Nodal Count, Magnetic Potentials and Dirac Cones: Exploring The Connections-
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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.