Motion of Electrons in Adiabatically Perturbed Periodic Structures

Panati, Gianluca; Spohn, Herbert; Teufel, Stefan

doi:10.1007/3-540-35657-6_22

Cited by 14 publications

(5 citation statements)

References 29 publications

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“…Let us also remark that while we take the more explicit approach of using Bloch waves in a modified FGA ansatz for periodic media, as in (3.8). The same approximation can be also derived by first projecting the whole Schrödinger equation using a super-adiabatic projection as developed in [29,30] and then apply the frozen Gaussian approximation to the resulting dynamics. We will not go into the details in this work.…”

Section: Formulation and Main Resultsmentioning

confidence: 99%

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Delgadillo

Yang

2018

ASY

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Propagation of high-frequency wave in periodic media is a challenging problem due to the existence of multiscale characterized by short wavelength, small lattice constant and large physical domain size. Conventional computational methods lead to extremely expensive costs, especially in high dimensions. In this paper, based on Bloch decomposition and asymptotic analysis in the phase space, we derive the frozen Gaussian approximation for high-frequency wave propagation in periodic media and establish its converge to the true solution. The formulation leads to efficient numerical algorithms, which are presented in a companion paper [5].1 2 RICARDO DELGADILLO, JIANFENG LU, AND XU YANG avoid the boundary effects. Therefore the total number of grid points is huge, which usually leads to unaffordable computational cost, especially in high (d > 1) dimensions.An alternative efficient approach is to solve (1.1) asymptotically by the Bloch decomposition and modified WKB methods [3,4,7], which lead to eikonal and transport equations in the semi-classical regime. An advantage of this method is that the computational cost is independent of ε. However, the eikonal equation can develop singularities which make the method break down at caustics. The Gaussian beam method (GBM) [31] was then introduced by Popov to overcome this drawback at caustics. The idea is to allow the phase function to be complex and choose the imaginary part properly so that the solution has a Gaussian profile; see [15-20, 28, 37, 38] for recent developments. Similar ideas can be also found in the Hagedorn wave packet method [8,9]. Unlike the geometric optics based method, the Gaussian beam method allows for accurate computation of wave function around caustics [6,33]. But the problem is that the constructed beam must stay near the geometric rays to maintain accuracy. This becomes a drawback when the solution spreads [23,28,32].The Herman-Kluk propagator [11,21,22] was proposed for Schrödinger equation without the oscillatory periodic background potential. The method was rigorously analyzed in [35,36] and further extended as the frozen Gaussian approximation (FGA) for general high frequency wave propagation in [23][24][25]. The FGA method uses Gaussian functions with fixed widths, instead of using those that might spread over time, to approximate the wave solution. Despite its superficial similarity with the Gaussian beam method, it is different at a fundamental level. FGA is based on phase plane analysis, while GBM is based on the asymptotic solution to a wave equation with Gaussian initial data. In FGA, the solution to the wave equation is approximated by a superposition of Gaussian functions living in phase space, and each function is not necessarily an asymptotic solution, while GBM uses Gaussian functions (called beams) in physical space, with each individual beam being an asymptotic solution to the wave equation. The main advantage of FGA over GBM is that the problem of beam spreading no longer exists.In this paper, we extend FGA for computation of hi...

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Section: Formulation and Main Resultsmentioning

confidence: 99%

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Delgadillo

Yang

2018

ASY

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“…denotes the so-called Berry phase term (Carles et al 2004, Panati, Spohn andTeufel 2006), which is found to be purely imaginary β m (t, x) ∈ (iR) d . It is, importantly, related to the quantum Hall effect (Sundaram and Niu 1999).…”

Section: Two-scale Wkb Approximationmentioning

confidence: 99%

Mathematical and computational methods for semiclassical Schrödinger equations

Jin¹,

Markowich²,

Sparber³

2011

Acta Numerica

150

145

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We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime. * Colour online available at journals.cambridge.org/anu.

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“…Note that in (3.10e) A m = ∇ ξ Φ m , Φ m is the so-called Berry-connection [12,35]. Notice that we have gauge freedom in choosing Bloch functions, i.e.…”

Section: The Bloch Decomposition-based Gaussian Beam Methodsmentioning

confidence: 99%

“…To consider the Schrödinger equation in the quasi-momentum space, we introduce a new functionψ ε by taking the Bloch transformation [35,21] T of the wave function ψ ε as…”

Section: The Schrödinger Equation Under the Bloch Transformationmentioning

confidence: 99%

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A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space

Chai

Jin

Markowich

2018

CiCP

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We propose a hybrid method which combines the Bloch decomposition-based time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with periodic potentials in the case of band-crossings. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band-crossing a BDTS method is used to capture the inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. We also compare the Schrödinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number ε → 0, the Schrödinger equation converges to the Dirac equations when the external potential is zero or small, but for general external potentials there is an O(1) difference in between the solutions of the Schrödinger equation and its Dirac approximation.

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Motion of Electrons in Adiabatically Perturbed Periodic Structures

Cited by 14 publications

References 29 publications

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Mathematical and computational methods for semiclassical Schrödinger equations

A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space

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