Rigorous Approximations for Many-Body Quantum Systems

Introduction

From first principles of quantum mechanics, physical properties of many-body quantum systems are usually encoded into Schrödinger equations. However, since the complexity of the Schrödinger equations grows so fast with the number of particles, it is generally impossible to solve them by current numerical techniques.

Approximate Theories

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.

Project Goals

The overall goal of RAMBAS is to justify key effective approximations used in many-body quantum physics, including:

1. Mean-field approximation
2. Quasi-free approximation
3. Random-phase approximation

Additionally, RAMBAS aims to derive subtle corrections in critical regimes.

Methodology

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.
3. Employ the bosonization structure of Fermi gases to study the many-body quantum dynamics in long time scales, thus deriving quantum kinetic equations.

Mathematical Techniques

By applying and suitably inventing mathematical techniques from:

• Functional analysis
• Spectral theory
• Calculus of variations
• Partial differential equations

RAMBAS will take standard approximations of quantum systems to the next level, with special focus on those particularly challenging situations where particle correlation plays a central role but is yet not adequately addressed.

Contribution to the Physics Community

RAMBAS will thereby provide the physics community with crucial mathematical tools, which are at the same time rigorous and applicable.