Quantum transport in crystals: effective-mass theorem and k.p Hamiltonians
aa r X i v : . [ m a t h - ph ] N ov Quantum Transport in Crystals:Effective Mass Theorem and K · P Hamiltonians
Luigi Barletti and Naoufel Ben Abdallah Dipartimento di Matematica, Universit`a di Firenze, Viale Morgagni 67/A, 50134 Firenze,Italy, [email protected]fi.it Institut de Math´ematiques de Toulouse, Universit´e de Toulouse Univ. Paul Sabatier, 118route de Narbonne, 31062 Toulouse, France, [email protected]
Abstract
In this paper the effective mass approximation and k · p multi-band models, describingquantum evolution of electrons in a crystal lattice, are discussed. Electrons are assumed tomove in both a periodic potential and a macroscopic one. The typical period ǫ of the periodicpotential is assumed to be very small, while the macroscopic potential acts on a much biggerlength scale. Such homogenization asymptotic is investigated by using the envelope-functiondecomposition of the electron wave function. If the external potential is smooth enough,the k · p and effective mass models, well known in solid-state physics, are proved to be close(in strong sense) to the exact dynamics. Moreover, the position density of the electrons isproved to converge weakly to its effective mass approximation. The effective mass approximation is a common approximation in solid state physics [7, 6, 21] andstates roughly speaking that the motion of electrons in a periodic potential can be replaced witha good approximation by the motion of a fictitious particle in vacuum but with a modified masscalled the effective mass of the electron. This approximation is valid when the lattice period issmall compared to the observation length scale, it relies on the Bloch decomposition theorem forthe Schr¨odinger equation with a periodic potential. The effective mass is actually a tensor anddepends on the energy band in which the electron “live’s’. One of the most important referencesin the Physics literature on the subject is the paper of Kohn and Luttinger [14] which datesback to 1955. As for rigorous mathematical treatment of this problem, we are aware of the workof Poupaud and Ringhofer [16] and that of Allaire and Piatnitski [3]. The aim of the presentwork is to provide an alternative mathematical treatment which is based on the original workof Kohn and Luttinger. Like in [3] (see also [2] and [4] for related problems), we consider thescaled Schr¨odinger equation i∂ t ψ ( t, x ) = (cid:18) −
12 ∆ + 1 ǫ W L (cid:16) xǫ (cid:17) + V (cid:16) x, xǫ (cid:17)(cid:19) ψ ( t, x ) , where W L ( z ) is a periodic potential with the periodicity of a lattice L , representing the crystalions, while V ( x, z ) represents an external potential. The latter is assumed to act both on the1acroscopic scale x and on the microscopic scale z = x/ǫ , and to be L -periodic with respect to z . The small parameter ǫ is interpreted as the so-called “lattice constant”, that is the typicalseparation between lattice sites. Note that the scaling of the Schr¨odinger equation (2.30) is ahomogenization scaling [3, 16]. As mentioned above, the analysis of the limit ǫ → ǫ → · p Hamiltonianas an intermediate model between the original Schr¨odinger equation and its limiting effectivemass approximation.The outline of the paper is as follows. Section 2 is devoted to the presentation of the functionalsetting, notations as well as the main result of the paper. As mentioned above, the Schr¨odingerequation is reformulated as an infinite system of coupled Schr¨odinger equations, where thecoupling comes both from the differential part and from the potential part. In Section 3, weconcentrate on the potential part and analyze its limit. Section 4 is devoted to the diagonaliza-tion of the differential part and to the expansion of the corresponding eigenvalues in the Fourierspace. In Section 5, we analyze the convergence of the solution of the Schr¨odinger equationtowards its effective mass approximation. The method relies on the definition of intermediatemodels and the comparison of their respective dynamics. Some comments are done in Section 6while some proofs are postponed to Section 7. 2 Notations and main results
Let us consider the operator H ǫ L = − ∆ + ǫ W L (cid:16) xǫ (cid:17) , (2.1)where W L is a bounded L -periodic potential where the lattice L is defined by L = n Lz (cid:12)(cid:12) z ∈ Z d o ⊂ R d , (2.2)where L be a d × d matrix with det L = 0. The centered fundamental domain C of L is, bydefinition, C = (cid:26) Lt (cid:12)(cid:12)(cid:12) t ∈ h − , i d (cid:27) . (2.3)Note that the volume measure |C| of C is given by |C| = | det L | . The reciprocal lattice L ∗ is, bydefinition, the lattice generated by the matrix L ∗ such that L T L ∗ = 2 πI. (2.4)The Brillouin zone B is the centered fundamental domain of L ∗ , i.e. B = (cid:26) L ∗ t (cid:12)(cid:12)(cid:12) t ∈ h − , i d (cid:27) . (2.5)Thus, we clearly have |C| |B| = (2 π ) d . (2.6)We assume without loss of generality that the periodic potential is larger than one ( W L ≥ W L is interpreted as the electrostatic potential generated by the ions ofthe crystal lattice [6]. With the change of variables z = x/ǫ , the operator H ǫ L turns to ǫ H L ,where H L is given by (2.1) with ǫ = 1. This operator has a band structure which is given bythe celebrated Bloch theorem [18]. Definition 2.1
