E. S'Er'E | Université Paris Dauphine - PSL (original) (raw)
Papers by E. S'Er'E
This note is a written version of the talk given at the French-Italian conference which took plac... more This note is a written version of the talk given at the French-Italian conference which took place in Torino in July 2006. In this talk I presented an overwiew of various results about the characterization of eigenvalues of self-adjoint operators in a spectral gap and its application to relativistic quantum mechanics, in order to define and compute the eigenvalues of Coulomb-Dirac type operators, with or without external magnetif field. The talk contained information about rigorous mathematical results and also about numerical computations done for some model problems in atomic and molecular physics. All the results described here can be found in [2, 4, 3, 1] and a very complete review about all this kind of problems for strongly indefinite operators, in particular the Dirac operator, can be found in [5].
Journées Équations aux dérivées partielles, 2012
Physical Review Letters, 2000
Numerical methods avoiding the problem of variational collapse and the appearance of spurious roo... more Numerical methods avoiding the problem of variational collapse and the appearance of spurious roots are proposed for the computation of the eigenvalues and eigenstates of one-particle Dirac Hamiltonians. They are based on exact characterizations of the eigenvalues by a direct minimization of the corresponding Rayleigh quotient on a set described by a nonlinear constraint.
Physical Review A, 2007
We study a mean-field relativistic model which is able to describe both the behavior of finitely ... more We study a mean-field relativistic model which is able to describe both the behavior of finitely many spin-1/2 particles like electrons and of the Dirac sea which is self-consistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the photon field. All our results are non-perturbative and mathematically rigorous.
Journal of Physics A: Mathematical and General, 2005
Journal of Functional Analysis, 2000
International Journal of Quantum Chemistry, 2003
Communications in Mathematical Physics, 2005
Calculus of Variations and Partial Differential Equations, 2000
In this paper we give two different variational characterizations for the eigenvalues of H + V wh... more In this paper we give two different variational characterizations for the eigenvalues of H + V where H denotes the free Dirac operator and V is a scalar potential. The first one is a min-max involving a Rayleigh quotient. The second one consists in minimizing an appropriate nonlinear functional. Both methods can be applied to potentials which have singularities as strong as the Coulomb potential.
Archive for Rational Mechanics and Analysis, 2013
Archive for Rational Mechanics and Analysis, 2009
This chapter is a review of some methods used for the computation of relativistic atomic and mole... more This chapter is a review of some methods used for the computation of relativistic atomic and molecular models based on the Dirac equation. In the linear case, we briefly describe finite basis set approaches, including ones that are generated numerically, perturbation theory and effective Hamiltonians procedures, direct variational methods based on nonlinear transformations, min-max formulations and constrained minimizations. In the atomic case, we describe the MCDF method and some ways to solve numerically the homogeneous and inhomogeneous Dirac-Fock equations. Finally, we describe also some numerical methods relevant to the case of molecules. Key words: Quantum chemistry, relativistic models for atoms and molecules, computational methods, no-pair Hamiltonian, relativistic many-body perturbation theory, relativistic random phase approximation, Dirac equation, continuous spectrum, eigenvalues, variational collapse, spurious states, finite basis, numerical basis sets, perturbation the...
Calculus of Variations and Partial Differential Equations, 1996
The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic fi... more The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.
Journal of Functional Analysis, 2000
This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the es... more This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential.
Proceedings of the London Mathematical Society
Stochastic Processes and their Applications
We consider the stochastic control problem of a financial trader that needs to unwind a large ass... more We consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form.
Journal of Functional Analysis
We prove local smoothing estimates for the massless 3D Dirac equation with a Coulomb potential. O... more We prove local smoothing estimates for the massless 3D Dirac equation with a Coulomb potential. Our strategy is inspired by [9] and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.
