# The Dirac Equation

Jul 6 – 8, 2022
Institut de Mathématiques de Bordeaux
Europe/Paris timezone

# Badreddine Benhellal: Poincaré-Steklov operator for the MIT bag model.

"In this talk, I will discuss the pseudodifferentiel properties of the Poincaré-Steklov (PS) operator associated with the MIT bag operator on a smooth domain $\O\subset\rr^3$ with a compact boundary $\partial\O$. This operator can be seen as the analog of the Dirichlet-to-Neumann mapping, where the free Dirac operator  $D_m=-i\alpha\cdot\nabla +m\beta$ plays the role of the Laplace operator, and the Dirichlet and the Neumann traces are replaced by orthogonal projections of the Dirichlet traces along the boundary $\partial\O$. In the first part of this talk, I will explain how the PS operator fits well into the framework of classical pseudodifferential operators and determine its principal symbol.  In the second part, I will discuss the properties of the PS operator when the mass $m$ becomes large enough. Namely,  I will show that it is a $1/m$-pseudodifferential operator and I will give its main properties, in particular its semiclassical principal symbol.  Then we apply these results to establish a Krein-type resolvent formula for the Dirac operator $H_M= D_m+ M\beta 1_{\rr^3\setminus\overline{\O}}$  in terms of the resolvent of the MIT bag operator when $M>0$ is large enough.   With its help, we show that in the large coupling limit $M\rightarrow\infty$, the operator  $H_M$ convergences toward the MIT bag operator in the norm-resolvent sense with a convergence rate of $\mathcal{O}(M^{-1})$."

This talk is based on joint work with Vincent Bruneau and Mahdi Zreik.

# Matteo Capoferri: The massless Dirac operator: global propagator and applications.

"In my talk I will present a global, invariant and explicit construction of hyperbolic propagators on closed Riemannian manifolds, with a special focus on the massless Dirac operator in dimension 3. I will show that the propagator can be written, modulo an operator with infinitely smooth kernel, as the sum of two oscillatory integrals, global in space and in time, and that this can be done in an invariant geometric fashion. I will then analyse the results through the prism of pseudodifferential techniques developed in a series of recent joint papers by Dmitri Vassiliev and myself, which, among other things, allow one to extend the construction to the Lorentzian setting. Time permitting, I will discuss applications to spectral theory and quantum field theory."

The talk is based on joint work with Dmitri Vassiliev (UCL) and Simone Murro (Genova).

# Biagio Cassano: General δ-shell interactions for the two-dimensional Dirac operator.

"In this talk we will consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials."

This is a joint work with V. Lotoreichik, A. Mas and M.Tušek.

# Jérôme Cayssol: Dirac materials.

"In this talk, I will explain how the Dirac equation can be used to describe the physics of materials like graphene and topological insulators. I will present briefly the topological band theory for Bloch electrons in crystalline materials. Specific examples of tight-binding models giving rise to lattice versions of the Dirac equation in various space dimension will be discussed in various space dimensions : 1D (Su–Schrieffer–Heeger and Rice–Mele models), 2D (graphene, boron nitride, Haldane model) and 3D (Weyl semi-metals). I will show how the Dirac equation provides a description of the physics near specific points in reciprocal space, and also allows to predict edge/surface states between various Dirac materials."

J. Cayssol and J.N. Fuchs, "Topological and geometrical aspects of band theory", J. Phys. Mater. 4, 034007 (2021).

# Orville Damaschke: An L2-index theorem for the Dirac operator on globally hyperbolic spacetimes.

"Index theory deals with solutions of certain differential equations, where the index roughly measures the difference between the number of kernel solutions and constraints coming from inhomogeneities. The famous Atiyah-Singer index theorem states, that for an elliptic operator this number can be expressed with topological data of the underlying (compact) Riemannian manifold - generalizations to singular and non-compact Riemannian spaces are known and well studied. Next to an analytical interest the index also appears formally in the study of anomalies in relativistic quantum field theories, where the underlying manifold is Lorentzian and the operator of interest usually hyperbolic. A rigorous treatment of these anomalies were not clear until the groundbreaking result of Bär and Strohmaier in 2015. Since then several extensions and applications have been discussed and are supposed to play a crucial role in the future analysis of quantum anomalies on globally hyperbolic spacetimes as well as differential geometry of pseudo-Riemannian manifolds."

# Brice Flamencourt: Dirac operators with large masses.

"We consider a particular class of Dirac operators with a potential interpreted as masses in separated regions of space. These operators appear naturally in the study of the MIT Bag model in dimension 3. We are interested in the behaviour of their eigenvalues when the masses become large.

