Slide 1

A Bloch Band Base Level Set Method in the Semi-classical Limit of the Schrödinger Equation

Hailiang Liu1 and Zhongming Wang2

Abstract

It is now known that one can use level setdescription to accurately capture multi-phases incomputation of high frequency waves. In thispaper, we develop a Bloch band based level setmethod for computing the semi-classical limit ofSchrödinger equations in periodic media. For theunderlying equation subject to a highly oscillatoryinitial data a hybrid of the WKB approximationand homogenization leads to the Bloch eigenvalueproblem and an associated Hamilton-Jacobisystem for the phase, with Hamiltonian being theBloch eigenvalues. We evolve a level setdescription to capture multi-valued solutions tothe band WKB system, and then evaluate totalposition density over a sample set of bands. Asuperposition of band densities is established overall bands and solution branches when away fromcaustic points. Numerical results with differentnumber of bands are provided to demonstrate thegood quality of the method.

Problem and Applications

We consider a general one dimensional linear Schrödingerequation

This type of Schrödinger equations is a fundamental modelin solid-state physics and also models the quantumdynamics of Bloch electrons subject to an external field.

In the semi-classical regime, where є tends to zero, directnumerical simulation can be prohibitively costly. Therefore,asymptotic models need to be considered.

Bloch Structure

Due to b(x/є) and V(x/є), homogenization is neededbesides WKB expansion, i.e.,

Using previous ansatz and balancing leading terms,we have

where En is the nth eigenvalue of the Bloch eigen-problem

with

being a differential operator parameterized by k.

In general, the resulting Hamilton-Jacobi systemdevelops singularities in finite time. After singularityformation, multi-valued solutions are the physicallyrelevant ones to capture correct physical phenomena.

Density

The wave solution over all band is asuperposition of wave fields on each band,

We have shown that the averaged density on nthband, , is a superposition of all multi-valueddensities, i.e.,

We have also shown that the averaged densityover all bands converges weakly, i.e.,

Here ρє is defined as

Numerical Issues

1. Initialization of ρn

where f is given in the initial wave Ψ. Numericalexperience shows that only 8 or 10 bands are sufficient.

2. Bloch eigenvalue problem

By Fourier transform of the differential operator H(k,y),the eigenvalue problem becomes

Where H is a Hermitian matrix and

Department of Mathematics

Level set method Formulation

Let Φ be a function in phase space whose zero levelset gives un, then

The multi-valued density ρn is computed by

where f solves

Note that we need to solve one level set equation andone conservative equation for each band. We willshow numerically that only a few bands are needed incomputation, which makes our method feasible.

Numerical Example 1