Blanc SIMI 1 - Sciences de l'information, de la matière et de l'ingénierie : Mathématiques et interactions

NOnlinear problems in Nuclear and Atomic Physics – NoNAP

The
essential mathematical tools are those from nonlinear analysis and partial differential equations, spectral theory and
numerical analysis. Relativistic effects based on the Dirac operator imply that either one has to deal with strongly
inde?nite functionals, leading to very hard problems of critical point theory, or one is led to consider systems of in?nitely
many particles, many of them being virtual (the one of the ‘Dirac sea’). The mathematical and physical description of
in?nite quantum systems is known to be very difficult. Some peculiarities of such systems were already stressed for the
BDF model in the works by Hainzl, Gravejat, Lewin, Séré and Solovej. Similar difficulties are expected to occur in the other
models that will be considered here. On the other hand, attractive models like the ones describing nuclear or stellar
matter are usually highly nonlinear. These nonlinearities need to be controlled if one wants to prove that the system is
stable and does not collapse. Also the occurence of Cooper pairing is a difficult effect to handle theoretically. We will
study the above models both from a theoretical and a numerical perspective. We hope to get some new insight on the
problem thanks to efficient and new numerical methods. Finally, we hope to develop some links with the physicists at CEA
who are specialists in making such computations.

The expected results are: publications of research articles in international scientific journals, participation of teh members of the project to international meetings, organization of meetings in France, training of young researchers (PhD and post-docs).

The project should improve the theoretical understanding of linear and nonlinear models arising in quantum physics and which serve as tools to describe matter at the microscopic and nanoscopic scales.

Several research articles have already been published. An updated list is maintained on the website of the project.

Submission summary

This project is concerned about the study, from a mathematical perspective, of linear and nonlinear models arising in quantum physics and which serve as tools to describe matter at the microscopic and nanoscopic scales. The project will focus on two main (non-independent) issues: the description of nuclear matter and the treatment of relativistic effects. The essential mathematical tools are those from nonlinear analysis and partial differential equations, spectral theory and numerical analysis.

Relativistic effects based on the Dirac operator imply that either one has to deal with strongly inde?nite functionals, leading to very hard problems of
critical point theory, or one is led to consider systems of in?nitely many particles, many of them being virtual (the one of the ‘Dirac sea’). The mathematical and physical description of in?nite quantum systems is known to be very difficult. Some peculiarities of such systems were already stressed for the BDF model in the works by Hainzl, Gravejat, Lewin, Séré and Solovej. Similar difficulties are expected to occur in the other models that will be considered here.

On the other hand, attractive models like the ones describing nuclear or stellar matter are usually highly nonlinear. These nonlinearities need to be
controlled if one wants to prove that the system is stable and does not collapse. Also the occurence of Cooper pairing is a difficult effect to handle
theoretically.

We will study the above models both from a theoretical and a numerical perspective. We hope to get some new insight on the problem thanks to efficient and new numerical methods. Finally, we hope to develop some links with the physicists at CEA who are specialists in making such computations.

Project coordination

Mathieu LEWIN (UNIVERSITE DE CERGY-PONTOISE) – mathieu.lewin@math.cnrs.fr

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

AGM UNIVERSITE DE CERGY-PONTOISE
CEREMADE UNIVERSITE PARIS IX [DAUPHINE]

Help of the ANR 200,000 euros
Beginning and duration of the scientific project: - 48 Months

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