April 24, 2014
WEYL LAW at 100
a workshop
at the Fields Institute
September 19-21, 2012

Organizer: Maciej Zworski, UC Berkeley

Supported by NSF award #: 1216660


Alexandrova, Ivana (State University of New York, Albany)

Resonances in Scattering by Two Magnetic Fields at Large Separation

We consider the problem of quantum resonances in scattering by two compactly supported magnetic fields at large separation. We develop a new complex scaling method to establish the existence of the resonances in this setting and determine their approximate locations. The model we study is motivated by the Aharonov-Bohm effect, which is considered one of the most important quantum mechanical phenomena. This is joint work with Hideo Tamura.

Bindel, David (Department of Computer Science, Cornell University)

Numerical Analysis of Resonances

Resonances can be naturally expressed via eigenvalues of an operator after a complex rescaling, poles of an extended resolvent, or as points where a operator-valued nonlinear function becomes singular. Each of these formulations leads to numerical methods for computing resonances. What can we say about the accuracy, stability, efficiency, and reliability of these methods? In this talk, we answer some of these questions, and we describe an approach to comparing different methods based on perturbation analysis of some related nonlinear eigenvalue problems. We also describe applications of our methods to a few favorite problems from microsystem engineering and quantum mechanics.

Dyatlov, Semyon (University of California, Berkeley)

Fractal Weyl laws for resonances

For an even asymptotically hyperbolic Riemannian manifold with hyperbolic geodesic flow on the trapped set, we give an upper bound on the number of resonances in disks of fixed size around the real line. The power in the estimate depends on the upper MInkowski dimension of the trapped set and is typically non-integer. In contrast with previous results of Sjostrand-Zworski and Guillope-Lin-Zworski, our class of manifolds includes general convex co-compact hyperbolic quotients; this is made possible by the recent work of Vasy. Using same methods together with the work of Faure - Sjostrand, we also prove a sharp upper bound on the number of Ruelle resonances for contact Anosov flows. Finally, we discuss a similar bound for the remainder in the asymptotic expansion for the scattering phase. Joint work with Kiril Datchev, Colin Guillarmou, and Maciej Zworski.

Frank, Rupert (Princeton University)

A microscopic derivation of Ginzburg- Landau theory

We describe the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semi-classical in nature, and semi-classical analysis, with minimal regularity assumptions, plays an important part in our proof.

The talk is based on joint work with C. Hainzl, R. Seiringer and J. P. Solovey

Hitrik, Michael

Tunnel effect and symmetries for non-self-adjoint operators

We study low lying eigenvalues and return to equilibrium for non-self-adjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and PT-symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low temperature) limit. This analysis also applies sometimes to chains of oscillators coupled to two heat baths, but according to a recent result, when the temperatures of the baths are different, the supersymmetric approach may break down and should then be replaced by more direct methods of semiclassical analysis.

This is joint work with Frédéric Hérau and Johannes Sjöstrand.

Jakobson, Dmitry

Nodal sets and negative eigenvalues in conformal geometry

This is joint work with Yaiza Canzani, Rod Gover and Raphael Ponge. We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, including the Yamabe operator (conformal Laplacian), and the Paneitz operator. We give several applications to curvature prescription problems. We establish a conformal version of Courant's Nodal Domain Theorem. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension $n\geq 3$.

Kuhl, Ulrich
(Laboratoire de Physique de la Matière Condensée, CNRS UMR 7733, Université de Nice Sophia-Antipolis)

Weyl asymptotics in microwave experiments: From closed to open systems

We present microwave experiments on the symmetry reduced 5-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed using the method of harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with an leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to a non integer value. For the open systems the extraction of resonances becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture.

Lebeau, Gilles
(Université de Nice)

Hypoelliptic random walks

Let $M$ be a compact manifold with volume form dx, and let $X_1,...,X_N$ be N divergence free vectors fields on M such that the Lie algebra generated by the $X_i$ at any point of $M$ is equal to the tangent space (Hörmander condition). Let $h$ be a small parameter. We study the following random walk on $M$: if the walk is at $x$, choose a vector field $X_i$ at random, and move to the point $y=exp(tX_i)(x)$ with $t\in [-h,h]$ random. We prove an exponential rate of convergence to equilibrium, and when $h$ goes to zero, a convergence result from the Markov Laplacian to the hypoelliptic Laplacian $\sum X_i^2$.

This talk is based on joint works with Laurent Michel (Université de Nice).

Manoharan, Hari (Stanford University)

A Tale of Two Spectra: Quantum Drums Beat as One

Embedded into the topology of our universe lurks a subtle yet far-reaching spectral ambiguity. There exist drum-like manifolds of different shape that resonate at identical frequencies, making it impossible to invert a measured spectrum of excitations into a unique physical reality. An ongoing mathematical quest has recently compacted this conundrum from higher dimensions to planar geometries. Inspired by these isospectral domains, we introduce a class of quantum nanostructures characterized by matching electronic structure but divergent physical structure. We perform quantum measurements (scanning tunneling spectroscopy) on these "quantum drums" (degenerate two-dimensional electrons confined by individually positioned molecules) to reveal that isospectrality provides an extra topological degree of freedom enabling the mapping of complete electron wavefunctions-including all internal quantum phase information normally obscured by direct quantum measurement [1].
The robustness of the technique stems from its connection to supersymmetric quantum mechanics, where inequivalent "superpartner" Hamiltonians produce equivalent energy spectra. The methods are general and extensible to other nanostructures and fabrication techniques, and we have recently used variants of these ideas to experimentally detect superposition phase and the Berry phase. We can also comment on the extension of the Weyl Law to these quantum systems.

