Nonlinear Schrödinger and Quantum-Boltzmann equations

This project in the field of modern mathematical physics is aimed at the application of advanced theory of partial differential equations to fundamental problems of quantum transport theory. It deals with the mathematical analysis and the derivation of new models (mainly) for the description of electrons in situations where Newtonian mechanics is no longer valid because of quantum effects or relativistic effects. Modern technology of semiconductor devices, fiber optics etc. increasingly involves equations of this kind. The recent developments of quantum kinetic theory, in particular Wigner transforms, where the proposer is an internationally recognized expert, make it possible to investigate quantum, relativistic and classical transport in a unified kinetic framework. Although, 100 years ago, Vienna was the cradle of kinetic theory with the revolutionary ideas of Boltzmann, this field has been largely neglected in Austria since world war II. This project resumes and extends this tradition to quantum kinetic theory. The 4 research topics of this proposal are 1.) Nonlinear Schrödinger equations, 2.) Quantum Boltzmann equations, 3.) The Dirac - Maxwell system, 4.) The Thomas-Fermi-von Weizsäcker-Dirac model. Wigner transforms give a structural link between the Schrödinger equation and the Liouville and Boltzmann equation. They are an appropriate tool for the derivation of Quantum- Boltzmann equations from (many body) Schrödinger equations. They are also a perfect method for (semi)classical limits of quantum mechanics: Asymptotic analysis is used to study the connection between model equations based on scaling limits, that is when certain problem parameters go to zero (e.g. scaled Planck constant) or go to infinity (e.g. scaled velocity of light, number of particles,...). It helps also deriving correction terms to the simpler limiting models. The classical and nonrelativistic limits from the nonlinear Dirac equation to Schrödinger equations and Vlasov equations and the high density limits from the N-particle Schrödinger equation (Hartree-Fock model) to local approximations useful in solid state physics and quantum chemistry complete our program. In an interdisciplinary approach both physical modelling, mathematical analysis and computer simulations will be intertwined. Influence of the proposed work on the development of the field This project boosts the interest in the field of mathematical physics to kinetic theory and will create a strong group in an important field. It will recast the use of Wigner functions for (asymptotic) analysis of nonlinear Schrödinger and Dirac equations appearing in quantum physics and nonlinear optics and also yield improved approximations for models used in quantum chemistry. The proposed derivations of Quantum-Boltzmann equations represent the first rigorous approach in a field where more or less heuristic "ansätze" prevail which allows both for an evaluation of existing models and the derivation of new ones. A successful application of Wigner transforms to general nonlinear Schrödinger equations would be a revolutionary progress. It would go far beyond the intrinsic limitations (e.g. one-dimensional only) of the currently used integrable systems (inverse scattering) approach. The problems chosen for this project have both theoretical and practical importance. This kind of applied mathematics is the indispensable basis for new developments in fields like quantum semiconductors, fiber optics, high energy particle physics and many more.

This project in the field of modern mathematical physics is aimed at the application of advanced theory of partial differential equations to fundamental problems of quantum transport theory. It deals with the mathematical analysis and the derivation of new models (mainly) for the description of electrons in situations where Newtonian mechanics is no longer valid because of quantum effects or relativistic effects. Modern technology of semiconductor devices, fiber optics etc. increasingly involves equations of this kind. The recent developments of quantum kinetic theory, in particular Wigner transforms, where the proposer is an internationally recognized expert, make it possible to investigate quantum, relativistic and classical transport in a unified kinetic framework. Although, 100 years ago, Vienna was the cradle of kinetic theory with the revolutionary ideas of Boltzmann, this field has been largely neglected in Austria since world war II. This project resumes and extends this tradition to quantum kinetic theory. The 4 research topics of this proposal are: 1. Nonlinear Schrödinger equations, 2. Quantum Boltzmann equations, 3. The Dirac - Maxwell system, 4. The Thomas-Fermi-von Weizsäcker-Dirac model. Wigner transforms give a structural link between the Schrödinger equation and the Liouville and Boltzmann equation. They are an appropriate tool for the derivation of Quantum- Boltzmann equations from (many body) Schrödinger equations. They are also a perfect method for (semi)classical limits of quantum mechanics: Asymptotic analysis is used to study the connection between model equations based on scaling limits, that is when certain problem parameters go to zero (e.g. scaled Planck constant) or go to infinity (e.g. scaled velocity of light, number of particles,...). It helps also deriving correction terms to the simpler limiting models. The classical and nonrelativistic limits from the nonlinear Dirac equation to Schrödinger equations and Vlasov equations and the high density limits from the N-particle Schrödinger equation (Hartree-Fock model) to local approximations useful in solid state physics and quantum chemistry complete our program. In an interdisciplinary approach both physical modelling, mathematical analysis and computer simulations will be intertwined. Influence of the proposed work on the development of the field This project boosts the interest in the field of mathematical physics to kinetic theory and will create a strong group in an important field. It will recast the use of Wigner functions for (asymptotic) analysis of nonlinear Schrödinger and Dirac equations appearing in quantum physics and nonlinear optics and also yield improved approximations for models used in quantum chemistry. The proposed derivations of Quantum-Boltzmann equations represent the first rigorous approach in a field where more or less heuristic "ansätze" prevail which allows both for an evaluation of existing models and the derivation of new ones. A successful application of Wigner transforms to general nonlinear Schrödinger equations would be a revolutionary progress. It would go far beyond the intrinsic limitations (e.g. one- dimensional only) of the currently used integrable systems (inverse scattering) approach. The problems chosen for this project have both theoretical and practical importance. This kind of applied mathematics is the indispensable basis for new developments in fields like quantum semiconductors, fiber optics, high energy particle physics and many more.