# Workshop Programme

## for period 30 July - 3 August 2012

### New developments in relativistic quantum mechanics and applications

30 July - 3 August 2012

Timetable

Monday 30 July | ||||

09:30-09:50 | Registration | |||

09:50-10:00 | Welcome from the Deputy Director | |||

10:00-10:45 | Séré, E (Université Paris-Dauphine) |
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Construction of the self-consistent Dirac vacuum in electromagnetic fields | Sem 1 | |||

This is joint work with Philippe Gravejat, Christian Hainzl and Mathieu Lewin (arxiv.org/1204.2893). Using the Pauli-Villars regularization and arguments from convex analysis, we construct the self-consistent polarized Dirac vacuum, in the presence of small external electromagnetic fields. We describe the electrons by a Hartree-Fock-type theory and the photons by a self-consistent classical magnetic potential. The resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics. |
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10:45-11:15 | Morning Coffee | |||

11:15-12:00 | Könenberg, M (Universität Wien) |
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Enhanced binding due to the quantized radiation field | Sem 1 | |||

We consider the non- and semi-relativistic Pauli-Fierz models for a single electron in an electrostatic potential interacting with the quantized radiation field. We prove that the ability of the system to bind the electron is enhanced in both models in the sense that coupling constant thresholds for certain classes of weak potentials are strictly decreased by the radiation field. Furthermore, we show that binding energies are always increased by the radiation field. The new achievementis that the occurrence of these effects is established, for arbitrary large values of the fine-structure constant and the ultra-violet cutoff. As a byproduct we determine the leading asymptotics of the ground state energy of the free, translation invariant semi-relativistic Pauli-Fierz Hamiltonian,as the ultra-violet cutoff goes to infinity. |
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12:30-13:30 | Lunch at Wolfson Court | |||

13:45-14:30 | Boulton, L (Heriot-Watt University) |
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Certified bounds for the eigenvalues of the Dirac operator | Sem 1 | |||

The estimation of certified bounds for the eigenvalues of the Dirac operator by finite-basis projection methods is a well-known challenge in Numerical Spectral Theory. In this talk various techniques for the effective computation of these certified bounds are examined. The applicability of these techniques will be demonstrated and the outcomes of various numerical experiments performed on benchmark potentials will be reported. |
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14:30-15:30 | Marquardt, FK (Universität Erlangen-Nürnberg) |
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A gentle introduction to the physics of graphene (mini-course 1) | Sem 1 | |||

In this talk I will give an elementary introduction to the physics of graphene. Though the talk represents the perspective of a physicist, it will be geared towards a mathematics oriented audience. |
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15:30-16:00 | Afternoon Tea | |||

16:00-17:00 | Marquardt, FK (Universität Erlangen-Nürnberg) |
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A gentle introduction to the physics of graphene (mini-course 2) | Sem 1 | |||

In this talk I will give an elementary introduction to the physics of graphene. Though the talk represents the perspective of a physicist, it will be geared towards a mathematics oriented audience. |
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17:00-18:00 | Welcome Drinks Reception | |||

18:15-19:15 | Dinner at Wolfson Court |

Tuesday 31 July | ||||

09:45-10:30 | Siedentop, H (Ludwig-Maximilians-Universität München) |
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Spectral Theory of Graphen Quantum Dots | Sem 1 | |||

We will give an estimate the number of electrons which a graphen quantum dot can support. |
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10:30-11:00 | Morning Coffee | |||

11:00-11:45 | Lewin, M (CNRS and University of Cergy-Pontoise) |
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Ground state properties of graphene in Hartree-Fock theory | Sem 1 | |||

In this talk I will discuss the Hartree-Fock approximation of graphene in infinite volume, with instantaneous Coulomb interactions. First I will explain how to construct the translation-invariant ground state. Due to the exchange term, the resulting effective Fermi velocity is logarithmically divergent at zero momentum. Then I will discuss the existence of ground states in the presence of local defects and some properties of the associated nonlinear response. Joint work with C. Hainzl and C. Sparber. Related Links: http://arxiv.org/abs/1203.5016 - arXiv paper |
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11:45-12:30 | Elton, DM (Lancaster University) |
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Spectral properties of a Dirac operator arising in models of graphene | Sem 1 | |||

We consider a Dirac operator which arises in modeling conduction within potential channels in graphene. For long uniform channels this reduces to a 1-dimensional linear spectral pencil problem for a Dirac operator with mass and a potential representing the channel cross section; a coupling constant in front of the potential is considered as the spectral parameter. Basic spectral properties are studied, together with the spectral asymptotics for large coupling constants. The latter show a surprisingly subtle dependence on the variation of the potential's sign and regions on which it is identically zero. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Hainzl, C (University of Tübingen) |
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Microscopic derivation of the Ginzburg-Landau model | Sem 1 | |||

14:45-15:30 | Rota-Nodari, S (Université Paris-Dauphine) |
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Mathematical study of the nonrelativistic limit of a relativistic mean-field model for nucleons | Sem 1 | |||

