Project description
Improved approaches to approximating the behaviour of quantum many-body systems
The physical properties of quantum many-body systems are usually described using Schrödinger equations. However, it is generally impossible to solve these equations with current numerical techniques. Therefore, physicists often use approximation theories in practice, which concentrate on just a few collective behaviours of the described systems. Using mathematical analyses, they confirm whether the chosen models effectively describe the behaviour of the systems. The overall goal of the EU-funded RAMBAS project is to justify certain key effective approximations used in many-body quantum physics. Leveraging new techniques from functional analysis, spectral theory, calculus of variations and partial differential equations, RAMBAS expects to raise standard approximations of quantum systems to the next level, providing physicists with new mathematical tools.
Objective
From first principles of quantum mechanics, physical properties of many-body quantum systems are usually encoded into Schroedinger equations. However, since the complexity of the Schroedinger equations grows so fast with the number of particles, it is generally impossible to solve them by current numerical techniques.
Therefore, in practice approximate theories are often applied, which focus only on some collective behaviors of the systems in question.
The corroboration of such effective models largely depends on mathematical methods. The overall goal of RAMBAS is to justify key effective approximations used in many-body quantum physics, including the mean-field, quasi-free, and random-phase approximations, as well as to derive subtle corrections in critical regimes.
Building on my unique expertise in mathematical physics, I will 1) develop general techniques to understand corrections to the mean-field and Bogoliubov approximations for dilute Bose gases, 2) introduce rigorous bosonization methods and combine them with existing techniques from the theory of Bose gases to understand Fermi gases, and 3) employ the bosonization structure of Fermi gases to study the many-body quantum dynamics in long time scales, thus deriving quantum kinetic equations.
By applying and suitably inventing mathematical techniques from functional analysis, spectral theory, calculus of variations and partial differential equations, RAMBAS will take standard approximations of quantum systems to the next level, with special focus on those particularly challenging situations where the particle correlation plays a central role but is yet not adequately addressed. RAMBAS will thereby provide the physics community with crucial mathematical tools, which are at the same time rigorous and applicable.
Fields of science
- natural sciencesphysical sciencesquantum physics
- natural sciencesmathematicsapplied mathematicsmathematical physics
- natural sciencesmathematicspure mathematicsmathematical analysisfunctional analysis
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
Keywords
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
80539 MUNCHEN
Germany