For any k ∈ B , the fiber Hamiltonian H L ( k ) = 12 | k | − ik · ∇ − ∆ + W L . (2.7) defined on L ( C ) with periodic boundary condition has a compact resolvent. Its eigenfunctionsform an orthonormal sequence of periodic solutions ( u n,k ) n ∈ N ) solving the eigenvalue problem H L ( k ) u n,k = E n ( k ) u n,k (2.8) The functions u n,k are the so-called Bloch functions and the eigenvalues E n ( k ) are the energybands of the crystal. For each fixed value of k ∈ B , the set { u n,k | n ∈ N } is a Hilbert basis of L ( C ) [8, 18]. The Bloch waves defined for k ∈ B and n ∈ N by X b n,k ( x ) = |B| − / B ( k ) e ik · x u n,k ( x ) In solid state physics the Brillouin zone used has a slightly different definition. However, the two definitionsare equivalent to our purposes. orm a complete basis of L ( R d ) and satisfy the equation H L X b n,k = E n ( k ) X b n,k . The scaled Bloch functions are given by X b ,ǫn,k ( x ) = |B| − / B /ǫ ( k ) e ik · x u n,ǫk ( x ) and they satisfy H ǫ L X b ,ǫn,k = E n ( ǫk ) ǫ X b ,ǫn,k . In order to analyze the limit ǫ →
0, the usual starting point is to decompose the wave functionon the Bloch wave functions. This decomposition was in particular used in [3]. This has thebig advantage of completely diagonalizing the periodic Hamiltonian, but since the wave vectorappears both in the plane wave e ik · x and in the standing periodic function u n,ǫk , the separationbetween the fast oscillating scale and the slow motion carried by the plane wave is not immediate.We follow in this work the idea of Kohn and Luttinger [14] who decompose the wave functionon the basis X lk n,k ( x ) = |B| − / B ( k ) e ik · x u n, ( x ) . (2.9)The family X lk n,k is also a complete orthonormal basis of L ( R ) but only partially diagonalizes H L since H L X lk n,k = |B| − / B /ǫ ( k ) e ik · x (cid:20) | k | − ik · ∇ + E n (0) (cid:21) u n, = |B| − / B /ǫ ( k ) e ik · x X n ′ (cid:20) | k | δ nn ′ − ik · P nn ′ + E n δ nn ′ (cid:21) u n ′ , = X n ′ (cid:20) | k | δ nn ′ − ik · P nn ′ + E n δ nn ′ (cid:21) X lk n ′ ,k . (2.10)Here, E n = E n (0) and P nn ′ = Z C u n, ( x ) ∇ u n ′ , ( x ) dx (2.11)are the matrix elements of the gradient operator between Bloch functions. The interest of theLuttinger-Kohn wave functions is that the wave vector k only appears in the plane wave and notin the standing periodic part u n, . This will allow us to decompose the wave function in a niceway for which we will prove some Hilbert analysis type results. This is the envelope functiondecomposition that we detail in the following section. In the following, we shall use the symbol F to denote the Fourier transformation on L ( R d ) F ψ ( k ) = 1(2 π ) d/ Z R d e − ix · k ψ ( x ) dk (2.12)and F ∗ = F − for the inverse transformation. We shall use a hat, ˆ ψ = F ψ , for the Fouriertransform of ψ . 4 efinition 2.2 We define L B ( R d ) ⊂ L ( R d ) to be the subspace of L -functions supported in B : L B ( R d ) = n f ∈ L ( R d ) (cid:12)(cid:12) supp (cid:0) f (cid:1) ⊂ B o . (2.13) Thus, F ∗ L B ( R d ) is the space of L -functions whose Fourier transform is supported in B . The envelope function decomposition is defined by the following theorem.
Theorem 2.3
Let v n : R d → C be L -periodic functions such that { v n | n ∈ N } is an orthonormalbasis of L ( C ) . For every ψ ∈ L ( R d ) there exists a unique sequence { f n ∈ F ∗ L B ( R d ) | n ∈ N } such that ψ = |C| / X n f n v n . (2.14) We shall denote f n = π n ( ψ ) . The decomposition satisfies the Parseval identity h ψ, ϕ i L ( R d ) = X n h π n ( ψ ) , π n ( φ ) i L ( R d ) . (2.15) For any ǫ > we shall consider the scaled version f ǫn = π ǫn ( ψ ) of the envelope function decom-position as follows: ψ ( x ) = |C| / X n f ǫn ( x ) v ǫn ( x ) , (2.16) with ˆ f ǫn ∈ L B /ǫ ( R d ) , where v ǫn ( x ) = v n (cid:16) xǫ (cid:17) . (2.17) We still have the Parseval identity h ψ, ϕ i L ( R d ) = X n h π ǫn ( ψ ) , π ǫn ( ϕ ) i L ( R d ) . (2.18) Finally, the Fourier transforms of the ǫ -scaled envelope functions are given by ˆ f ǫn ( k ) = Z R d X ǫn,k ( x ) ψ ( x ) dx, (2.19) where, for x ∈ R d , k ∈ R d , n ∈ N , X ǫn,k ( x ) = |B | − / B /ǫ ( k ) e ik · x v ǫn ( x ) . (2.20)The proof of this theorem is postponed to Section 7. Remark 2.4
Note that the above result is a variant of the so-called Bloch transform. In [5],the function b ψ ( x, k ) = |C| / X n b f n ( k ) v n ( x ) is referred to as the Bloch transform of ψ . We also refer to [1], [18] and [13] for Bloch wavemethods in periodic media. Definition 2.5
The functions f n = π n ( ψ ) of Theorem 2.3 will be called the envelope functions of ψ relative to the basis { v n | n ∈ N } , while f ǫn = π ǫn ( ψ ) will be called the ǫ -scaled envelopefunction relative to the basis { v n | n ∈ N } Theorem 2.6
Let us consider the ǫ -scaled envelope function decomposition (2.16) of ψ ∈ L ( R d ) .Then, for every θ ∈ L ( R d ) such that ˆ θ ∈ L ( R d ) , we have lim ǫ → Z R d θ ( x ) " | ψ ( x ) | − X n | f ǫn ( x ) | dx = 0 . (2.21)The proofs of this theorem is also postponed to Section 7.5 .3 Functional spaces In this section, we define some functional spaces which will be used all along the paper.