This problem admits a generalization in higher dimensions, and it can also be consider from the point of view of spin geometry. We recall the construction of the Dirac operator on spin manifolds, so we can define a generalized MIT Bag Dirac operator for which we obtain convergence results in certain asymptotic regimes."

# Albert Mas: Spectral analysis of a confinement model in relativistic quantum mechanics.

"In this talk we will focus on the Dirac operator on domains of R^3 with confining boundary conditions of scalar and electrostatic type. This operator is a generalization of the MIT-bag operator, which is used as a simplified model for the confinement of quarks in hadrons that has interested many scientists in the last decades. It is conjectured that, under a volume constraint, the ball is the domain which has the smallest first positive eigenvalue of the MIT-bag operator. I will describe our results -in collaboration with N. Arrizabalaga (U. País Vasco), T. Sanz-Perela (U. Autónoma de Madrid), and L. Vega (U. País Vasco and BCAM)- on the spectral analysis of the generalized operator. I will discuss on the parameterization of the eigenvalues, their symmetry and monotonicity properties, the optimality of the ball for large values of the parameter, and the connection to boundary Hardy spaces."

# Pablo Miranda: Asymptotic behavior of the Spectral Shift Function for a discrete Dirac type operator in Z^2.

"In this talk, we consider a Dirac type operator in the graph Z^2. This is a matrix difference operator defined on the vertices and edges of Z^2, together with a perturbation given by a potential that decays at infinity. We are interested in the spectral properties of this operator, which we will study through the analysis of the spectral shift function. Our main theorem describes the asymptotic behavior of this function near the thresholds in the spectrum.

The main novelty of this work is related to the nature of the thresholds of our model, for which the spectral shift function has not been studied before. In particular, we consider parabolic and hyperbolic thresholds as well as Dirac points."

This part of a joint work with Daniel Parra and Georgi Raikov.

# Rémi Mokdad: Scattering of Dirac Fields in the Interior of Black Holes.

" Lately, more and more of the attention of the mathematical GR communities is being given to the cosmic censorship conjecture (CCC). In this context, there have been recently some studies focusing in particular on energy estimates and scattering theories in the interior of black holes. In this talk, I will discuss the results of two works on the scattering of Dirac fields in the interior of spherically symmetric charged black holes that are Reissner-Nördstrom-like, namely, the scattering between the outer event horizon and the inner Cauchy horizon. In the first paper, we show asymptotic completeness for the massive charged Dirac equation in the aforementioned interior region of a sub-extremal ((Anti-) De Sitter) Reissner-Nordström black hole. This is done by first decomposing the Dirac equation using the Newman-Penrose formalism and obtaining analytic scattering in a dynamical framework via the wave operators. The analytical results are then re-interpreted geometrically to define the trace operators. In the second paper the conformal scattering theory for the same settings is constructed and we obtain the geometrical results by directly solving the characteristic Cauchy problem using what I refer to as the 'waves re-interpretation' method."

# Fabio Pizzichillo: Boundary value problems for  2-D Dirac operator on Corner domains and the Coulomb interaction.

"This talk aims to present results on the self-adjoint extensions of Dirac operators on plane domains with corners in dimension two. We consider the case of infinite-mass boundary conditions and we obtain explicitly the self-adjoint extensions of the operator. It turns out that the presence of corners typically spoils the elliptic regularity known to hold for smooth boundaries.

Then we discuss the self-adjointness and some spectral properties of these operators in presence of a Coulomb-type potential with the singularity placed on the vertex."

This is a collaboration work with Hanne Van Den Bosch, Biagio Cassano and Matteo Gallone.

# Luz Roncal: Wavelet analysis and the Frisch--Parisi formalism for the Schrödinger equation.

"We consider the solution of the Schrödinger equation when the initial datum tends to the Dirac comb. It is known that the fluctuations associated to this equation can be expressed via a simplification of the Riemann non-differentiable function. We prove, using wavelet analysis, that the Frisch--Parisi multifractal formalism holds in this context."

Joint work with Sandeep Kumar, Felipe Ponce-Vanegas, and Luis Vega.

# Simona Rota Nodari: The Dirac-Klein-Gordon system in the strong coupling limit.

"In this talk, I will present a recent result on the behaviour of the solutions of a Dirac-Klein-Gordon system in the limit of strong coupling and large masses of the Klein-Gordon fields. I will prove convergence of the solutions to the system to those of a cubic non-linear Dirac equation. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction."

This is a joint work with J. Lampart, L. Le Treust and J. Sabin.