Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation

Abstract

Abstract. This work is concerned with asymptotic approximations of the semi-classical Schrödinger equation in periodic media using Gaussian beams. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the Gaussian beam approximation and homogenization leads to the Bloch eigenvalue problem and associated evolution equations for Gaussian beam components in each Bloch band. We formulate a superposition of Blochband based Gaussian beams to generate high frequency approximate solutions to the original wave field. For initial data of a sum of finite number of band eigen-functions, we prove that the first-order Gaussian beam superposition converges to the original wave field at a rate of ǫ 1/2 , with ǫ the semiclassically scaled constant, as long as the initial data for Gaussian beam components in each band are prepared with same order of error or smaller. For a natural choice of initial approximation, a rate of ǫ 1/2 of initial error is verified. IntroductionWe consider the semiclassically scaled Schrödinger equation with a periodic potential:subject to the two-scale initial condition:where Ψ(t, x) is a complex wave function, ε is the re-scaled Planck constant, V e (x)-smooth external potential, S 0 (x)-real-valued smooth function, g(x, y) = g(x, y + 2π)-smooth function, compactly supported in x, i.e., g(x, y) = 0, x ∈ K 0 , K 0 -is a bounded set. V (y) is periodic with respect to the crystal lattice Γ = (2πZ) d , it models the electronic potential generated by the lattice of atoms in the crystal [14].A typical application arises in solid state physics where (1.1) describes the quantum dynamics of Bloch electrons moving in a crystalline lattice (generated by the ionic cores) [39]. The asymptotics of (1.1) as ε → 0+ is a well-studied two-scale problem in the physics and mathematics literature [8,16,20,41,23,13,33,1,14]. On the other hand, the computational challenge because of the small parameter ε has prompted a search for asymptotic model based numerical methods, see e.g., [29,37].1991 Mathematics Subject Classification. 35A21, 35A35, 35Q40.

General superpositions of Gaussian beams and propagation errors

“…In this paper we investigate issues related to the accuracy of Gaussian beam approximations to high frequency wave propagation. This is related to recent results on Gaussian beam methods in [5][6][7][8][9][10][11]13]. Our model equation is the acoustic wave equation…”

Sobolev and max norm error estimates for Gaussian beam superpositions

“…One of the reasons have been to give a rigorous foundation for the beam based numerical methods above. For the time-dependent case error estimates were first derived for the initial data [18,37], and later for the solution of scalar hyperbolic equations and the Schrödinger equation [23,24,26,41,22]. For the Helmholtz equation estimates have been given in [29,25].…”