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. We compute the characteristic function for the $L^2$ (Ebin) distance to the reference metric. Next, we study Lipschitz-type distance between Riemannian metrics, and give applications to the diameter and eigenvalue functionals.

Colin Guillarmou (Paris ENS)

"Ruelle-Pollicott resonances for axiom A flows"

Abstract to follow.

Iosif Polterovich (Montréal)

"Spectral geometry of the Steklov problem"

The Steklov problem is an eigenvalue problem with spectral parameter in the boundary conditions. It has many physical applications, particularly to hydrodynamics, vibration theory, etc. Recently, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. I will discuss some recent advances, in particular on the study of spectral rigidity and eigenvalue estimates. The talk is based on joint works with David Sher (University of Michigan) and Alexandre Girouard (Universite Laval).

Steven Zelditch (Northwestern)

"Manifolds with maximal eigenfunction growth"

The pointwise Weyl law implies that the sup norm of an L2 normalized eigenfunction of the Laplacian of eigenvalue lambda on a compact Riemannian manifold (M, g) is bounded by C_b lambda^{(n-1)/4}). On the round sphere and on any surface of revolution, the zonal eigenfunctions achieve this bound. We say that (M, g) has maximal eigenfunction growth if it has a sequence of eigenfunctions attaining the bound. In 2002 Sogge and I proved that such a manifold must have a self-focal point, i.e a point p for which a positive measure of the geodesics starting at p loop back to p at some time (measure in the usual sense on S^*_p M). However this is not sharp since umbilic points of ellipsoids are self-focal but the eigenfunctions do not have maximal sup norms. In recent work (still being written), Sogge and I prove that if (M, g) is real analytic and has maximal eigenfunction growth, then the first return map on S^*_p M for geodesic loops must have an invariant L^2 function. This is in a certain sense a sharp result because there is a converse for quasi-modes.

Pär Kurlberg (KTH)

"Quantum Ergodicity for point scatterers on arithmetic tori"

We investigate eigenfunctions of the Laplacian, perturbed by a delta-potential, on arithmetic 2D-tori. For a full density subsequence of "new" eigenfunctions we prove decay of matrix coefficients associated with pure momentum observables. This, together with previous work by Rudnick-Ueberschaer, allows us to conclude that quantum ergodicity holds for the set of "new" eigenfunctions.

Roman Schubert (Bristol)

"What is the semiclassical limit of non-Hermitian time evolution?"

In recent work with E.M. Graefe we studied the semiclassical limit of non-Hermitian time evolution using coherent states and found that, contrary to what one would expect, the classical dynamics which emerges is not the complexified Hamiltonian one. Instead a combination between a Hamiltonian and a gradient vector field governs the dynamics. In this talk we describe how to put these observations onto a rigorous footing and rederive them from complex WKB theory. Complex WKB theory is based on complex Hamiltonian dynamics, but the semiclassical dominant terms in a WKB wave function come from the local minima of the imaginary part of the wave function, and these local minima do not follow Hamiltonian trajectories, but a different dynamics which we derive and which coincides with our previous results.


Last modified: Tue Jan 14 16:53:13 GMT 2014