Definition 2.7
We define the space L = ℓ (cid:0) N , L ( R d ) (cid:1) as the Hilbert space of sequences g =( g , g , . . . ) , g n = g n ( k ) , with g n ∈ L ( R d ) , such that k g k L = X n k g n k L ( R d ) < ∞ . (2.22) Moreover, for µ ≥ let L µ be the subspace of all sequences g ∈ L such that k g k L µ = k (1 + | k | ) µ/ g k L = X n k (1 + | k | ) µ/ g n k L < ∞ (2.23) and let H µ = ℓ (cid:0) N , H µ ( R d ) (cid:1) , with k f k H µ = X n k f n k H µ = X n k (1 + | k | ) µ/ b f n k L < ∞ . (2.24) It is readily seen that f ∈ H µ if and only if b f ∈ L µ . Let us redefine the eigenpairs ( E n , v n ) ofthe operator H L = − ∆ + W L with periodic boundary conditions by − ∆ v n + W L v n = E n v n , on C Z C | v n | dx = 1 , v n periodic (2.25) (note that v n = u n, , according to Definition 2.1). The sequence E n is increasing and tends to + ∞ . Let us now define the functional spaces for the external potential: W µ = n V ∈ L ∞ ( R d ) (cid:12)(cid:12)(cid:12) V ( · , z + λ ) = V ( · , z ) , λ ∈ L , k V k W µ < ∞ o , (2.26)where k V k W µ = 1(2 π ) d/ ess sup z ∈C Z R d (1 + | k | ) µ | ˆ V ( k, z ) | dk (2.27)and b V ( k, z ) = (2 π ) − d/ Z R d e − ik · x V ( x, z ) dx .We finally define for any positive constant γ the truncation operator T γ ( f ) = F ∗ ( γ B b f ) . (2.28)It is now readily seen that the truncation operator satisfies for any nonnegative real numbers s, µ , k f − T γ f k H s ≤ Cγ − µ k f k H s + µ , (2.29)where C > γ .6 .4 Main Theorem We announce in this section the main theorem of our paper. We recall that ( v n , E n ) are definedby (2.25). Theorem 2.8
Assume that W L ∈ L ∞ and that all the eigenvalues E n = E n (0) are simple. Let ψ in,ǫ be an initial datum in L ( R d ) , let f in,ǫn = π ǫn ( ψ in,ǫ ) be its scaled envelope functions relativeto the basis v n . Assume that the sequence f in,ǫ belongs to H µ , with a uniform bound for thenorm as ǫ vanishes, and that it converges in L as ǫ tends to zero to an initial datum f in . Let ψ ǫ be the unique solution of i∂ t ψ ǫ ( t, x ) = (cid:18) −
12 ∆ + 1 ǫ W L (cid:16) xǫ (cid:17) + V (cid:16) x, xǫ (cid:17)(cid:19) ψ ǫ ( t, x ) ,ψ ( t = 0) = ψ in,ǫ , (2.30) and assume that V ∈ W µ for a positive µ . Then for any θ ∈ L ( R d ) such that b θ ∈ L ( R d ) , wehave the following local uniform convergence in time Z | ψ ǫ ( t, x ) | θ ( x ) dx → Z X n | h n ( t, x ) | θ ( x ) dx where the envelope function h n is the unique solution of the homogenized Schr¨odinger equation i∂ t h n = −
12 div (cid:0) M − n ∇ h n (cid:1) + V nn ( x ) h n , h n ( t = 0) = f inn , with V nn = Z C V ( x, z ) | v n ( z ) | dz and M − n = ∇ ⊗ ∇ E n ( k ) | k =0 = I − X n ′ = n P nn ′ ⊗ P n ′ n E n − E n ′ . (effective mass tensor of the n -th band). · p model Let ψ ǫ ( t, x ) be the solution of the Schr¨odinger equation (2.30) and let f ǫn ( t, x ) be its ǫ -scaledenvelope function relative to the basis v n defined in (2.25) and (2.17): ψ ǫ ( t, x ) = |C| / X n f ǫn ( t, x ) v ǫn ( x )Let us define g ǫn ( t, k ) = b f n ( t, k ) . From now on, we will reserve the notation f for functions of the position variable x , while g willbe used for functions of the wavevector k . Multiplying the Schr¨odinger equation by X ǫn,k ( x ) (seeEq. (2.20)) and integrating over k leads to the following equation i∂ t g ǫn ( t, k ) = 12 | k | g ǫn ( t, k ) − iǫ X n ′ k · P nn ′ g ǫn ′ ( t, k ) + 1 ǫ E n g ǫn ( t, k )+ X n ′ Z R d U ǫnn ′ ( k, k ′ ) g ǫn ′ ( t, k ′ ) dk ′ , (3.1)7here the kernel U nn ′ ( k, k ′ ) is given by U ǫnn ′ ( k, k ′ ) = Z R d X ǫn,k ( x ) V (cid:16) x, xǫ (cid:17) X ǫn ′ ,k ′ ( x ) dx = |B | − B /ǫ ( k ) Z R d B /ǫ ( k ′ ) e − i ( k − k ′ ) · x v nǫ ( x ) V (cid:16) x, xǫ (cid:17) v ǫn ′ ( x ) dx. By writing V ( x, z ) v n ( z ) = X n ′ V n ′ n ( x ) v n ′ ( z ) , where V n ′ n ( x ) = Z C v n ′ ( z ) v n ( z ) V ( x, z ) dz = V nn ′ ( x ) , (3.2)we can express U ǫnn ′ ( k, k ′ ) in the form U ǫnn ′ ( k, k ′ ) = B /ǫ ( k ) |B| X m Z R d B /ǫ ( k ′ ) e − i ( k − k ′ ) · x v nǫ ( x ) V mn ′ ( x ) v ǫm ( x ) dx (3.3)In position variables, the envelope functions satisfy the system i∂ t f ǫn ( t, x ) = E n ǫ f n ( t, x ) − ∆ f ǫn ( t, x ) − ǫ X n ′ ∈ N P nn ′ · ∇ f ǫn ′ ( t, x ) + X n ′ ∈ N Z R d V ǫnn ′ ( x, x ′ ) f ǫn ′ ( t, x ′ ) dx ′ , (3.4)where V ǫnn ′ ( x, x ′ ) = 1(2 π ) d |B| Z B /ǫ dk Z R d dy Z B /ǫ dk ′ ×× n e ik · x e − i ( k − k ′ ) · y v ǫn ( y ) V (cid:16) y, yǫ (cid:17) v ǫn ′ ( y ) e − ik ′ · x ′ o (3.5)From equation (3.4) we see that the fast oscillation scales are different for different envelopefunctions. This will naturally lead to adiabatic decoupling (see [11, 15, 19, 20]). Definition 3.1
