CAREER: Adiabatic theory for nonlinear Schrodinger equations with applications in complex quantum systems

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The mathematical language of modern physics is to a large extent based on partial differential equations (PDEs). These equations describe the variations of certain physical observables, such as energy, momentum, etc. with respect to more basic variables like space and time. A particularly important feature of PDE-based descriptions is that problems with rather diverse scientific backgrounds can often be described by mathematically similar equations. Many insights about the underlying scientific problem can thus be gained by studying the PDEs themselves using analytical and/or numerical techniques. The current project studies a certain class of nonlinear partial differential equations that arise in the description of collective phenomena in quantum mechanics. Quantum mechanics is one of the two fundamental building blocks in modern physics (the other one being Einstein's theory of general relativity) and it concerns the study of mechanical systems whose dimensions are close to, or even smaller than, the atomic scale. The mathematical formulation pioneered by Erwin Schrödinger, Paul Dirac, and especially John von Neumann is to a large extent based on PDEs for so-called wave functions, which describe the state of the quantum particles in a probabilistic interpretation. It is not surprising that more than 100 years after its discovery, quantum theory has reached a stage of enormous complexity. Modern experimental techniques have made it possible to design, manipulate, and closely monitor quantum mechanical systems involving thousands of particles interacting with each other. In order to obtain a better understanding of these highly complicated systems, it is necessary to clearly separate the slow and fast degrees of freedom intrinsically present within such systems. The main objective of the current project is to develop a mathematical theory which allows to do so by means of several different analytical techniques. Its outcome will be a much simpler mathematical description in which the fast degrees of freedom have been taken into account through their total cummulative effects only. The effective model equation thereby obtained becomes accessible for a more detailed mathematical study and for accurate numerical simulations using the current state of computing power. All of the equations studied arise as important physical models in the description of various quantum mechanical phenomena. Answers to the questions raised will therefore have a strong impact as proven statements in a wide range of theoretical physics, as well as in the analysis of PDEs. In addition, the PI proposes an integrated plan of educational activities related to the proposed research: A series of mini-courses and workshops for advanced graduate students and recent PhDs will be held. They will bring together students with renowned specialists from different fields of mathematics. Furthermore, the PI will organize and support a weekly student seminar for graduate and strong undergraduate students interested in applied analysis and related fields. Finally, the PI has proposed new graduate courses on topics at the juncture of nonlinear dispersive equations, mathematical physics and asymptotic analysis.Separation of scales plays a fundamental role in the understanding of the dynamical behavior of complex systems in physics and other natural sciences. By identifying the slow and fast degrees of freedom, it is often possible to derive simple laws for certain slow variables, which in turn serve as an approximate description of the full multi-scale dynamics of the problem. A particular example is the classical adiabatic theorem of quantum mechanics, going back to M. Born and V. Fock in 1928, which can be loosely stated as follows: A quantum mechanical system subjected to slowly changing external conditions retains its basic form (at least approximately). For quantum systems described by linear Schrödinger equations there is a wealth of mathematical results which give a rigorous meaning to this basic idea. The current project is devoted to the development of a rigorous adiabatic theory for partial differential equations of nonlinear Schrödinger type (and related models). The intellectual relevance of the proposed research stems from the importance of nonlinear Schrödinger equations as mean-field models of complex many-body quantum systems. Applications can be found in the mathematical description of Bose-Einstein condensates, quantum wave guides, and semiconductor graphene layers. All of these applications are at the forefront of modern experimental and theoretical physics. The mathematical theory to be developed will provide a qualitative description of the solutions to the respective model equation in physically relevant multi-scale regimes, and in addition establish deep connections to other areas, such as semiclassical analysis, homogenization theory, and to the study of dispersive equations on manifolds.