A quantum two-day meeting with Green talks, Green functions, and threshold behavior

Schedule for Wednesday, April 27th

13:30-14:15  Robert Stolz, University of the Virgin Islands 

Title:  Building Versatile  Reef Connectivity Graphs  with Ocean Currents.

Abstract: In this work, we demonstrate a flexible technique to build connectivity graphs of ocean reefs in the Virgin Islands with ocean currents. A key part of building the connectivity graphs uses a numerical approximation of non-linear diffusion advection equations based on random walks. Using the connectivity graphs we can identify critical reef neighborhoods and pathways for larvae dispersal or disease spread.

14:30-15:15    Arne Jensen, Aalborg University

Title: Discrete approximations to Dirac operators and norm resolvent convergence.

Abstract: We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We investigate whether the proposed discrete models converge in norm resolvent sense  to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. Both positive and negative results are obtained.

Joint work with H. Cornean (Aalborg) and H. Garde (Aarhus).

15:30-16:15   Edgardo Stockmeyer, Pontificia Universidad Catolica de Chile

Title: On the spectral stability of the nonlinear Dirac equation of Soler-type

Abstract: Consider the time dependent Dirac equation with a non-linearity of mass-type. (This may be considered as a generalized Soler model.) As apposed to Schrödinger-like models, there is very little known concerning the stability of standing wave solutions for non-linear Dirac equations. In this talk I will present some partial results concerning the spectral stability of such solutions in the 1-D case. In particular, we give bounds for the spectrum of the (non-symmetric) linearized operator.

16:30-17:15  Hynek Kovarik,  University of Brescia

Title: Resolvent expansions for Pauli and Dirac operators in dimension two.

Abstract: In this talk I will consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main result to be discussed provides resolvent expansions, including all the singular terms, of these operators in the vicinity of their thresholds. In particular it will be shown that the nature of these expansions is fully determined by the flux of the magnetic field.

17:30-18:15  Dirk Hundertmark,  Karlsruhe Institute of Technology

Title: Quantum Systems at The Brink: Sharp criteria for existence and nonexistence of zero energy bound states

Abstract: The existence of ground states below the essential spectrum is well understood. Far less understood is the question whether Schrödinger operators can have zero energy ground states at the edge of the essential spectrum. We present a family of sharp criteria which styles this question in all dimensions and which shows the existence of a spectral phase transition with critical dimension 4. This is joint work with Michal Jex (Prague) and Markus Lange (Sissa).

Schedule for Thursday, April 28th

09:30-10:15    Fumio Hiroshima,  Kyushu University

Title: Angle operators and phase operators associated with 1D-harmonic oscillators

Abstract: Let $h$ be the 1D-harmonic oscillator. The time operator $T$ is defined by an operator satisfying  $[h,T]=-i$. This may be called the angle operator or the phase operator in physics. As far as we know, however, the firm mathematical investigation of $T$ is scarce so far. We discuss (1) the domain of $T$, (2) continuous limit of $T$ and (3) the matrix representation of $T$.

10:30-11:15  Domenico Monaco, Sapienza University of Rome

Title: Purely linear response of the quantum Hall current to space-adiabatic perturbations

Abstract: Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost stationary state, we give a new proof that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime. In particular, we prove that, in quantum Hall systems and Chern insulators, the transverse response current is quantized up to any order in the strength of the inducing electric field. The latter is introduced as a perturbation to a periodic, spectrally gapped equilibrium Hamiltonian by means of a linear potential; existing proofs of the exactness of Kubo formula rely instead on a time-dependent magnetic potential. The result applies to both continuum and discrete crystalline systems modelling the quantum (anomalous) Hall effect.

This is joint work with Giovanna Marcelli (arXiv:2112.03071).

11:30-12:15  Jean-Marie Barbaroux, University of Toulon

Title: New results in progress concerning Van der Waals - London forces.

Abstract: TBA

Sponsored by: Institut for Matematiske Fag, AAU, and Danmarks Frie Forskningsfond bevilling 8021-00084B