Nodal count, magnetic potentials and Dirac cones: exploring the connections

量子粒子的可能状态可以用数学物理方程的解来描述，比如薛定谔方程。在许多情况下，粒子的能量是“量子化的”，即只能假设特殊的离散值。这些值取决于问题的参数，例如存在的电势和磁势。每个特殊值对应于一个解（一个特征函数），它描述了在空间中任何给定位置找到粒子的概率振幅。特别地，特征函数的零集是粒子完全避免的一组位置。在一些重要的应用中，例如晶体结构，能量不再是离散的，而是粒子运动方向的函数（称为色散关系）。该项目的目标是研究以下三个问题之间的可能联系：(a)磁场对薛定谔方程能级的影响，(b)本征函数零集的性质，(c)色散关系表面上圆锥形包涵体（“狄拉克点”）的存在性和稳定性。除了这三个都是数学物理理论研究的活跃领域之外，后一个问题对新材料的研究有直接的影响，比如石墨烯和碳纳米管，以及设计具有理想物理性质的新材料，这些新材料通常由狄拉克点的存在所控制。直到几年前，人们才知道(b)地区与(a)和(c)地区之间是否存在任何联系。预计这三个领域中的每一个都将受益于对其他两个领域的进一步了解。该项目侧重于流形和图（离散和量子）上薛定谔型算子的谱分析。该项目的具体目标如下：描述磁场的空间构型，该磁场减少了量子粒子的第n个本征态的能量，并将其与第n个本征函数的零集联系起来；研究流形上Aharonov-Bohm通量线的“最优”位置；研究在周期结构能带边缘计算的（广义）特征函数的性质；给出了类石墨烯结构中狄拉克点在减少对称性的扰动下保持的条件；设计了一种预测狄拉克点位置和能量带极值点位置的构造方法；研究色散曲面上的奇点数与在该曲面上的特殊点计算出的特征函数的零点数之间的联系。

The possible states of a quantum particle can be described by solutions to equations of mathematical physics, such as the Schroedinger equation. In many circumstances, the energy of a particle is "quantized", i.e. can assume only special discrete values. These values depend on the parameters of the problem, such as the electric and magnetic potentials present. Each special value corresponds to a solution (an eigenfunction), which describes the probability amplitude of finding the particle in any given location in space. In particular, the zero set of an eigenfunction is a set of positions that the particle avoids completely. In some important applications, such as crystal structures, the energy is no longer discrete, but is a function (called the dispersion relation) of the direction of the particle's motion. The goal of the project is to investigate possible connections among the following three questions: (a) effect of the magnetic field on the energy levels of Schroedinger equation, (b) properties of the zero sets of eigenfunctions, and (c) existence and stability of conical inclusions ("Dirac points") in the surface of the dispersion relation. In addition to all three being active areas of theoretical research in mathematical physics, the latter question has direct implications to the study of novel materials, such as graphenes and carbon nanotubes, and to design of new materials with desired physical properties, which are often governed by the presence of the Dirac points. The existence of any connection between area (b) on one hand and areas (a) and (c) on the other was not known until few years ago. Each of the three areas is expected to benefit from an improved understanding of the other two.The project focuses on spectral analysis of Schroedinger-type operators on manifolds and graphs (both discrete and quantum). Among the particular aims of the project are the following: describe the spatial configuration of the magnetic field that reduces the energy of the n-th eigenstate of a quantum particle and relate it to the zero set of the n-th eigenfunction; investigate the "optimal" placement of an Aharonov-Bohm flux-line on a manifold; investigate the properties of the (generalized) eigenfunctions calculated at the edges of the energy bands of periodic structures; formulate the conditions for the persistence of the Dirac points in graphene-like structures under perturbations reducing the symmetry; design a constructive method for predicting the location of Dirac points and of extremal points of energy bands; investigate the connection between the number of singularities in the dispersion surface and the number of zeros of eigenfunctions calculated at special points of the surface.