Bloch Eigenvalue Research Articles

Homogenization of the Schrödinger Equation and Effective Mass Theorems

We study the homogenization of a Schrodinger equation with a large periodic potential: denoting by ∈ the period, the potential is scaled as ∈−2. We obtain a rigorous derivation of so-called effective mass theorems in solid state physics. More precisely, for well-prepared initial data concentrating on a Bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating Bloch eigenfunction and of a slowly varying solution of an homogenized Schrodinger equation. The homogenized coefficients depend on the chosen Bloch eigenvalue, and the homogenized solution may experience a large drift. The homogenized limit may be a system of equations having dimension equal to the multiplicity of the Bloch eigenvalue. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.

Spectral Properties of the Periodic Magnetic Schr�dinger Operator in the High-Energy Region. Two-Dimensional Case

The goal is to investigate spectral properties of the operator H=(−i∇ +a(x))2+a0(x) in the two-dimensional situation, a(x), a0(x)) being periodic. We construct asymptotic formulae for Bloch eigenvalues and eigenfunctions in the high-energy region, describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the Bethe-Sommerfeld conjecture.

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)l +V inRn, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.

Scattering theory for the perturbations of periodic Schrödinger operators

In this article, we study the short- and long-range perturbations of periodic Schrödinger operators. The asymptotic completeness is proved in the short-range case by referring to known results on the stationary approach and more explicitly with the time-dependent approach. In the long-range case, one is able to construct modified wave operators. In both cases, the asymptotic observables can be defined as elements of a commutative $C^{*}$-algebra of which the spectrum equals or is contained in the Bloch variety. Especially, the expression of the mean velocity as the gradient of the Bloch eigenvalues is completely justified in this framework, even when the Bloch variety presents singularities.

PERTURBATION THEORY FORMULAS FOR THE SCHRÖDINGER EQUATION WITH A NONSMOOTH PERIODIC POTENTIAL

Series from perturbation theory are constructed for the Bloch eigenvalues and eigenfunctions for the periodic Schr?dinger operator in . An extensive set of quasimomenta on which the series converge is described. It is shown that the series have asymptotic character at high energies. They are infinitely differentiable with respect to the quasimomentum, and preserve their asymptotic character under such differentiation.

Perturbation theory for a polyharmonic operator with nonsmooth periodic potential

The operator , where V is a real periodic potential, is considered in L2(Rn), n>1. A method is described for constructing convergent series of perturbation theory, in powers of α, for the Bloch eigenfunctions and eigenvalues on a rich set of quasimomenta. These series are asymptotic at high energies.