High-Frequency Dynamics for the Schrödinger Equation, with Applications to Dispersion and Observability

Macià, Fabricio

doi:10.1007/978-3-319-19015-0_4

Cited by 6 publications

(2 citation statements)

References 105 publications

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“…Here the situation is more complicated, as interactions between projections on different Bloch bands may occur. This regime involving performing simultaneously the semi-classical and long time limit has been useful in other contexts, we refer the reader to the survey articles [5,43,44]. In Section 3.1 below, we recall elements of the theory of semi-classical measures that we apply to (ψ ε ) ε>0 in the next two sections.…”

Section: Semi-classical Approach To the Energy Dynamicsmentioning

confidence: 99%

Effective Mass Theorems with Bloch Modes Crossings

Chabu

Kammerer²,

Macià

2022

Arch Rational Mech Anal

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We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.

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Section: Semi-classical Approach To the Energy Dynamicsmentioning

confidence: 99%

Effective Mass Theorems with Bloch Modes Crossings

Chabu

Kammerer²,

Macià

2022

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…Here the situation is more complicated, as interactions between projections on different Bloch bands may occur. This regime involving performing simultanously the semi-classical and long time limit has been useful in other contexts, we refer the reader to the survey articles [5,42,43].…”

Section: Semi-classical Approach To the Energy Dynamicsmentioning

confidence: 99%

Effective mass theorems with Bloch modes crossings

Chabu¹,

Kammerer²,

Macià³

2020

Preprint

Self Cite

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We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.

show abstract

Geometric Control of Eigenfunctions of Schrödinger Operators

Macià

2022

Research in PDEs and Related Fields

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High-Frequency Dynamics for the Schrödinger Equation, with Applications to Dispersion and Observability

Cited by 6 publications

References 105 publications

Effective Mass Theorems with Bloch Modes Crossings

Effective Mass Theorems with Bloch Modes Crossings

Effective mass theorems with Bloch modes crossings

Geometric Control of Eigenfunctions of Schrödinger Operators

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