Let us define the operator U ǫ on L as follows: for any element g = ( g , g , . . . ) of L ( U ǫ g ) n ( k ) = X n ′ Z R d U ǫnn ′ ( k, k ′ ) g ǫn ′ ( k ′ ) dk ′ . (3.6) Let us also define the operator V ǫ on the position space L by ( V ǫ f ) n ( x ) = X n ′ Z R d V ǫnn ′ ( x, x ′ ) f ǫn ′ ( x ′ ) dx ′ . (3.7) We obviously have \ V ǫ ( f ) = U ǫ ( b f ) . Since v n and v m are L -periodic, the formal limit of U ǫnn ′ ( k, k ′ ) is given by U nn ′ ( k, k ′ ) = X m h v n , v m i|B||C| Z R d e − i ( k − k ′ ) · x V mn ′ ( x ) dx = 1(2 π ) d/ d V nn ′ ( k − k ′ ) . Therefore the formal limit of U ǫ is the operator U defined by (cid:0) U g (cid:1) n ( k ) = X n ′ π ) d/ Z R d ˆ V nn ′ ( k − k ′ ) g n ′ ( k ′ ) dk ′ , (3.8)8hich means that the in position space the limit of V ǫ is the non diagonal multiplication operator V defined by (cid:0) V f (cid:1) n ( x ) = X n ′ V nn ′ ( x ) f n ′ ( x ) . (3.9)The operators become diagonal in n if V ( x, z ) does not depend on z . Indeed, in this case V nn ′ ( x ) = V ( x ) δ nn ′ . The k · p approximation found in semiconductor theory [21], consists inreplacing the operator U ǫ by U . Let us now analyze the departure of U ǫ from U . Lemma 3.2
Let the external potential V ( x, z ) be in L ∞ . Then, for any ǫ ≥ , U ǫ is a boundedoperator on L and we have the uniform bound kU ǫ k ≤ k V k L ∞ , ∀ ǫ ≥ . (3.10) Proof
Let us begin with the case ǫ = 0. We remark that U g = \ V ( f ) , where f = F ∗ ( g ). Let G be another element of L , and let F be its back Fourier transform. Wehave (cid:12)(cid:12)(cid:10) U g, G (cid:11)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:10) V f, F (cid:11)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X nn ′ Z V nn ′ ( x ) f n ′ ( x ) F n ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X nn ′ Z V ( x, z ) v n ′ ( z ) v n ( z ) f n ′ ( x ) F n ( x ) dx dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z V ( x, z ) "X n f n ( x ) v n ( z ) n F n ( x ) v n ( z ) dx dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k V k L ∞ "Z (cid:12)(cid:12)(cid:12) X n f n ( x ) v n ( z ) (cid:12)(cid:12)(cid:12) dx dz "Z (cid:12)(cid:12)(cid:12) X n F n ( x ) v n ( z ) (cid:12)(cid:12)(cid:12) dx dz ≤ k V k L ∞ k f k L k F k L = k V k L ∞ k g k L k G k L . Since the result holds for any g and G in L , this implies that kU ( g ) k L ≤ k V k L ∞ k g k L . For ǫ > U ǫ is unitarily equivalent to the multiplication operator by V ( x, xǫ ) in position space. More precisely, defining f ǫ ( x ) = F ∗ ( B /ǫ g ) and defining ψ ǫ ( x ) = P n f ǫn ( x ) v ǫn ( x ) so that f ǫn = π ǫn ( ψ ǫ ), then it follows from the definition of U ǫ that( U ǫ g ) n = F h π ǫn (cid:16) V (cid:16) x, xǫ (cid:17) ψ ǫ (cid:17)i . It is now readily seen that kU ǫ ( g ) k L = k V (cid:16) x, xǫ (cid:17) ψ ǫ k L ≤ k V k L ∞ k ψ ǫ k L ≤ k V k L ∞ k g k L . (cid:4) Lemma 3.3
For any γ > let γ B be the set of γk where k is in B . Then γ B + β B = ( γ + β ) B . Moreover Let k ∈ B and k ′ ∈ B . Let λ a non vanishing element of the reciprocal lattice L ∗ .Then k − k ′ + λ / ∈ B . B is the linear deformation of ahypercube, see definition (2.5)) and is left to the reader. Lemma 3.4
Let V ∈ W and g ∈ L be such that supp (cid:0) ˆ V nm (cid:1) ⊂ ǫ B and supp( g n ) ⊂ ǫ B , forall n, m ∈ N . Then, in this case, U ǫ g = U g . Proof
Let us first notice that {|C| − / e iη · x | η ∈ L ∗ } is a orthonormal basis of L ( C ) (theFourier basis). We first deduce from (3.3) and from the identity v n ( y ) = 1 |C| / X λ ∈L ∗ v n,λ e iλ · x where v n,λ = h v n , e iλ · x |C| / i that( U ǫ g ) n ( k ) = X λ,λ ′ ∈L ∗ X m,n ′ Z R d × R d e − i ( k − k ′ + λ − λ ′ ǫ ) · x B /ǫ ( k ′ ) B /ǫ ( k ) ×× V nm ( x ) v m,λ v n ′ ,λ ′ g n ′ ( k ′ ) dx dk ′ = B /ǫ ( k )(2 π ) d/ X λ,λ ′ ∈L ∗ X m,n ′ Z B /ǫ b V nm (cid:16) k − k ′ + λ − λ ′ ǫ (cid:17) v m,λ v n ′ ,λ ′ g n ′ ( k ′ ) dx dk ′ . Since the support of g n ′ is included in B / ǫ and k ∈ B /ǫ , Lemma 3.3 implies that the onlycontributing terms to the above sum are those for which λ = λ ′ . Therefore, we are lead toevaluate P λ v m,λ v n ′ ,λ which is equal to h v n ′ , v m i = δ mn ′ because of the orthonormality of thefamily ( v n ). Therefore( U ǫ g ) n ( k ) = (2 π ) − d/ B /ǫ ( k ) X n ′ Z B /ǫ d V nn ′ ( k − k ′ ) g n ′ ( k ′ ) dx dk ′ . Now, we can remove B /ǫ ( k ) from the right hand side of the above identity, since both thesupport of g n ′ and that of d V nn ′ are in ǫ B . Hence( U ǫ g ) n ( k ) = (2 π ) − d/ X n ′ Z R d d V nn ′ ( k − k ′ ) g n ′ ( k ′ ) dxdk ′ = (cid:0) U g (cid:1) n ( k ) . (cid:4) Theorem 3.5
