Finite Moment Problems and Applications to Multiphase Computations in Geometric Optics

Gosse, Laurent; Runborg, Olof

doi:10.4310/cms.2005.v3.n3.a5

Cited by 10 publications

(27 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We still consider the potential (2) and we use (18) to investigate numerically its eigenvalues in case k = "New bands" might really be a quasi-periodic phenomenon which appearance stems only from the set of basic frequencies present in the potential V . In both cases given by (2) and (15), basic frequencies are ω = (1, k) at any fixed time t; "new bands" could develop better with |k| getting bigger, meaning that the (small) perturbation term oscillates at a frequency higher than the one of the lattice at rest.…”

Section: K Is Smaller Than Onementioning

confidence: 99%

“…Generally, semi-classical approximation restricts itself to considering only the propagation of charge carriers in valence and conduction bands: lower ones correspond to inner shells electrons strongly tied to the atomic cores and higher ones are usually empty except in special cases of intense excitation. It has been observed that the use of modern shock-capturing numerical techniques allow for quite satisfying computations of wave packets in increasingly complex environments including possibly self-interactions: see [14][15][16][17][18]. (right).…”

mentioning

confidence: 99%

See 1 more Smart Citation

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Gosse¹

2008

Nonlinear Analysis: Real World Applications

Self Cite

View full text Add to dashboard Cite

This paper investigates a simple one-dimensional model of incommensurate "harmonic crystal" in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional space. It is shown that both approaches lead to essentially the same results; that is, the lower spectrum is splitted between "Cantor-like zones" and "impurity bands" to which correspond critical and extended eigenstates, respectively. These "new bands" seem to emerge inside the band gaps of the unperturbed problem when certain conditions are met and display a parabolic nature. Numerical tests are extensively performed on both steady and time-dependent problems.Key words: Quasi-periodic potential, impurity band, quasi-Bloch wave, quantum chaos, spectral algorithm, deformed crystal, Charge-Density Wave (CDW). 1991 MSC: 81Q05, 81Q20 IntroductionThis article is a step further in the numerical study of electronic motion in a one-dimensional model of "harmonic crystal", that is to say, a modeling in solid-state physics going beyond the rigid periodic lattice picture. Indeed, it consists in assuming that, within the BornOppenheimer approximation (the motion of nucleons is much slower than the electron's ones and thus decouples adiabatically), ionic cores are linked together by springs and vibrate like classical coupled oscillators; consult the book [1] and the recent paper [14] to which we are about to borrow notation.In [14], the following rescaled problem describing the motion of a free electron inside a 1-D lattice of atoms which display single-mode vibrations has been investigated within the * Corresponding Author Email address: l.gosse@ba.iac.cnr.it (Laurent Gosse). Preprint submitted to Elsevier ScienceWKB semi-classical framework [5,19] for the following Schrödinger equation,and under the fundamental assumption that k and Ω(k) = | sin(kπ)| are rational and commensurate. This is perhaps the simplest non-trivial model available which describes the effects of phonons onto electronic conduction and is sometimes referred to as the "modulated crystal" [1,9,25,26,32]. Since the potential in (1) isn't of the form cos(x − 1 2 at 2 ), the unitary transform propsed in [35] and leading to Bloch oscillations doesn't apply here (see also [33] for an AC-type modulation). A particular feature pointed out in [39] and called lattice tracking which refers to the dragging effect of heavy nucleons onto the much lighter electron was easily observed in the numerical results.The commensurability hypothesis is of critical importance in this global picture in order to keep on applying the classic Bloch decomposition for (1); we assume the reader familiar with this theory and refer only to [1,5,15,19] for details. The first step in this line of thinking is the derivation of "energy bands", which are dispersion relations for electrons around certain levels of excitation. Solids are clas...

show abstract

Section: K Is Smaller Than Onementioning

confidence: 99%

mentioning

confidence: 99%

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Gosse¹

2008

Nonlinear Analysis: Real World Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…We close this section mentioning that the map R K m → u is called the "finite Markov moment problem"; this delicate inversion has been recently tackled in [13].…”

Section: General Proceduresmentioning

confidence: 99%

“…for which the multivalued (or geometric) solution is to be sought according to the moment system (13). As soon as one completes this program, the principal intensity |a 0 | 2 can be easily recovered; indeed, at any time t > 0, one deduces from (9) that…”

Section: Moment Systems and Intensity Recoverymentioning

confidence: 99%

See 1 more Smart Citation

Multiphase semiclassical approximation of the one-dimensional harmonic crystal: I. The periodic case

Gosse¹

2006

J. Phys. A: Math. Gen.

Self Cite

View full text Add to dashboard Cite

Kinetic Scheme with Reflections and Linear Geometric Optics

Gosse

2013

Computing Qualitatively Correct Approximations of Balance Laws

View full text Add to dashboard Cite

Finite Moment Problems and Applications to Multiphase Computations in Geometric Optics

Cited by 10 publications

References 34 publications

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Multiphase semiclassical approximation of the one-dimensional harmonic crystal: I. The periodic case

Kinetic Scheme with Reflections and Linear Geometric Optics

Contact Info

Product

Resources

About