We study the inverse scattering for Schrodinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
In physical literatures, the 2-dimensional Dirac operator is usually adopted as a mathematical model for the graphene (see e.g. @cite_18 , @cite_48 , @cite_10 ). Therefore, our discrete Laplacian on the hexagonal lattice is regarded as a discretization of this Dirac operator. For the mathematical model of carbon nano-tube, see @cite_44 , @cite_32 . An experimental result for the defects in graphite is seen in @cite_55 .
@cite_18 @cite_48 @cite_55 @cite_32 @cite_44 @cite_10
2061935292 2087402748 2105685140 1980428773
We show that in graphene, modelled by the two-dimensional Dirac operator, charge distributions with non-vanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power. The electronic spectrum of sheets of graphite (plane honeycomb lattice) folded into regular polyhedra is studied. A continuum limit valid for sufficiently large molecules and based on the tight-binding approximation is derived. It is found that a Dirac equation describes the flat graphite lattice. Curving the lattice by insertion of odd-numbered rings can be mimicked by coupling effective gauge fields. In particular the C60 and related molecules are well described by the Dirac equation on the surface of a sphere coupled to a color monopole sitting at its center. This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented. When an algorithm in dimension one is extended to dimension d, in nearly every case its computational cost is taken to the power d. This fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality. For numerical analysis in dimension d, we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy. Basic linear algebra operations can be performed in this representation using one-dimensional operations, thus bypassing the exponential scaling with respect to the dimension. Although not all operators and algorithms may be compatible with this representation, we believe that many of the most important ones are. We prove that the multiparticle Schrodinger operator, as well as the inverse Laplacian, can be represented very efficiently in this form. We give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground-state eigenfunction for a simplified Schrodinger operator with 30 particles. We conjecture and provide numerical evidence that functions of operators inherit this property, in which case numerical operator calculus in higher dimensions becomes feasible.