In this talk we consider a model for nucleons interacting with the omega and sigma mesons in the atomic nucleus. The model is relativistic, but we study it in the nuclear physics nonrelativistic limit, which is of a very different nature from the one of the atomic physics. This particular nonrelativistic limit naturally contains a relativistic correction linked to the spin-orbit interaction. I will present some existence results for this model and I will show that, for a good choice of parameters, the very striking shapes of mesonic densities inside and outside the nucleus are well described by the solutions of our model. The talk is based on joint works with Maria J. Esteban. |
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15:30-16:00 | Afternoon Tea | |||

16:00-16:45 | Stockmeyer, E (Ludwig-Maximilians-Universität München) |
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Confinement-deconfinement transitions of two-dimensional Dirac particles | Sem 1 | |||

It is well known that , as opposed to the non-relativistic case, particles described by the Dirac operator can not be confined through electric potential walls. However, one can produce a strongly confined system by adding a magnetic field growing at infinity. In this talk I will present some results describing confinement-deconfinement transitions of two-dimensional Dirac particles obtained by modulating the electro-magnetic field at infinity. This is joint work with Josef Mehringer. |
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18:15-19:15 | Dinner at Wolfson Court |

Wednesday 1 August | ||||

09:45-10:30 | Esteban, MJ (CNRS & Université Paris-Dauphine) |
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Symmetry breaking in atomic multi-configuration calculations | Sem 1 | |||

The multiconfiguration Dirac-Fock method allows to calculate the bound states of relativistic electrons in atoms or molecules. For a long time, this method has been known to provide certain "wrong" predictions in the nonrelativistic limit. By describing in detail the nonlinear model obtained in the nonrelativistic limit for Berilyum-like atoms, we will show the origin of this phenomenon which is actually linked to symmetry breaking in the angular momentum of the configurations. (These results are contained in a joint work with Mathieu Lewin and Andreas Savin). |
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10:30-11:00 | Morning Coffee | |||

11:00-11:45 | Umeda, T (University of Hyogo) |
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Low energy spectral and scattering theory for relativistic Schroedinger operators | Sem 1 | |||

We shall report some recent progress on spectral and scattering theory at low energy for the relativistic Schroedinger operators in the massless case. Some striking properties of the operator will be exhibited: (1) the absence of the zero-energy resonances; (2) the zero energy is not an accumulation point of embedded eigenvalues of the operator if 0 is not an eigenvalue. Low energy behavior of the wave operators and of the scattering operator will be presented. (This talk is based on joint work with Serge Richard.) |
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11:45-12:30 | Sabin, J (Université de Cergy-Pontoise) |
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Electron-Positron Pair Creation in a Nonlinear Dirac Model | Sem 1 | |||

Electron-Positron Pair Creation is a famous consequence of Dirac's interpretation of the relativistic vacuum. In this talk, I will present some new mathematical results about this phenomenon. These results are obtained within the Hartree-Fock approximation of Quantum Electrodynamics, a model introduced recently by Gravejat, Hainzl, Lewin, Séré, and Solovej. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-17:00 | Free Afternoon | |||

19:30-22:00 | Conference Dinner at Emmanuel College |

Thursday 2 August | ||||

09:45-10:30 | Derezinski, J (Uniwersytet Warszawski) |
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General properties of Bogoliubov transformations | Sem 1 | |||

Bogoliubov transformations (bosonic and fermionic) play an important role in many body quantum physics and quantum field theory. I believe it is useful to discuss them from the abstract point of view. I will review their basic algebraic properties, including the implementability in a Fock space and the choice of the phase factor. |
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10:30-11:00 | Morning Coffee | |||

11:00-11:45 | Nam, PT (Université de Cergy-Pontoise) |
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Bogoliubov spectrum of interacting Bose gases | Sem 1 | |||

We prove that Bogoliubov theory indeed predicts the ground state energy and the excitation spectrum of a very general class of interacting Boses gases. More precisely, we will show that the lower spectrum of the N-body Hamiltonian, in the limit as N tends to infinity, by the spectrum of a quadratic Hamiltonian in Fock space, which is the so-called Bogoliubov Hamiltonian. Such kind of result was already obtained for weakly interacting Bose gases by Seiringer and Grech. Our method is different and it works for a larger class of interactions. This is joint work with M. Lewin, S. Serfaty, and J. P. Solovej |
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11:45-12:30 | Morozov, S (Aarhus Universitet) |
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Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators | Sem 1 | |||

I am going to present a recent result in which a high-energy asymptotic expansion of the integrated density of states is obtained for a wide class of multidimensional almost-periodic pesudo-differential operators. Important particular applications are magnetic Schroedinger operators and operators with relativistic kinetic energies, for which the existence of such asymptotics was not known even in the periodic setting. The talk is based on a joint work with Leonid Parnovski (University College London) and Roman Shterenberg (University of Alabama in Birmingham). |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:45 | Frank, R (Princeton University) |
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Uniqueness and nondegeneracy of ground states for non-local equations in 1D | Sem 1 | |||

