A mathematical approach to the effective Hamiltonian in perturbed periodic problems

Gérard, Christian; Martinez, André; Sjöstrand, Johannes

doi:10.1007/bf02102061

Cited by 65 publications

(85 citation statements)

References 8 publications

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“…The asymptotics of (1.1) as ε → 0+ is a well-studied two-scale problem in the physics and mathematics literature [8,16,20,41,23,13,33,1,14]. On the other hand, the computational challenge because of the small parameter ε has prompted a search for asymptotic model based numerical methods, see e.g., [29,37].…”

Section: Introductionmentioning

confidence: 99%

Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation

Liu¹,

Pryporov²

2015

Spectral Theory and Partial Differential Equations

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Abstract. This work is concerned with asymptotic approximations of the semi-classical Schrödinger equation in periodic media using Gaussian beams. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the Gaussian beam approximation and homogenization leads to the Bloch eigenvalue problem and associated evolution equations for Gaussian beam components in each Bloch band. We formulate a superposition of Blochband based Gaussian beams to generate high frequency approximate solutions to the original wave field. For initial data of a sum of finite number of band eigen-functions, we prove that the first-order Gaussian beam superposition converges to the original wave field at a rate of ǫ 1/2 , with ǫ the semiclassically scaled constant, as long as the initial data for Gaussian beam components in each band are prepared with same order of error or smaller. For a natural choice of initial approximation, a rate of ǫ 1/2 of initial error is verified. IntroductionWe consider the semiclassically scaled Schrödinger equation with a periodic potential:subject to the two-scale initial condition:where Ψ(t, x) is a complex wave function, ε is the re-scaled Planck constant, V e (x)-smooth external potential, S 0 (x)-real-valued smooth function, g(x, y) = g(x, y + 2π)-smooth function, compactly supported in x, i.e., g(x, y) = 0, x ∈ K 0 , K 0 -is a bounded set. V (y) is periodic with respect to the crystal lattice Γ = (2πZ) d , it models the electronic potential generated by the lattice of atoms in the crystal [14].A typical application arises in solid state physics where (1.1) describes the quantum dynamics of Bloch electrons moving in a crystalline lattice (generated by the ionic cores) [39]. The asymptotics of (1.1) as ε → 0+ is a well-studied two-scale problem in the physics and mathematics literature [8,16,20,41,23,13,33,1,14]. On the other hand, the computational challenge because of the small parameter ε has prompted a search for asymptotic model based numerical methods, see e.g., [29,37].1991 Mathematics Subject Classification. 35A21, 35A35, 35Q40.

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Section: Introductionmentioning

confidence: 99%

Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation

Liu¹,

Pryporov²

2015

Spectral Theory and Partial Differential Equations

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“…As it is well known (see, e.g., [Ba,DiSj1,GMS,Ma2]), with such a type of quantization is associated a full and explicit symbolic calculus that permits to handle these operators in a very easy and pleasant way. In particular, we have the following results: Proposition A.2 (Composition).…”

Section: Smooth Peudodifferential Calculus With Operator-valued Symbolmentioning

confidence: 99%

Twisted pseudodifferential calculus and application to the quantum evolution of molecules

Martinez¹,

Sordoni²

2009

Memoirs of the AMS

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Abstract. We construct an abstract pseudodifferential calculus with operatorvalued symbol, suitable for the treatment of Coulomb-type interactions, and we apply it to the study of the quantum evolution of molecules in the BornOppenheimer approximation, in the case of the electronic Hamiltonian admitting a local gap in its spectrum. In particular, we show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, we study the propagation of certain wave packets up to long time values of Ehrenfest order. (This work has been accepted for publication as part of the Memoirs of the American Mathematical Society and will be published in a future volume.) Contents IntroductionIn quantum physics, the evolution of a molecule is described by the initial-value Schrödinger system,where ϕ 0 is the initial state of the molecule and H stands for the molecular Hamiltonian involving all the interactions between the particles constituting the molecule (electron and nuclei). In case the molecule is imbedded in an electromagnetic field, the corresponding potentials enter the expression of H, too. Typically, the interaction between two particles of positions z and z ′ , respectively, is of Coulomb type, that is, of the form α|z − z ′ | −1 with α ∈ IR constant.In the case of a free molecule, a first approach for studying the system (1.1) consists in considering bounded initial states only, that is, initial states that are eigenfunctions of the Hamiltonian after removal of the center of mass motion. More precisely, one can split the Hamiltonian into,where the two operators H CM (corresponding to the kinetic energy of the center of mass) and H Rel (corresponding to the relative motion of electrons and nuclei) commute. As a consequence, the quantum evolution factorizes into,where the (free) evolution e −itHCM of the center of mass can be explicitly computed (mainly because H CM has constant coefficients), while the relative motion e −itH Rel still contains all the interactions (and thus, all the difficulties of the problem). Then, taking ϕ 0 of the form,where α 0 depends on the position of the center of mass only, and ψ j is an eigenfunction of H Rel with eigenvalue E j , the solution of (1.1) is clearly given by,Therefore, in this case, the only real problem is to know sufficiently well the eigenelements of H Rel , in order to be able to produce initial states of the form (1.2).In 1927, M. Born and R. Oppenheimer [BoOp] proposed a formal method for constructing such an approximation of eigenvalues and eigenfunctions of H Rel . This method was based on the fact that, since the nuclei are much heavier than the electrons, their motion is slower and allows the electrons to adapt almost instantaneously to it. As a consequence, the motion of the electrons is not really perceived

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“…However this is not practically accessible in most cases. The method to perform such a reduction goes as follows [13,61,56] for atoms with one valence electron. The single atom Schrödinger equation…”

Section: Electrons and Phononsmentioning

confidence: 99%

The Noncommutative Geometry of Aperiodic Solids

Bellissard

2003

Geometric and Topological Methods for Quantum Field Theory

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A mathematical approach to the effective Hamiltonian in perturbed periodic problems

Cited by 65 publications

References 8 publications

Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation

Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation

Twisted pseudodifferential calculus and application to the quantum evolution of molecules

The Noncommutative Geometry of Aperiodic Solids

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