The numerical spectrum of a one-dimensional Schrödinger operator with two competing periodic potentials

Abstract

Abstract. We are concerned with the numerical study of a simple one-dimensional Schrödinger operator − 1 2 ∂xx + αq(x) with α ∈ R, q(x) = cos(x) + εcos(kx), ε > 0 and k being irrational. This governs the quantum wave function of an independent electron within a crystalline lattice perturbed by some impurities whose dissemination induces long-range order only, which is rendered by means of the quasi-periodic potential q. We study numerically what happens for various values of k and ε; it turns out that for k > 1 and ε 1, that is to say, in case more than one impurity shows up inside an elementary cell of the original lattice, "impurity bands" appear and seem to be k-periodic. When ε grows bigger than one, the opposite case occurs.

Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients II

“…In [20], a numerical study of the one-dimensional Schrödinger operator with the potential q(x) = cos(x) + ε cos(kx) is considered, where ε > 0 and k are irrational. This governs the quantum wave function of an independent electron within a crystalline lattice perturbed by some impurities whose dissemination induces long-range order only, which is rendered by means of the quasi-periodic potential q.…”

Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials

“…For the computations of multivalued solutions to this problem see also [10][11][12][13]. The problems with all these semiclassical methods is that q m defined in (2.13) blows up at caustics since @ n / m ¼ 0.…”

On the Bloch decomposition based spectral method for wave propagation in periodic media

“…Similarly, Kevorkian and Bosley [16] considered hyperbolic conservation laws with rapidly varying, spatially periodic fluctuations by multiple asymptotic analysis. Numerical approaches on the related problem of linear Schrödinger equation with a periodic potential were studied in [11,12]. However, by passing to an effective (homogenized) equation, one usually looses all details of the underlying microscopic dynamics.…”