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

量子粒子的可能状态可以通过数学物理方程的解来描述，例如薛定谔方程。 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（本征函数），它描述了在空间中任何给定位置找到粒子的概率幅度。 特别地，本征函数的零点集是粒子完全避免的位置的集合。 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）。 该项目的目标是调查以下三个问题之间可能存在的联系：（a）磁场对薛定谔方程能级的影响，（B）本征函数零点集的性质，（c）色散关系曲面中圆锥形内含物（“狄拉克点”）的存在和稳定性。除了这三个领域都是数学物理理论研究的活跃领域外，后一个问题对新材料的研究有直接的影响，如石墨烯和碳纳米管，以及设计具有所需物理性质的新材料，这些物理性质通常由狄拉克点的存在所决定。 直到几年前，人们才知道（B）区与（a）和（c）区之间是否存在任何联系。这三个领域中的每一个都将受益于对其他两个领域的更好理解。该项目的重点是流形和图（离散和量子）上Schroedinger型算子的谱分析。 该项目的具体目标包括：描述减少量子粒子第n个本征态能量的磁场的空间结构，并将其与第n个本征函数的零点集联系起来;研究Aharonov-Bohm通量线在流形上的“最佳”位置;研究的性质在周期结构的能带边缘计算的（广义）本征函数;提出了在对称性降低的扰动下，类石墨烯结构中Dirac点持续存在的条件，设计了一种预测Dirac点和能带极值点位置的构造性方法;研究色散曲面中奇点的数目与在曲面特殊点处计算的本征函数零点数目之间的关系。

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.