No abstract

No abstract

No abstract

No abstract

We prove that the two-dimensional Schrödinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.for some complex constant λ 7 ‰ 0. A consequence is that for wave-packet initial conditions with spectral components which are concentrated near these vertices, the effective evolution equation governing the wave-packet envelope is the two-dimensional Dirac wave equation, the equation of evolution for massless relativistic fermions [14,1]. Hence, these special vertex quasi-momenta associated with the hexagonal lattice are often called Dirac points. In contrast, wave-packets concentrated at spectral band edges, bordering a spectral gap where the dispersion relation is typically quadratic, behave as massive non-relativistic particles; the effective wave-packet envelope equation

No abstract

ABSTRACT. If a function ip(x) is mostly concentrated in a box Q, while its Fourier transform $>(£) is concentrated mostly in Q', then we say ijj is microlocalized in Q X Q' in (x, £)-space. The uncertainty principle says that Q X Q' must have volume at least 1. We will explain what it means for ip to be microlocalized to more complicated regions S of volume ~ 1 in (x, £)-space. To a differential operator P(x, D) is associated a covering of (x, £)-space by regions {B a } of bounded volume, and a decomposition of L 2 -functions u as a sum of "components" it a microlocalized to B a -This decomposition u -» (uot) diagonalizes P(x,D) modulo small errors, and so can be used to study variable-coefficient differential operators, as the Fourier transform is used for constant-coefficient equations. We apply these ideas to existence and smoothness of solutions of PDE, construction of explicit fundamental solutions, and eigenvalues of Schrodinger operators. The theorems are joint work with D. H. Phong. CHAPTER I: THE SAK PRINCIPLEThe uncertainty principle says that a function if), mostly concentrated in \x -Xo\ < 6 X , cannot also have its Fourier transform «0 mostly concentrated in l£~~ £o| < % unless 8 X -8^ > 1. This simple fact has far-reaching consequences for PDE, but until recently it was used only in a very crude form. The most significant classical application concerned the eigenvalues of a self adjoint differential operator

No abstract