Assume that V ∈ W µ for some µ ≥ . Then, a constant c µ > , independent of ǫ , exists such that kU ǫ g − U g k L ≤ ǫ µ c µ k V k W µ k g k L µ (3.11) for all g ∈ L µ and for all ǫ > . Proof
Let the smoothed potential V ǫs be defined byˆ V ǫs ( k, z ) = B / ǫ ( k ) ˆ V ( k, z ) . (3.12)Moreover, let U ǫs denote the operator U ǫ with the potential V s . Let us assume firstly thatsupp ( g n ) ⊂ B / ǫ for all n ∈ N . Then, from Lemma 3.4 we have U ǫs g = U s g and we can write kU ǫ g − U g k L ≤ kU ǫ g − U ǫs g k L + kU s g − U g k L . (3.13)10sing (3.10) and the linearity of U ǫ and U with respect to the potential, we have kU ǫ g − U ǫs g k L ≤ k V − V ǫs k W k g k L . ǫ ≥ , Recalling the definition (2.27), we also have k V − V ǫs k W = 1(2 π ) d/ ess sup z ∈C Z R d \B / ǫ | ˆ V ( k, z ) | dk ≤ π ) d/ ess sup z ∈C Z k / ∈B / ǫ (cid:18) | ǫk | R (cid:19) µ | ˆ V ( k, z ) | dk ≤ (cid:18) ǫR (cid:19) µ k V k W µ where R > B . Then (still in the case supp( g n ) ⊂ B / ǫ ),from (3.13) we get kU ǫ g − U g k L ≤ (cid:18) ǫR (cid:19) µ k V k W µ k g k L . (3.14)Now, if g ∈ L µ (Definition 2.7), we can write (using c = 1 − ) kU ǫ g − U g k L ≤ kU ǫ c B / ǫ g k L + k ( U ǫ − U ) B / ǫ g k L + kU c B / ǫ g k L (3.15)From (3.10) we have kU ǫ c B / ǫ g k L ≤ k V k W k c B / ǫ g k L , for all ǫ ≥
0. But k c B / ǫ g k L = X n Z k / ∈B / ǫ | g n ( k ) | dk ≤ X n Z k / ∈B / ǫ (cid:18) | ǫk | R (cid:19) µ | g n ( k ) | dk ≤ (cid:18) ǫR (cid:19) µ k g k L µ and so we can estimate the first and third term in the right hand side of (3.15) as follows: kU ǫ c B / ǫ g k L + kU c B / ǫ g k L ≤ (cid:18) ǫR (cid:19) µ k V k W k g k L µ . Moreover, since Eq. (3.14) holds for B / ǫ g , then we can estimate also the second term: k ( U ǫ − U ) B / ǫ g k L ≤ (cid:18) ǫR (cid:19) µ k V k W µ k g k L . Since k V k W ≤ k V k W µ and k g k L ≤ k g k L µ , then from (3.15) we conclude that (3.11) holds,with c µ = 4(3 /R ) µ (note that R does not depend on ǫ ). (cid:4) · p Hamiltonian In this section, we consider the case V ( x, z ) = 0 and concentrate on the diagonalization ofthe k · p Hamiltonian. The envelope function dynamics are then given in Fourier variables byEq. (3.1) which we rewrite under the form iǫ ∂ t g n ( t, k ) = 12 ǫ | k | g n ( t, k ) − iǫ X n ′ k · P nn ′ g n ′ ( t, k ) + E n g n ( t, k ) . (4.1)Putting ξ = ǫk , we are therefore led to consider, for any fixed ξ ∈ R d , the following operators,acting in ℓ ≡ ℓ ( N , C ) and defined on their maximal domains:( A ) nn ′ = E n δ nn ′ , ( A ( ξ )) nn ′ = − iξ · P nn ′ , ( A ( ξ )) nn ′ = 12 | ξ | δ nn ′ . (4.2)11oreover, we put A ( ξ ) = A + A ( ξ ) + A ( ξ ), so that( A ( ξ )) nn ′ = E n δ nn ′ − iξ · P nn ′ + 12 | ξ | δ nn ′ (4.3)is the operator at the right-hand side of Eq. (4.1) (with ξ = ǫk ). Lemma 4.1
The following properties hold: (a) for any given ξ ∈ R d , A ( ξ ) is A -bounded with A -bound less than 1, which implies that A ( ξ ) = A + A ( ξ ) + A ( ξ ) is self-adjoint on the (fixed) domain of A , that is D ( A ) = n g ∈ ℓ (cid:12)(cid:12)(cid:12) X n | E n g n | < ∞ o ; (4.4)(b) { A ( ξ ) | ξ ∈ R d } is a holomorphic family of type (A) of self-adjoint operators [12]; (c) for any given ξ ∈ R d , A ( ξ ) has compact resolvent, which implies that A ( ξ ) has a sequence ofeigenvalues λ ( ξ ) ≤ λ ( ξ ) ≤ λ ( ξ ) ≤ · · · , with λ n ( ξ ) → ∞ , and a corresponding sequence ϕ (1) ( ξ ) , ϕ (2) ( ξ ) , ϕ (3) ( ξ ) . . . of orthonormal eigenvectors . Proof (a) We first recall (see (2.25)) that ( v n , E n ) is an eigencouple of H L = − ∆ + W L onthe domain H ( C ) (the subscript “per” denoting periodic boundary conditions). The operator A is the representation in the basis ( v n ) of the operator H L , while A ( ξ ) is the representationin the same basis of − iξ · ∇ with domain H ( C ): D ( A ) ≡ H ( C ) ⊂ H ( C ) ≡ D ( A ( ξ )) . Then, for any given sequence ( g n ), denoting g ( x ) = P n g n v n ( x ), we have12 Z C |∇ g ( x ) | dx + Z C W L ( x ) | g ( x ) | dx = (cid:10) H L g, g (cid:11) L ( C ) = X n E n | g n | . Since W L is bounded and W L ≥