In these experiments we utilize the exciting technology of atomic and molecular manipulation: a custom-built scanning tunneling microscope, operating at low temperature in ultrahigh vacuum, is used to assemble nanostructures atom-by-atom to generate versatile quantum laboratories at the spatial limit of condensed matter.

[1] C. R. Moon, L. S. Mattos, B. K. Foster, G. Zeltzer, W. Ko, and H. C. Manoharan, "Quantum phase extraction in isospectral electronic nanostructures," Science 319, 782-787 (2008).

Melrose, Richard (Massachusetts Institute of Technology)

Eigenvalues, transmission and transition

I will talk about the `articulated manifolds' which corresponds to a component of the boundary of a manifold with corners with the analogue of a Dirac operator induced from the interior. With relatively minor modifications, well-known results (including those by Ivrii) on eigenvalue asymptotics for boundary problems can be extended to this setting, at least when the articulation is in codimension one. The transition part of the title refers to the possibility of examining the passage from a smooth manifold to the articulated case.

Safarov, Juri (King's College, London)

Ergodicity of branching billiards

If the Riemannian metric has a jump discontinuity on a surface of codimension one then a geodesic hitting the surface admits two possible continuations called the reflected and retracted rays. In this situation, instead of the usual geodesic flow, one has to deal with a branching (ray-splitting) billiard dynamics. It is not immediately clear how to define ergodicity for such billiard systems. On the other hand, quantum ergodicity of eigenfunctions of the corresponding Laplacian can be defined in the usual way. We investigate the link between ergodic properties of eigenfunctions and characteristics of branching billiards and introduce a new notion of ergodicity in terms of a dynamical system on the space of functions on the cotangent bundle. In order to do this, we study Fourier integral operators associated with reflected and refracted rays, calculate their principal symbols and develop a symbolic calculus giving explicit formulae for principal symbols of their compositions and adjoints.

This is a joint work with D. Jakobson and A. Strohmaier.

Sogge, Chris
(Johns Hopkins University}

A couple of endpoint restriction theorems for eigenfunctions

In this joint work with Xuehua Chen we prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general $3$-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $R$, we are able to improve the corresponding $L^2$-restriction bounds of Burq, Gerard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for restrictions to geodesics, which, by Holder's inequality and interpolation, implies improved $L^p$-bounds for all exponents $p\ge 2$. We do this by using oscillatory integral theorems of Hormander, Greenleaf and Seeger and Phong and Stein, along with a simple geometric lemma about properties of the mixed-Hessian of the Riemannian distance function restricted to pairs of geodesics in Riemannian surfaces.

Tacy, Melissa
(Northwestern University}

Directional Localization and Toral Eigenfunctions

To obtain information on eigenfunctions of the Laplace-Beltrami operator on general manifolds it is customary to use the status of such a function as a stationary state and invoke an invariance under propagation property. However it is sometimes the case that in specific examples further information is available through other means. In this talk I will focus on the case of the flat torus. In this case we have a complete algebraic description of the eigenfunctions; $$u=\sum_{k}c_{k}e^{i\lambda{}k\cdot{}x}$$ where $\lambda{}k_{1}$ and $\lambda{}k_{2}$ are integers and $|k|=1$. We ask how can we transfer this information to an analytic setting?

Tataru, Daniel
(UC Berkeley)

Sharp Lp bounds on spectral clusters for rough metrics

We consider the problem of obtaining Lp bounds for spectral cluster associated to the Laplace-Beltrami operator on manifolds with rough metrics (below C^2). We use wave packet methods to establish sharp estimates in two space dimensions. This is joint work with Herbert Koch (U. Bonn) and Hart Smith (U. Washington).

Toth, John (McGill University)

Intersection Bounds for Nodal Sets of Planar Neumann Eigenfunctions with Interior Analytic Curves

Vasy, András (Stanford University)

The Laplacian on differential forms on asymptotically hyperbolic spaces

I will explain how to obtain the analytic continuation of the resolvent of the Laplacian on differential forms on even asymptotically hyperbolic spaces by ``continuation across the boundary''. This method also supplies high energy resolvent estimates for this continuation in strips, including microlocal estimates. Such estimates in the scalar setting have been useful in recent results on fractal Weyl laws by Datchev and Dyatlov, and in the description of microlocal limits of Eisenstein functions by Dyatlov and Guillarmou. I will also relate this extended problem to asymptotically Minkowski-like spaces.

Zelditch, Steve (Northwestern University)

Ergodicity and intersections of nodal sets and geodesics

When the geodesic flow is ergodic, it is conjectured that nodal sets of eigenfunctions become equidistributed wrt volume in the high eigenvalue limit. This is much too hard, but the equidistribution problem becomes more tractable in the complex domain. In this talk, I discuss equidistribution of intersection points of a nodal hypersurface with a geodesic in the complexification of a real analytic Riemannian manifold. Under certain assumptions on the geodesic, they become concentrated in the real points of the geodesic and become uniformly distributed with respect to arc-length along it.

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