We prove uniqueness of energy minimizing solutions Q for the nonlinear equation (-\Delta)^s Q + Q - Q^{\alpha+1}= 0 in 1D, where 0 < s < 1 and 0 < \alpha < \frac{4s}{1-2s} for s < 1/2 and 0 < \alpha < \infty for s \geq 1/2. Here (-\Delta)^s is the fractional Laplacian. As a technical key result, we show that the associated linearized operator is nondegenerate, in the sense that its kernel is spanned by Q'. This solves an open problem posed by Weinstein and by Kenig, Martel and Robbiano. The talk is based on joint work with E. Lenzmann. |
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14:45-15:30 | Lenzman, E (Københavns Universitet (University of Copenhagen)) |
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Minimizers and blowup in relativistic Hartree-Fock-Bogoliubov theory | Sem 1 | |||

In this talk, we will discuss both time-independent and time-dependent models of Hartree-Fock-Bogoliubov (HFB) type that arise in the description of relativistic self-gravitating fermionic systems. In particular, we will examine the variational calculus attached to this problem, as well as results on the singularity formation for the corresponding dynamical equations of HFB type. Time permitting, we will also discuss some open problems too. This is joint work with M. Lewin, C. Hainzl, and B. Schlein. |
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15:30-16:00 | Afternoon Tea | |||

16:00-16:45 | Ben-Artzi, M (Hebrew University of Jerusalem) |
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From Crystal Optics to Dirac Operators--A Spectral Theory | Sem 1 | |||

A unified approach to first-order systems of classical physics is presented. Dirac operators are included, but there is no need for the fact that the square is the Laplacian (+ constant). The main tool is a derivation of the Limiting Absorption Principle, using the geometry of the characteristic surfaces and the trace lemma. As an application, global spacetime estimates are derived, in weighted Sobolev spaces. |
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18:15-19:15 | Dinner at Wolfson Court |

Friday 3 August | ||||

09:45-10:30 | Schmidt, KM (Cardiff University) |
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Some Spectral Properties of Massless Dirac Operators | Sem 1 | |||

The one-dimensional massless Dirac operator not only does not have a central gap in its spectrum, but it is even unitarily equivalent to the free massless operator and hence has purely absolutely continuous spectrum covering the whole real line. The talk reports on work regarding the question to what extent similar general statements about the essential spectrum and the absolutely continuous spectrum are possible in the higher-dimensional case. Under the assumption of spherical symmetry, it can be shown that the spectrum is purely absolutely continuous outside the limit range of the potential, thus giving an a priori set containing all eigenvalues. This can be partly extended to a far more general situation by a virial technique. Moreover, the essential spectrum is the whole real line for a large class of potentials satisfying a local separability condition. This is joint work with T. Umeda. |
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10:30-11:00 | Morning Coffee | |||

11:00-11:45 | Matte, O (Aarhus Universitet) |
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The mass shell in the semi-relativistic Pauli-Fierz model | Sem 1 | |||

We consider the semi-relativistic Pauli-Fierz model for a single free electron interacting with the quantized radiation field. By translation invariance the corresponding Hamiltonian can be written as a direct fiber integral with respect to different values of the total momentum of the system. Employing a variant of Pizzo's iterative analytic perturbation theory we prove that the mass shell, i.e. the ground state energies of the fiber Hamiltonians considered as a function of the total momentum, is twice continuously differentiable and strictly convex on balls about the origin. The ground state energy at total momentum zero turns out to be an eigenvalue of the corresponding fiber Hamiltonian while there are no ground state eigenvalues at non-vanishing total momenta. These results hold true, for sufficiently small coupling constants depending on an ultraviolet cutoff and the radii of the balls. The talk is based on joint work with Martin Koenenberg (Vienna). |
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11:45-12:30 | Vassiliev, D (University College London) |
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Spectral theory of first order systems: an interface between analysis and geometry | Sem 1 | |||

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the following objects: a) the propagator (time-dependent operator which solves the Cauchy problem for the dynamic equation), b) the spectral function (sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda) and c) the counting function (number of eigenvalues between zero and a positive lambda). We derive explicit two-term asymptotic formulae for all three. For the propagator "asymptotic" is understood as asymptotic in terms of smoothness, whereas for the spectral and counting functions "asymptotic" is understood as asymptotic with respect to the parameter lambda tending to plus infinity. In performing this analysis we establish that all previous publications on the subject are either incorrect or incomplete, the underlying issue being that there is simply too much differential geometry involved in the application of microlocal techniques to systems. We then focus our attention on the special case of the massless Dirac operator in dimension 3 and provide simple spectral theoretic characterisations of this operator and corresponding action (variational functional). [1] O.Chervova, R.J.Downes and D.Vassiliev. The spectral function of a first order system. Preprint arXiv:1204.6567. |
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12:30-13:30 | Lunch at Wolfson Court | |||

18:15-19:15 | Dinner at Wolfson Court |