1, then for g ∈ D ( A ) we obtain k A ( ξ ) g k ℓ ≤ | ξ | k∇ g k L ( C ) ≤ | ξ | X n E n | g n | , (4.5)where we used the notation g for both g ( x ) = P n g n v n ( x ) and for the sequence g = ( g n ) ∈ ℓ .Since E n → ∞ , then, for any given 0 < b <
1, a positive integer n ( ξ ) exists such that 2 | ξ | E n 1, which proves point (a). The proofof the remaining points is standard (see Refs. [8, 12, 18]). (cid:4) Remark 4.2 Recalling Definition 2.1 and Eq. (2.10) we see that A ( ξ ) is nothing but the ex-pression of the fiber Hamiltonian H L ( ξ ) in the Bloch basis v n = u n, . Then, the diagonalizationof A ( ξ ) corresponds to the diagonalization of H L ( ξ ) and, therefore, the eigenvalues λ n ( ξ ) co-incide with the energy bands E n ( ξ ) inside the Brillouin zone. Moreover, ϕ ( n ) ( ξ ) is clearly thecomponent expression of u n,ξ in the basis u n, , i.e. ϕ ( n ) ( ξ ) = h u n,ξ , u n, i L ( C ) . λ n ( ξ ) have been numbered in increasing order for each ξ ; this means that, whena eigenvalue crossing occurs, then the smoothness of λ n ( ξ ) (and of ϕ ( n ) ( ξ )) is lost. However,since we are assuming that λ n (0) = E n are simple, then λ n ( ξ ) and ϕ ( n ) ( ξ ) are analytic in aneighborhood of the origin. Of course, such neighborhood depends of n . Next lemma allows toestimate the growth of the eigenvalues and, consequently, the size of the analyticity domain. Lemma 4.3 For any given ξ ∈ R d , an integer n ( ξ ) ≥ exists such that | λ n ( ξ ) − E n | ≤ | ξ | p E n + 12 | ξ | , for all n ≥ n ( ξ ) . (4.6) Proof The behavior of the eigenvalues λ n ( ξ ) for large n will be investigated by means of the max min principle, which holds for increasingly-ordered eigenvalues, [17]. Since the operators A ( ξ ) have compact resolvent, the max min principle reads as follows: λ n ( ξ ) = max S ∈ M n − min g ∈ S ⊥ ∩D ( A ) , k g k =1 h A ( ξ ) g, g i ℓ , where M n denotes the set of all subspaces of dimension n . In particular, λ n (0) = E n = max S ∈ M n − min g ∈ S ⊥ ∩D ( A ) , k g k =1 h A g, g i ℓ . Let g ∈ D ( A ) with k g k ℓ = 1. From (4.5) we have k A ( ξ ) g k ℓ ≤ | ξ | X n E n | g n | = 2 | ξ | h A g, g i ℓ and, therefore, |h A ( ξ ) g, g i ℓ | ≤ k A ( ξ ) g k ℓ ≤ √ | ξ |h A g, g i / ℓ , which, using A ( ξ ) = A + A ( ξ )+ A ( ξ ), yields |h A ( ξ ) g, g i ℓ − h A g, g i ℓ | ≤ √ | ξ | h A g, g i / ℓ + 12 | ξ | . (4.7)From (4.7) we get, in particular, h A ( ξ ) g, g i ℓ ≤ h A g, g i ℓ + √ | ξ | h A g, g i / ℓ + 12 | ξ | . which allows us to estimate λ n ( ξ ) from above. In fact, since x + √ | ξ | x / + | ξ | is an increasingfunction of x , we can writemax min h A ( ξ ) g, g i ℓ ≤ max min (cid:26) h A g, g i ℓ + √ | ξ |h A g, g i / ℓ + 12 | ξ | (cid:27) ≤ max min h A g, g i ℓ + √ | ξ | max min h A g, g i / ℓ + 12 | ξ | , that is λ n ( ξ ) ≤ E n + 2 | ξ | E / n + | ξ | , (4.8)which holds for all n ∈ N . We now estimate λ n ( ξ ) from below, at least for large n . From (4.7)we get h A ( ξ ) g, g i ℓ ≥ h A g, g i ℓ − √ | ξ |h A g, g i / ℓ − | ξ | and we remark that x − √ | ξ | x / − | ξ | / x for x ≥ | ξ | / 2. Thus,let n ( ξ ) be such that E n ( ξ ) ≥ | ξ | / n ≥ n ( ξ ). Let us define S n − = span { e (1) , e (2) , . . . e ( n − } , { e ( n ) | n ∈ N } is the canonical basis of ℓ (eigenbasis of A ). We therefore havemin g ∈ S ⊥ n − ∩D ( A ) , k g k =1 h A g, g i ℓ = E n , because S ⊥ n − = span { e ( n ) , e ( n +1) , . . . } . Thus, for every g ∈ S ⊥ n − ∩ D ( A ) with k g k ℓ = 1, wecan write h A ( ξ ) g, g i ℓ ≥ h A g, g i ℓ − √ | ξ | h A g, g i / ℓ − | ξ | ≥ E n − √ | ξ | E / n − | ξ | , (because E n ≥ E n ( ξ ) ≥ | ξ | / g ∈ S ⊥ n − ∩D ( A ) , k g k =1 h A ( ξ ) g, g i ℓ ≥ E n − √ | ξ | E / n − | ξ | . Since S n − ∈ M n − , we conclude that λ n ( ξ ) ≥ E n − √ | ξ | E / n − | ξ | , n ≥ n ( ξ ) , (4.9)which, together with (4.8), yields (4.6). (cid:4) From (4.6) we see that, for fixed ξ , the sequences E n and λ n ( ξ ) are asymptotically equivalent.Moreover it is not difficult to prove the following. Corollary 4.4 A constant C , independent of n , exists such that λ n ( ξ ) ≥ λ n − ( ξ ) for all | ξ | ≤ C ( E n +1 − E n ) / √ E n . Then, the first N bands do not cross each other in a ball of radius R N = C max { E n +1 − E n | n ≤ N + 1 } / p E N +1 . Let us now consider the family of diagonalization operators { T ( ξ ) : ℓ → ℓ | ξ ∈ R d } , i.e. theunitary operators that map 1-1 the basis { e ( n ) | n ∈ N } onto