MCQM22 Third School and Workshop

Mathematical Challenges in Quantum Mechanics

Como, Italy -- June 13 - 18, 2022

Sponsors

Laboratorio Ypatia

di Scienze Matematiche

The mini-workshop is scheduled for the dates 14-15 June 2021, it will be streamed online through Zoom.

Please register here.

Schedule

14/6/2021

15:30  Eric Séré (Université Paris-Dauphine)

Title: Dirac-Coulomb operators with general charge distribution: results and open problems

Abstract: This talk is based on joint works with M.J. Esteban and M. Lewin. Consider an electron moving in the attractive Coulomb potential generated by a non-negative finite measure representing an external charge density. If the total charge is fixed, it is well known that the lowest eigenvalue of the corresponding Schrodinger operator is minimized when the measure is a delta. We investigate the conjecture that the same holds for the relativistic Dirac-Coulomb operator. First we give conditions ensuring that this operator has a natural self-adjoint realisation and that its eigenvalues are given by min-max formulas. Then we define a critical charge such that, if the total charge is fixed below it, then there exists a measure minimising the first eigenvalue of the Dirac-Coulomb operator. Moreover this optimal measure concentrates on a compact set of Lebesgue measure zero. The last property is proved using a new unique continuation principle for Dirac operators.

17:00  Michael I. Weinstein (Columbia University)

Title: Tight binding approximation of continuum 2D quantum materials

Abstract: We consider 2D quantum materials, modeled by a continuum Schroedinger operator whose potential is composed of an array of identical potential wells centered on the vertices of a discrete subset, \Omega, of the plane. 

We study the low-lying spectrum in the regime of very deep potential wells.

We present results on scaled resolvent norm convergence to a discrete (tight-binding) operator and, in the translation invariant case, corresponding results on the scaled convergence of low-lying dispersion surfaces.

Examples include the single electron model for bulk graphene (\Omega=honeycomb lattice), and  a sharply terminated graphene half-space, interfaced with the vacuum along an arbitrary line-cut. 

We also apply our methods to the case of strong constant perpendicular magnetic fields. 

This is joint work with CL Fefferman and J Shapiro.

A detailed analysis of the spectrum of the limiting tight binding model on a honeycomb lattice, which is terminated along an arbitrary rational line-cut (joint work with CL Fefferman and S Fliss), will be presented in the upcoming lecture of CL Fefferman.

15/6/2021

15:30  Ari Laptev (Imperial College London)

Title: Symmetry Results in Two-Dimensional Inequalities for Aharonov–Bohm Magnetic Fields.

Abstract: We study functional and spectral properties of perturbations of a magnetic second order differential operator on a circle.

This operator appears when considering the restriction to the unit circle of a two dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove some Hardy-type inequalities and sharp Keller-Lieb-Thirring inequalities.

17:00  Charles Fefferman (Princeton University)

Title: Graphene edge states in a tight binding model

Abstract: We study a standard tight binding model of graphene, sharply terminated along an edge. It is well known that zero energy (“flat band”)

 edge states arise for a "zigzag" edge, while an "armchair" edge gives rise to no edge states.

We present joint work with S. Fliss and M. Weinstein that determines which rational edges give rise to flat band edge states, and exhibits formulas for such edge states when they exist. The joint work includes also preliminary results on non-flat-band edge states.

Thanks to results presented in Michael Weinstein's lecture, flat bands for a tight binding model give rise to almost flat band edge states for a continuum model.