Homogenization of the Schrödinger equation and effective mass theorems

DOI10.1007/s00220-005-1329-2zbMath1081.35092OpenAlexW1975358805MaRDI QIDQ2575367

Grégoire Allaire, Andrey L. Piatnitski

Publication date: 9 December 2005

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00220-005-1329-2

zbMATH Keywords

Bloch waves Bloch spectrum Bloch eigenfunction Bloch eigenvalue homogenized Schrödinger equation

Mathematics Subject Classification ID

Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) PDEs in connection with quantum mechanics (35Q40) Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics (82C31) Homogenization in context of PDEs; PDEs in media with periodic structure (35B27)

Related Items

Classical limit for the varying-mass Schrödinger equation with random inhomogeneities ⋮ Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results ⋮ Propagation of one- and two-dimensional discrete waves under finite difference approximation ⋮ Effective wave factorization for a stochastic Schrödinger equation ⋮ Gaussian beam construction for adiabatic perturbations ⋮ Diffractive behavior of the wave equation in periodic media: weak convergence analysis ⋮ Bloch wave homogenisation of quasiperiodic media ⋮ Homogenization of a nonlinear elliptic problem with large nonlinear potential ⋮ Wave Mechanics in Media Pinned at Bravais Lattice Points ⋮ Spectral approach to homogenization of nonstationary Schrödinger-type equations ⋮ Effective mass theorems with Bloch modes crossings ⋮ Continuum limit of the lattice quantum graph Hamiltonian ⋮ Multiscale analysis and numerical algorithm for the Schrödinger equations in heterogeneous media ⋮ Effective approximation for a nonlocal stochastic Schrödinger equation with oscillating potential ⋮ Reflection from a semi-infinite stack of layers using homogenization ⋮ Mathematical aspects of the Kubo formula for electrical conductivity with dissipation ⋮ Second-Order Homogenization of Periodic Schrödinger Operators with Highly Oscillating Potentials ⋮ Homogenization of Schrödinger equations. Extended effective mass theorems for non-crystalline matter ⋮ High-energy homogenization of a multidimensional nonstationary Schrödinger equation ⋮ On a characterization of the Rellich-Kondrachov theorem on groups and the Bloch spectral cell equation ⋮ Homogenisation for elastic photonic crystals and dynamic anisotropy ⋮ Second-order three-scale asymptotic analysis and algorithms for Steklov eigenvalue problems in composite domain with hierarchical cavities ⋮ Effective masses for Laplacians on periodic graphs ⋮ Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature ⋮ Wave packets in honeycomb structures and two-dimensional Dirac equations ⋮ Wigner measures and effective mass theorems ⋮ Quantum transport in crystals: Effective mass theorem and k$\cdot$p Hamiltonians ⋮ Homogenization of a coupled system with periodic oscillating coefficients ⋮ Spectral Band Degeneracies of $\frac{\pi}{2}$-Rotationally Invariant Periodic Schrödinger Operators ⋮ Homogenization of periodic Schrödinger-type equations, with lower order terms ⋮ Elliptic operators with honeycomb symmetry: Dirac points, edge states and applications to photonic graphene ⋮ Multiscale Algorithms and Computations for the Time-Dependent Maxwell--Schrödinger System in Heterogeneous Nanostructures ⋮ Multiscale numerical algorithm for the elliptic eigenvalue problem with the mixed boundary in perforated domains ⋮ Multi-scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates ⋮ Bloch Waves in Bubbly Crystal Near the First Band Gap: A High-Frequency Homogenization Approach ⋮ AN EFFECTIVE MASS MODEL FOR THE SIMULATION OF ULTRA-SCALED CONFINED DEVICES ⋮ MultiscaleComputations fortheMaxwell–Schrödinger System in Heterogeneous Nanostructures ⋮ An effective mass theorem for the bidimensional electron gas in a strong magnetic field ⋮ Homogenization limit for a multiband effective mass model in heterostructures ⋮ Homogenization of periodic systems with large potentials ⋮ A goal-oriented error estimator for a class of homogenization problems ⋮ DIFFRACTION OF BLOCH WAVE PACKETS FOR MAXWELL'S EQUATIONS ⋮ Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows ⋮ Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients ⋮ Quantum transport in a crystal with short-range interactions: the Boltzmann-Grad limit ⋮ Homogenization for the cubic nonlinear Schrödinger equation on R2 ⋮ Eigenvalue Asymptotics in a Twisted Waveguide ⋮ Low regularity justification results for envelope approximations of nonlinear wave packets in periodic media ⋮ Variational formulation of dynamical homogenization towards nonlocal effective media ⋮ Asymptotic spectral analysis in semiconductor nanowire heterostructures ⋮ Multiscale Analysis and Computation for a Stationary Schrödinger--Poisson System in Heterogeneous Nanostructures ⋮ A Quasi-Optimal Factorization Preconditioner for Periodic Schrödinger Eigenstates in Anisotropically Expanding Domains

Cites Work

This page was built for publication: Homogenization of the Schrödinger equation and effective mass theorems