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

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 set description to accurately capture multi-phases in computation of high frequency waves. In this paper, we develop a Bloch band based level set method for computing the semi-classical limit of Schrödinger equations in periodic media. For the underlying equation subject to a highly oscillatory initial data a hybrid of the WKB approximation and homogenization leads to the Bloch eigenvalue problem and an associated Hamilton-Jacobi system for the phase, with Hamiltonian being the Bloch eigenvalues. We evolve a level set description to capture multi-valued solutions to the band WKB system, and then evaluate total position density over a sample set of bands. A superposition of band densities is established over all bands and solution branches when away from caustic points. Numerical results with different number of bands are provided to demonstrate the good quality of the method. Bloch Structure Due to b(x/є) and V(x/є), homogenization is needed besides 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 system develops singularities in finite time. After singularity formation, multi-valued solutions are the physically relevant ones to capture correct physical phenomena. Level set method Formulation Let Φ be a function in phase space whose zero level set gives un, then The multi-valued density ρn is computed by where f solves Note that we need to solve one level set equation and one conservative equation for each band. We will show numerically that only a few bands are needed in computation, which makes our method feasible. Problem and Applications We consider a general one dimensional linear Schrödinger equation This type of Schrödinger equations is a fundamental model in solid-state physics and also models the quantum dynamics of Bloch electrons subject to an external field. In the semi-classical regime, where є tends to zero, direct numerical simulation can be prohibitively costly. Therefore, asymptotic models need to be considered. Density The wave solution over all band is a superposition of wave fields on each band, We have shown that the averaged density on nth band, , is a superposition of all multi-valued densities, i.e., We have also shown that the averaged density over 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. Numerical experience 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 Numerical Example 1 Numerical Example 2 Future work: Remove the unboundness of the density at caustics Reference: H.-L. Liu and Z.-M. Wang, A Bloch band based level set method in the semi-classical limit of the Schrödinger equation, submitted • Department of Mathematics, Iowa State University • Presenter, Department of Mathematics, University of California, San Diego Department of Mathematics