
SCIENTIFIC PROGRAMS AND ACTIVITIES 

October 3, 2024  
AbstractsAlexandrova, 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 AharonovBohm 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 Dyatlov, Semyon (University of California,
Berkeley) 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 noninteger. In contrast with previous results of SjostrandZworski and GuillopeLinZworski, our class of manifolds includes general convex cocompact 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 GinzburgLandau (GL) theory, starting from the microscopic BardeenCooperSchrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical 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 Tunnel effect and symmetries for nonselfadjoint operators We study low lying eigenvalues and return to equilibrium for nonselfadjoint
semiclassical differential operators, where symmetries play an important
role. In the case of the KramersFokkerPlanck operator, we show
how the presence of certain supersymmetric and PTsymmetric 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. Weyl asymptotics in microwave experiments: From closed to open systems We present microwave experiments on the symmetry reduced 5disk
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. Embedded into the topology of our universe lurks a subtle yet farreaching
spectral ambiguity. There exist drumlike 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 twodimensional electrons
confined by individually positioned molecules) to reveal that isospectrality
provides an extra topological degree of freedom enabling the mapping
of complete electron wavefunctionsincluding all internal quantum
phase information normally obscured by direct quantum measurement
[1]. [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, 782787 (2008). 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, wellknown 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) A couple of endpoint restriction theorems for eigenfunctions Directional Localization and Toral Eigenfunctions To obtain information on eigenfunctions of the LaplaceBeltrami
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? Sharp Lp bounds on spectral clusters for rough metrics Toth, John (McGill 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 Minkowskilike spaces. Zelditch, Steve (Northwestern University) 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 arclength along it. 
