scholarly journals Parseval Frames of Exponentially Localized Magnetic Wannier Functions

2019 ◽  
Vol 371 (3) ◽  
pp. 1179-1230 ◽  
Author(s):  
Horia D. Cornean ◽  
Domenico Monaco ◽  
Massimo Moscolari
Keyword(s):  
Wannier Functions ◽  
Parseval Frames

scholarly journals Discrete Nonlinear Schrödinger Systems for Periodic Media with Nonlocal Nonlinearity: The Case of Nematic Liquid Crystals

2021 ◽  
Vol 11 (10) ◽  
pp. 4420
Author(s):  
Panayotis Panayotaros
Keyword(s):  
Differential Equation ◽  
Infinite System ◽  
Linear Part ◽  
Optical Power ◽  
Explicit Construction ◽  
Translation Invariance ◽  
Nonlocal Nonlinearity ◽  
Wannier Functions ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger Systems

We study properties of an infinite system of discrete nonlinear Schrödinger equations that is equivalent to a coupled Schrödinger-elliptic differential equation with periodic coefficients. The differential equation was derived as a model for laser beam propagation in optical waveguide arrays in a nematic liquid crystal substrate and can be relevant to related systems with nonlocal nonlinearities. The infinite system is obtained by expanding the relevant physical quantities in a Wannier function basis associated to a periodic Schrödinger operator appearing in the problem. We show that the model can describe stable beams, and we estimate the optical power at different length scales. The main result of the paper is the Hamiltonian structure of the infinite system, assuming that the Wannier functions are real. We also give an explicit construction of real Wannier functions, and examine translation invariance properties of the linear part of the system in the Wannier basis.

scholarly journals Linear scaling calculation of maximally localized Wannier functions with atomic basis set

2006 ◽  
Vol 124 (23) ◽  
pp. 234108 ◽  
Author(s):  
H. J. Xiang ◽  
Zhenyu Li ◽  
W. Z. Liang ◽  
Jinlong Yang ◽  
J. G. Hou ◽  
...  
Keyword(s):  
Basis Set ◽  
Linear Scaling ◽  
Wannier Functions

scholarly journals Construction and solution of a Wannier-functions based Hamiltonian in the pseudopotential plane-wave framework for strongly correlated materials

2008 ◽  
Vol 65 (1) ◽  
pp. 91-98 ◽  
Author(s):  
Dm. Korotin ◽  
A. V. Kozhevnikov ◽  
S. L. Skornyakov ◽  
I. Leonov ◽  
N. Binggeli ◽  
...  
Keyword(s):  
Plane Wave ◽  
Strongly Correlated ◽  
Wannier Functions ◽  
Strongly Correlated Materials

Spontaneous polarization and piezoelectric properties of AlN nanowires: Maximally localized Wannier functions analysis

2015 ◽  
Vol 111 (6) ◽  
pp. 67003 ◽  
Author(s):  
Haibo Niu ◽  
Guangde Chen ◽  
Yelong Wu ◽  
Honggang Ye ◽  
Youzhang Zhu
Keyword(s):  
Spontaneous Polarization ◽  
Piezoelectric Properties ◽  
Wannier Functions

Burst Erasures and the Mean-Square Error for Cyclic Parseval Frames

2011 ◽  
Vol 57 (7) ◽  
pp. 4622-4635 ◽  
Author(s):  
Bernhard G. Bodmann ◽  
Pankaj K. Singh
Keyword(s):  
Mean Square Error ◽  
Mean Square ◽  
Parseval Frames ◽  
The Mean

Peierls’ substitution via minimal coupling and magnetic pseudo-differential calculus

2019 ◽  
Vol 31 (03) ◽  
pp. 1950008
Author(s):  
Horia D. Cornean ◽  
Viorel Iftimie ◽  
Radu Purice
Keyword(s):  
Magnetic Field ◽  
Differential Calculus ◽  
Spectral Band ◽  
Spatial Decay ◽  
Minimal Coupling ◽  
Field Perturbation ◽  
Wannier Functions ◽  
Weak Magnetic Fields ◽  
The Magnetic Field ◽  
Magnetic Field Perturbation

We revisit the celebrated Peierls–Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic [Formula: see text]-periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by [Formula: see text] exponentially localized composite Wannier functions. Then we show that the effective magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix living in [Formula: see text]. In addition, if the magnetic field perturbation is slowly variable in space, then the perturbed spectral island is close (in the Hausdorff distance) to the spectrum of a Weyl quantized minimally coupled symbol. This symbol only depends on [Formula: see text] and is [Formula: see text]-periodic; if [Formula: see text], the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.

scholarly journals Effective Hamiltonians and Wavefunctions for Electrons in Deformed Crystals

1983 ◽  
Vol 36 (3) ◽  
pp. 321 ◽  
Author(s):  
RA Brown
Keyword(s):  
Hilbert Space ◽  
Probability Density ◽  
Physical Interpretation ◽  
Wannier Function ◽  
Deformation Potential ◽  
Effective Hamiltonian ◽  
Wannier Functions ◽  
Modulating Functions ◽  
Strain Dependent ◽  
Effective Hamiltonians

An effective Hamiltonian for electrons in in homogeneously deformed crystals is derived by expanding the wavefunction in terms of Wannier functions of the homogeneously deformed crystal. The physical interpretation of the modulating functions which determine the amplitude of each Wannier function in the expansion, and which are governed by the effective Hamiltonian, is investigated. This leads to strain-dependent expressions for the probability density and current, averaged over the fluctuations within each unit cell. The operators which represent, in the Hilbert space of the . modulating functions, similarly averaged physical observables are introduced and explicit straindependent expressions for the velocity and momentum operators are obtained. Applications of the theory are foreshadowed and its relationship to previous deformation-potential theories is examined.

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