the basis { ϕ ( n ) ( ξ ) | n ∈ N } , so thatΛ( ξ ) = T ∗ ( ξ ) A ( ξ ) T ( ξ ) = λ ( ξ ) 0 0 · · · λ ( ξ ) 0 · · · λ ( ξ ) · · · ... ... ... . . . (4.10)For any given ǫ ≥ T ǫ on the space L (see Definition 2.7) by (cid:0) T ǫ g (cid:1) ( k ) = T ( ǫk ) g ( k ) . (4.11) Theorem 4.5 For every ǫ ≥ , the operator T ǫ : L → L is unitary, with T = I . Moreover, if g ∈ L µ for some µ > , then lim ǫ → k T ǫ g − g k L = 0 . Proof The first part of the statement is clear, because Z R d k T ( ǫk ) g ( k ) k ℓ dk = Z R d k g ( k ) k ℓ dk = k g k L and λ n (0) = E n . Now, let Π N be the projection operator in ℓ on the N -dimensional sub-spacespanned by e (1) , e (2) , . . . , e ( N ) (in other words, the cut-off operator after the N -th component).14ince the first N bands do not cross in a ball of radius R N (see Corollary 4.4), then ξ T ( ξ )Π N is unitary analytic from span (cid:8) e (1) , e (2) , . . . , e ( N ) (cid:9) to span (cid:8) ϕ (1) ( ξ ) , ϕ (2) ( ξ ) , . . . , ϕ ( N ) ( ξ ) (cid:9) , in | ξ | ≤ R N . Let g ∈ L µ and put g ( N ) = Π N g, g ( N ) c = g − g ( N ) , so that k ( T ǫ − I ) g k L ≤ k ( T ǫ − I ) g ( N ) k L + k ( T ǫ − I ) g ( N ) c k L . Let ǫ > r > ǫr ≤ R N . Then, using the analyticity of T ( ǫk )Π N in | ǫk | ≤ ǫr ≤ R N , we can write k ( T ǫ − I ) g ( N ) k L = Z R d k ( T ( ǫk ) − I ) g ( N ) ( k ) k ℓ dk = Z | k |≤ r k ( T ( ǫk ) − I ) g ( N ) ( k ) k ℓ dk + Z | k | >r k ( T ( ǫk ) − I ) g ( N ) ( k ) k ℓ dk ≤ L N Z | k |≤ r | ǫk | k g ( N ) ( k ) k ℓ dk + 4 r µ Z | k | >r | ǫk | µ k g ( N ) ( k ) k ℓ dk, for some Lipschitz constant L N > 0. Now, it can be easily verified that the inequality | k | n ≤ (1 + | k | µ ) r max { n − µ, } (4.12)holds for any r > n ≥ µ ≥ 0, and | k | ≤ r . From this (with n = 1) we get Z | k |≤ r | ǫk | k g ( N ) ( k ) k ℓ dk ≤ ǫ r { − µ, } Z | k |≤ r (1 + | ǫk | µ ) k g ( N ) ( k ) k ℓ dk and, therefore, k ( T ǫ − I ) g ( N ) k L ≤ (cid:0) L N ǫ r { − µ, } + 4 r − µ (cid:1) k g k L µ . Choosing r = R N /ǫ we obtain k ( T ǫ − I ) g ( N ) k L ≤ ǫ min { µ, } C ( µ, N ) k g k L µ , (4.13)where C ( µ, N ) = (cid:16) L N R { − µ, } N + 4 R − µN (cid:17) / . Moreover, k ( T ǫ − I ) g ( N ) c k L ≤ k T ǫ g ( N ) c k L + k g ( N ) c k L ≤ k g ( N ) c k L . Since k g ( N ) c k L → N → ∞ , we can fix N and, then, ǫ in inequality (4.13) so that k ( T ǫ − I ) g k L is arbitrarily small, which proves the limit. (cid:4) Remark 4.6 From inequality (4.13) we see that, when a finite number N of bands is considered,the distance between T ǫ and I is of order ǫ min { µ, } for g in ∈ L µ , with µ > . Let us now consider the second-order approximation of Λ( ξ ),Λ (2) ( ξ ) = λ (2)1 ( ξ ) 0 0 · · · λ (2)2 ( ξ ) 0 · · · λ (2)3 ( ξ ) · · · ... ... ... . . . (4.14)15here λ (2) n ( ξ ) is the second-order Taylor approximation of λ n ( ξ ): λ n ( ξ ) = λ (2) n ( ξ ) + O (cid:0) | ξ | (cid:1) . The approximated eigenvalues λ (2) n ( ξ ) can be computed by means of standard non-degenerateperturbation techniques, which yield λ (2) n ( ξ ) = E n + 12 ξ · M − n ξ, (4.15)where M − n = ∇ ⊗ ∇ λ n ( ξ ) | ξ =0 = I − X n ′ = n P nn ′ ⊗ P n ′ n E n − E n ′ (4.16)is the n -th band effective mass tensor [21] (we remind that P nn ′ = 0 if n = n ′ ). Note that the1st order term in (4.15) is zero.The operators Λ( ξ ) and Λ (2) ( ξ ), which are self-adjoint on their maximal domains, generate,respectively, the exact dynamics and the effective mass dynamics (in Fourier variables and inabsence of external fields). Theorem 4.7 Let g in ∈ L µ , for some µ > , and assume g in = Π N g in (i.e. the initial datumis confined in the first N bands). Then, a constant C ( µ, N, t ) ≥ , independent of ǫ , exists suchthat k (e − itǫ Λ( ǫk ) − e − itǫ Λ (2) ( ǫk ) ) g in k L ≤ ǫ min { µ/ , } C ( µ, N, t ) k g in k L µ . (4.17) Proof Note that, since Λ( ǫk ) and Λ (2) ( ǫk ) are diagonal, then both e − itǫ Λ( ǫk ) g in and e − itǫ Λ (2) ( ǫk ) g in remain confined in the first N bands at all times. Denoting g ǫ ( t, k ) = e − itǫ Λ (2) ( ǫk ) g in , the func-tion h ǫ ( t, k ) = (e − itǫ Λ( ǫk ) − e − itǫ Λ (2) ( ǫk ) ) g in satisfies the Duhamel formula h ǫ ( t, k ) = Z t e − i ( t − s ) ǫ Λ( ǫk ) Λ( ǫk ) − Λ (2) ( ǫk ) ǫ g ǫ ( s, k ) ds, so that k h ǫ ( t, k ) k ℓ ≤ Z t (cid:13)(cid:13)(cid:13) Λ( ǫk ) − Λ (2) ( ǫk ) ǫ g ǫ ( s, k ) (cid:13)(cid:13)(cid:13) ℓ ds Since λ ( ξ ) , . . . λ N ( ξ ) are analytic for | ξ | ≤ R N (see Corollary 4.4), then