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

现代物理学的数学语言在很大程度上是基于偏微分方程(PDE)的。这些方程描述了某些物理观察量，如能量、动量等相对于更基本的变量，如空间和时间的变化。基于偏微分方程的描述的一个特别重要的特征是，具有相当不同的科学背景的问题通常可以用数学上相似的方程来描述。因此，通过使用分析和/或数值技术研究PDE本身，可以获得关于潜在科学问题的许多见解。目前的项目研究的是在描述量子力学集体现象时出现的一类非线性偏微分方程组。量子力学是现代物理学的两个基本组成部分之一(另一个是爱因斯坦的广义相对论)，它涉及尺寸接近甚至小于原子尺度的机械系统的研究。Erwin Schrödinger，Paul Dirac，特别是John Von Neumann开创的数学公式在很大程度上是基于所谓的波函数的偏微分方程组，它以概率解释描述量子粒子的状态。在发现100多年后，量子理论已经达到了一个极其复杂的阶段，这并不令人惊讶。现代实验技术使设计、操纵和密切监控涉及数千个相互作用的粒子的量子力学系统成为可能。为了更好地了解这些高度复杂的系统，有必要清楚地区分这些系统中固有的慢自由度和快自由度。目前项目的主要目标是发展一种数学理论，使之能够通过几种不同的分析技术来实现这一点。其结果将是一个简单得多的数学描述，其中快速自由度仅通过它们的总累积效应被考虑在内。由此获得的有效模型方程可用于更详细的数学研究和使用计算能力的当前状态进行精确的数值模拟。所有这些方程都是描述各种量子力学现象的重要物理模型。因此，对提出的问题的回答将在广泛的理论物理中以及在偏微分方程的分析中产生强有力的影响。此外，国际和平研究所提出了与拟议研究有关的教育活动的综合计划：将为高级研究生和新近获得博士学位的学生举办一系列小型课程和讲习班。他们将把学生和来自不同数学领域的知名专家聚集在一起。此外，PI将为对应用分析和相关领域感兴趣的研究生和优秀本科生组织和支持每周一次的学生研讨会。最后，PI提出了新的研究生课程，涉及非线性色散方程、数学物理和渐近分析的交界处。标度分离在理解物理和其他自然科学中复杂系统的动力学行为方面发挥着基础性作用。通过识别慢自由度和快自由度，通常可以推导出某些慢变量的简单定律，这些定律反过来又用作对问题的全部多尺度动力学的近似描述。一个特别的例子是经典的量子力学绝热定理，可以追溯到M.Born和V.Fock在1928年，它可以松散地表述如下：一个受到缓慢变化的外部条件的量子力学系统保持其基本形式(至少是近似的)。对于由线性薛定谔方程描述的量子系统，有大量的数学结果赋予了这一基本概念严格的意义。本项目致力于发展非线性薛定谔偏微分方程组(及相关模型)的严格绝热理论。这项研究的智力相关性源于非线性薛定谔方程作为复杂多体量子系统平均场模型的重要性。可以在玻色-爱因斯坦凝聚体、量子波导层和半导体石墨烯层的数学描述中找到应用。所有这些应用都处于现代实验和理论物理学的前沿。将要发展的数学理论将提供对物理上相关的多尺度区域中各个模型方程的解的定性描述，并与其他领域建立深刻的联系，如半经典分析、齐次化理论，以及流形上的色散方程的研究。

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.