A non-homogeneous Hardy-like inequality has recently been found to be closely related to the know... more A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum. Hardy inequalities and Coulomb singularities The relationship between usual Hardy inequalities and spectra of elliptic operators is quite well known: the classical Hardy inequality in IR N − ∆ ≥ (N − 2) 2 4
This note is a written version of the talk given at the French-Italian conference which took plac... more This note is a written version of the talk given at the French-Italian conference which took place in Torino in July 2006. In this talk I presented an overwiew of various results about the characterization of eigenvalues of self-adjoint operators in a spectral gap and its application to relativistic quantum mechanics, in order to define and compute the eigenvalues of Coulomb-Dirac type operators, with or without external magnetif field. The talk contained information about rigorous mathematical results and also about numerical computations done for some model problems in atomic and molecular physics. All the results described here can be found in [2, 4, 3, 1] and a very complete review about all this kind of problems for strongly indefinite operators, in particular the Dirac operator, can be found in [5].
Journées Équations aux dérivées partielles, 2012
Physical Review Letters, 2000
Numerical methods avoiding the problem of variational collapse and the appearance of spurious roo... more Numerical methods avoiding the problem of variational collapse and the appearance of spurious roots are proposed for the computation of the eigenvalues and eigenstates of one-particle Dirac Hamiltonians. They are based on exact characterizations of the eigenvalues by a direct minimization of the corresponding Rayleigh quotient on a set described by a nonlinear constraint.
Physical Review A, 2007
We study a mean-field relativistic model which is able to describe both the behavior of finitely ... more We study a mean-field relativistic model which is able to describe both the behavior of finitely many spin-1/2 particles like electrons and of the Dirac sea which is self-consistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the photon field. All our results are non-perturbative and mathematically rigorous.
Journal of Physics A: Mathematical and General, 2005
Journal of Functional Analysis, 2000
International Journal of Quantum Chemistry, 2003
Communications in Mathematical Physics, 2005
Calculus of Variations and Partial Differential Equations, 2000
In this paper we give two different variational characterizations for the eigenvalues of H + V wh... more In this paper we give two different variational characterizations for the eigenvalues of H + V where H denotes the free Dirac operator and V is a scalar potential. The first one is a min-max involving a Rayleigh quotient. The second one consists in minimizing an appropriate nonlinear functional. Both methods can be applied to potentials which have singularities as strong as the Coulomb potential.
Archive for Rational Mechanics and Analysis, 2013
Archive for Rational Mechanics and Analysis, 2009
This chapter is a review of some methods used for the computation of relativistic atomic and mole... more This chapter is a review of some methods used for the computation of relativistic atomic and molecular models based on the Dirac equation. In the linear case, we briefly describe finite basis set approaches, including ones that are generated numerically, perturbation theory and effective Hamiltonians procedures, direct variational methods based on nonlinear transformations, min-max formulations and constrained minimizations. In the atomic case, we describe the MCDF method and some ways to solve numerically the homogeneous and inhomogeneous Dirac-Fock equations. Finally, we describe also some numerical methods relevant to the case of molecules. Key words: Quantum chemistry, relativistic models for atoms and molecules, computational methods, no-pair Hamiltonian, relativistic many-body perturbation theory, relativistic random phase approximation, Dirac equation, continuous spectrum, eigenvalues, variational collapse, spurious states, finite basis, numerical basis sets, perturbation the...
Calculus of Variations and Partial Differential Equations, 1996
The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic fi... more The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.
Journal of Functional Analysis, 2000
This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the es... more This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential.
Proceedings of the London Mathematical Society
Stochastic Processes and their Applications
We consider the stochastic control problem of a financial trader that needs to unwind a large ass... more We consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form.
Journal of Functional Analysis
We prove local smoothing estimates for the massless 3D Dirac equation with a Coulomb potential. O... more We prove local smoothing estimates for the massless 3D Dirac equation with a Coulomb potential. Our strategy is inspired by [9] and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.
A non-homogeneous Hardy-like inequality has recently been found to be closely related to the know... more A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum. Hardy inequalities and Coulomb singularities The relationship between usual Hardy inequalities and spectra of elliptic operators is quite well known: the classical Hardy inequality in IR N − ∆ ≥ (N − 2) 2 4