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

Jin, Shi; Wu, Hao; Yang, Xu; Huang, Zhongyi

doi:10.1016/j.jcp.2010.01.025

Cited by 39 publications

(32 citation statements)

References 36 publications

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“…This goes back to the geophysical applications in [12,21]. Recent work in this direction includes [35,44,36,28,42,31,37].…”

Section: Introductionmentioning

confidence: 99%

“…In order to handle more general initial data in this paper, we (i) present the approximation solution through beam superpositions over Bloch bands and initial points from which beams are issued; and (ii) estimate the error between the exact wave field and the asymptotic ones. Numerical results using this type of superpositions were presented in [37].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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

Liu¹,

Pryporov²

2015

Spectral Theory and Partial Differential Equations

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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.

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“…This goes back to the geophysical applications in [12,21]. Recent work in this direction includes [35,44,36,28,42,31,37].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Liu¹,

Pryporov²

2015

Spectral Theory and Partial Differential Equations

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“…Specifically, one decomposes the initial wave function into localized wave packets (Gaussian beams) which are then evolved individually along particle trajectories and finally summed up to construct the solution at a later time. It was first studied rigorously in [25], and has seen many recent developments in both Eulerian and Lagrangian frameworks [13,14,15,17,20,21], error estimates [2,19], and fast Gaussian wave decompositions [1,24]. A related approach, known as the Hagedorn wave packet method, was studied in [8,6].…”

Section: Introductionmentioning

confidence: 99%

A Gaussian Beam Method for High Frequency Solution of Symmetric Hyperbolic Systems with Polarized Waves

Jefferis¹,

Jin²

2015

Multiscale Model. Simul.

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Symmetric hyperbolic systems include many physically relevant systems of partial differential equations (PDEs) such as Maxwell's equations, the elastic wave equations and the acoustic equations [26]. In this paper we extend the Gaussian beam method to efficiently compute the high frequency solutions to such systems with constant degeneracy that corresponds to polarized waves, in which the dispersion matrix of the hyperbolic system has eigenvalues with constant degeneracy over the domain of computation. The new results in this paper include new Gaussian beam equations in the presence of eigenvalue degeneracy, improved error estimates for Gaussian beam summation and a new multi-directional Eulerian summation formula which maintains accuracy after the formation of caustics.

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“…Most of the methods were in the Lagrangian framework. More recently, Eulerian Gaussian beam methods have also received special attention for their advantage of uniform accuracy [10,11,12,13,16,17]. A major simplification of the Eulerian Gaussian beam method is that the Hessian matrices can be constructed by taking derivatives of the level set functions [10].…”

Section: Introductionmentioning

confidence: 99%

Gaussian beam methods for the Dirac equation in the semi-classical regime

Huang

Jin

et al. 2012

Communications in Mathematical Sciences

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Abstract. The Dirac equation is an important model in relativistic quantum mechanics. In the semi-classical regime ε ≪ 1, even a spatially spectrally accurate time splitting method [6] requires the mesh size to be O(ε), which makes the direct simulation extremely expensive. In this paper, we present the Gaussian beam method for the Dirac equation. With the help of an eigenvalue decomposition, the Gaussian beams can be independently evolved along each eigenspace and summed to construct an approximate solution of the Dirac equation. Moreover, the proposed Eulerian Gaussian beam keeps the advantages of constructing the Hessian matrices by simply using the derivatives of level set functions. Finally, several numerical examples show the efficiency and accuracy of the method.

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

Cited by 39 publications

References 36 publications

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

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

A Gaussian Beam Method for High Frequency Solution of Symmetric Hyperbolic Systems with Polarized Waves

Gaussian beam methods for the Dirac equation in the semi-classical regime

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