Category: seminar

Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, June 27, 17:00

Christian Sadel

Speaker: Leonid Parnovski. University College London

Title: Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons.

Place: PUC Chile, Campus San Joaquin, Fac. Matematicas, Sala 2

Abstract:
I will discuss recent results (joint with M.Levitin, I.Polterovich and D.Sher) on the asymptotic behaviour of Steklov eigenvalues on polygons and other two-dimensional domains with corners. The answer is completely unexpected and depends on the arithmetic properties of the angles. 

Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, June 6, 17:00

Christian Sadel

Expositor:  Horia Cornean, Aalborg University

Title:  Peierls’ substitution for low lying spectral energy windows 

Abstract: We consider a 2d periodic Schrödinger operator for which we assume that either the first Bloch eigenvalue remains isolated while its corresponding Riesz spectral projection family has a non-zero Chern number, or the first two Bloch eigenvalues have a conical crossing. The system is afterwards  perturbed by a weak magnetic field which slowly varies around a positive mean. Then we prove the appearance of a “Landau type” structure of spectral islands and gaps both at the bottom of the spectrum, and near the possible crossings.
This is joint (past and ongoing) work with B. Helffer (Nantes) and R. Purice (Bucharest).

http://www.mat.uc.cl/~graikov/seminar.html

seminar FisMat

FisMat Seminar, June 5, 15:45

Christian Sadel

Speaker: Horia Cornean. Aalborg University

Title: A Beals criterion for magnetic pseudo-differential operators proved with magnetic Gabor frames

Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 5

Abstract:
First, we give a new proof for the Beals commutator criterion for non-magnetic Weyl pseudo-differential operators based on classical Gabor tight frames. Second, by introducing a modified ‘magnetic’ Gabor tight frame, we naturally derive the magnetic analogue of the Beals criterion originally considered by Iftimie-Mantoiu-Purice. This is joint work with Bernard Helffer (Nantes) and Radu Purice (Bucharest). https://doi.org/10.1080/03605302.2018.1499777

Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, May 30, 17:00

Christian Sadel

Speaker: Dmitrii Shirokov. National Research University Higher School of Economics, Russia

Title: On Constant Solutions of Su(2) Yang-Mills Equations

Place: Pontificia Universidad Católica, Facultad de Matemáticas (Campus San Joaquin), Sala 2

Abstract:
We present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space of arbitrary finite dimension. We use the singular value decomposion method and the method of two-sheeted covering of orthogonal group by spin group to do this. Using hyperbolic singular value decomposition, we solve the same problem in arbitrary pseudo-Euclidean space. The case of Minkowski space is discussed in details. Nonconstant solutions of the Yang-Mills equations are considered in the form of series of perturbation theory.

http://www.mat.uc.cl/~graikov/seminar.html 

seminar FisMat

FisMat Seminar, May 29, 15:45

Christian Sadel

Speaker: Andrés Fernando Reyes Lega. Universidad de los Andes (Colombia)

Title: Emergent Gauge Symmetries, Quantum Operations and Anomalies

Place: Pontificia Universidad Católica, Facultad de Matemáticas (campus San Joaquin), Sala 5

Abstract:
The Gelfand-Naimark-Segal (GNS) construction is a fundamental tool for the study of the representation theory of operator algebras. It also plays a prominent role in the algebraic approach to quantum field theory. In this talk I will discuss some examples of applications of the algebraic approach to quantum physics to systems with a finite number of degrees of freedom. I will illustrate how the GNS construction naturally leads to interesting connections between gauge symmetries, anomalies and quantum-information concepts like entanglement entropy and quantum operations.

Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, May 16, 17:00

Christian Sadel

Frédéric Klopp. Institut de Mathématiques Jussieu – Paris Rive Gauche, Sorbonne
Title: Exponential decay for the 2 particle density matrix of disordered many-body fermions at zero and positive temperature.
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (campus San Joaquin), Sala 2
Abstract:
We will consider a simple model for interacting fermions in a random background at zero and positive temperature. At zero temperature, we prove exponential decay for the 2 particle density matrix of a ground state. At positive temperature we prove exponential decay for the 2 particle density matrix of the density operator in the grand canonical ensemble.
http://www.mat.uc.cl/~graikov/seminar.html

seminar FisMat

FisMat Seminar, May 15, 15:45

Christian Sadel

Massimo Moscolari. Aalborg University
Title: Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians
Place: Pontificia Universidad Católica, Facultad de Matemáticas (Campus San Joaquin), Sala 5
Abstract:
In 1978 Wannier discovered a Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. I will show how to extend this relation to a gap labelling theorem for any 2D Bloch-Landau Hamiltonian operator and to certain non-covariant systems having slowly varying magnetic fields. The integer slope will be interpreted as the Chern character of the projection onto the space of occupied states. The talk is based on a joint work with H. Cornean and D. Monaco.

Seminar Spectral Theory and PDE

Seminar Spectral Theory and PDE, May 2, 17:00

Christian Sadel

Georgi Raikov. Pontificia Universidad Católica de Chile
Title: Threshold singularities of the spectral shift function for geometric perturbations of a magnetic Hamiltonian
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (campus San Joaquin), Sala 2
Abstract:
I will consider the 3D Schrödinger operator H0 with constant magnetic field, and its perturbations H_+ (resp., H_-\, ) obtained from H_0  by imposing Dirichlet (resp., Neumann) conditions on an appropriate surface. I will introduce the Krein spectral shift function for the operator pairs (H_0;H_+) and (H_0;H_-), and will discuss its singularities at the Landau levels which play the role of thresholds in the spectrum of the unperturbed operator H_0. 
The talk is based on a joint work with V. Bruneau (Bordeaux).
http://www.mat.uc.cl/~graikov/seminar.html

seminar FisMat

FisMat Seminar, April 24, 15:45

Christian Sadel

Svetlana Jitomirskaya. University of California, Irvine
Title: Cantor spectrum of a model of graphene in magnetic field
Place: Pontificia Universidad Católica de Chile, Facultad de Matemáticas (Campus San Joaquin), Sala 5
Abstract:
We consider a quantum graph as a model of graphene in magnetic fi elds and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux  through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero and Hausdorff dimension bounded by 1/2.

Based on joint works with S. Becker, R. Han, and also I. Krasovsky

seminar FisMat

FisMat Seminar, April 17, 15:45

Christian Sadel

Walter de Siqueira Pedra. University of São Paulo
Title: Thermodynamical Stability and Dynamics of Lattice Fermions with Mean-Field Interactions
Place: Pontificia Universidad Católica, Facultad Matemáticas (Campus San Joaquin) Sala 5
Abstract:
For lattice fermions we study the thermodynamic limit of the time evolution of observables when the corresponding finite-volume Hamiltonians contain mean-field terms (like, e.g., the BCS model). It is well-known that, in general, this limit does not exist in the sense of the norm of observables, but may exist in the strong operator topology associated to a well-chosen representation of the algebra of observables. We proved that this is always the case for any cyclic representation associated to an invariant minimizer of the free energy density, if the Hamiltonians are invariant under translations. Our proof uses previous results on the structure of states minimizing the free energy density of mean-field models along with Lieb-Robinson bounds for the corresponding families of finite-volume time evolutions. This is a joint work with Jean-Bernard Bru, Sébastien Breteaux and Rafael Miada.