The Semiclassical Limit and Geometric Quantization

AbstractAward：DMS-0805878首席研究员：Alejandro Uribe首席研究员将研究量子力学对象的半经典行为（即普朗克常数趋于零）与相空间中的几何对应物之间的关系。 更具体地说，要研究的问题包括：在各种半经典环境中薛定谔算子的正谱和逆谱问题，在物理和量子化学社区中常用的半经典近似的严格分析（玻尔-索末菲条件，近似传播子和相关函数的计算），以及辛几何结构的量子力学类比（如辛切割）。 背景将是经典的（当相空间是欧几里德）和更多的几何（例如当相空间是紧致的Kahler流形或构形空间的余切丛时）.数学和物理之间的关系对于这些学科的理论和应用都是非常重要的.一方面，数学是物理的语言，它的方法可以有强大的应用;在另一方面，物理问题经常会引发新的数学问题，这些问题可能会非常富有成效。 拟议中的研究将研究量子力学和几何学之间的界面问题。 这些问题通常分为两类：光谱问题，涉及量子系统的能量;动力学问题，比较量子系统和经典系统的演化。光谱问题试图理解量子系统的可能能级和相应的经典系统的几何之间的关系。 动力学问题应该导致更好地理解量子化学中一些流行的计算方法的基础。

AbstractAward: DMS-0805878Principal Investigator: Alejandro UribeThe principal investigator will study aspects of the relationshipbetween the semiclassical behavior (i.e. as Planck's constanttends to zero) of quantum-mechanical objects with their geometriccounterpart in phase space. More specifically, problems to beinvestigated include: Direct and inverse spectral problems forSchrodinger operators in various semi-classical settings,rigorous analyses of semiclassical approximations commonly usedin the physics and quantum chemistry communities (Bohr-Sommerfeldconditions, approximate propagators and computation ofcorrelation functions), as well as quantum-mechanical analogiesof constructions in symplectic geometry (like symplecticcutting). The settings will be both classical (when the phasespace is Euclidean) and more geometric (e.g. when the phase spaceis a compact Kahler manifold or the cotangent bundle of aconfiguration space).The relationship between mathematics and physics is profoundlyimportant for both the theory and applications of these subjects.In one direction, mathematics is the language of physics and itsmethods can have powerful applications; in the other, physicalproblems often give rise to new mathematical problems that can beextremely fruitful. The proposed research will study problems atthe interface between quantum mechanics and geometry. Theproblems are generally of two kinds: spectral, meaning related tothe energy of a quantum system, and dynamical, comparing theevolution of a quantum system with its classical counterpart.The spectral problems seek to understand the relationship betweenthe possible energy levels of a quantum system and the geometryof the corresponding classical system. The dynamic problemsshould result in a better understanding of the underpinnings ofsome popular computational methods in quantum chemistry.