a Lipschitz constant L ′ N exists such that (cid:13)(cid:13)(cid:13) Λ( ǫk ) − Λ (2) ( ǫk ) ǫ g ǫ ( s, k ) (cid:13)(cid:13)(cid:13) ℓ ≤ ǫL N | k | k g ǫ ( k, s ) k ℓ = ǫL N | k | k g in ( k ) k ℓ for all k with | ǫk | ≤ R N (where we also used the fact that the ℓ norm of g ǫ is conserved duringthe unitary evolution). Now we can proceed as in the proof of Theorem 4.5: if r > ǫr ≤ R N , then we can write Z | k |≤ r k h ǫ ( t, k ) k ℓ dk ≤ ( L ′ N tǫ ) Z | k |≤ r | k | k g in ( k ) k ℓ dk and, using inequality (4.12) with n = 3, Z | k |≤ r k h ǫ ( t, k ) k ℓ dk ≤ (cid:16) L ′ N t ǫ r max { − µ, } (cid:17) k g in k L µ . Z | k | >r k h ǫ ( t, k ) k ℓ dk ≤ r µ Z | k | >r | k | µ k h ǫ ( t, k ) k ℓ dk ≤ r µ Z | k | >r | k | µ k g in ( t, k ) k ℓ dk ≤ r µ k g in k L µ , where we used the fact that k g ǫ ( t, k ) k ℓ = k g in ( k ) k ℓ for all t . Hence, k h ǫ ( t ) k ℓ ≤ (cid:20)(cid:16) L ′ N t ǫ r max { − µ, } (cid:17) + 4 r − µ (cid:21) k g in k L µ and, choosing r = R N /ǫ / , we obtain k h ǫ ( t ) k ℓ ≤ C ( µ, N, t ) ǫ min { µ/ , } k g in k L µ , that is inequal-ity (4.17), with C ( µ, N, t ) = (cid:20)(cid:16) L ′ N t R max { − µ, } N (cid:17) + 4 R − µ/ N (cid:21) / . (cid:4) Corollary 4.8 Let g in ∈ L µ , with µ > (but g in not necessarily confined in the first N bands),then lim ǫ → k (e − itǫ Λ( ǫk ) − e − itǫ Λ (2) ( ǫk ) ) g in k L = 0 , uniformly in bounded time intervals. Proof Like in the proof of the above theorem, we define h ǫ ( t, k ) = Z t e − i ( t − s ) ǫ Λ( ǫk ) Λ( ǫk ) − Λ (2) ( ǫk ) ǫ g ǫ ( s, k ) ds, For any given N we can write k h ǫ ( t ) k L ≤ k Π N h ǫ ( t ) k L + k Π cN h ǫ ( t ) k L , where Π cN = I − Π N . Recalling that the evolutions are diagonal, the first term at the right handside corresponds to the initial datum Π N g in , for which (4.17) holds. Using the fact that Π N commutes with both e − itǫ Λ( ǫk ) and e − itǫ Λ (2) ( ǫk ) , for the second term we have k Π cN h ǫ ( t ) k L ≤ k Π cN g in k L . Since Π cN g in → L as N → ∞ , this inequality, together with (4.17), shows that lim ǫ → k h ǫ ( t ) k L =0, uniformly in bounded t -intervals. (cid:4) We are now in position to exhibit the ensemble of models encountered and to compare theirrespective dynamics.We first started by the exact dynamics. Let the wave function ψ ǫ ( t, x ) be solution of the initialvalue problem (2.30). If we denote by f in,ǫn ( x ) the ǫ -scaled envelope functions of the initial wavefunction ψ in,ǫ , relative to the basis v n , and by g in,ǫn ( k ), their Fourier transform, then the Fouriertransformed envelope functions g ǫ of ψ ǫ ( t, x ) are the solutions of i∂ t g = A ǫ kp g + U ǫ g, g ǫ ( t = 0) = g in,ǫ (exact dynamics) (5.1)17here (cid:0) A ǫ kp g (cid:1) n ( k ) = 1 ǫ ( A ( ǫk ) g ( k )) n = E n ǫ + | k | ! g n ( k ) − iǫ X n ′ k · P nn ′ g n ′ ( k ) , (5.2)The k · p approximation consists in passing to the limit in U ǫ . Therefore, we define g ǫ kp ( t ) as thesolution of i∂ t g = A ǫ kp g + U g, g ( t = 0) = g in,ǫ (k · p model) (5.3)It is worth noting that the back Fourier transform of g ǫ kp ( t ) which we will denote by f ǫ kp ( t, x ) isa solution of system i∂ t f n = E n ǫ f n − 12 ∆ f n − ǫ X n ′ P nn ′ · ∇ f n ′ + X n ′ V nn ′ f n ′ ,f n ( t = 0) = f in,ǫn ( x ) . (5.4)The diagonalization of the operator A ǫ kp performed in the previous section leads to the effectivemass dynamics i∂ t g = A ǫ em g + U g, g ( t = 0) = g in,ǫ (effective mass model) (5.5)where ( A ǫ em g ) ( k ) = 1 ǫ (cid:16) Λ (2) ( ǫk ) g ( k ) (cid:17) n = (cid:18) E n ǫ + 12 k · M − n k (cid:19) g n ( k ) . (5.6)The solution of (5.5) will be denoted by g ǫ em ( t, k ) and its back Fourier transform f ǫ em ( t, x ) iseasily shown to be the solution of i∂ t f n = 1 ǫ E n f n − 12 div (cid:0) M − n ∇ f ǫn (cid:1) + X n ′ V nn ′ f ǫn ′ ,f n ( t = 0) = f in,ǫn ( x ) . (5.7)This equation is still involving oscillations in time. These oscillations can be filtered by setting f ǫn, em ( t, x ) = h ǫn, em ( t, x )e − iE n tǫ which will be a solution of i∂ t h ǫ em ,n = − 12 div (cid:0) M − n ∇ h ǫ em ,n (cid:1) + X n ′ e iω nn ′ t/ǫ V nn ′ h ǫ em ,n ′ ,h ǫ em ,n ( t = 0) = f in,ǫn ( x ) , (5.8)where ω nn ′ = E n − E n ′ . (5.9)The limit h em ,n of these function is the solution of the system i∂ t h em ,n = − 12 div (cid:0) M − n ∇ h em ,n (cid:1) + V nn h em ,n , h em ,n ( t = 0) = f inn ( x ) , (5.10)where f inn ( x ) is the limit as ǫ tends to zero of f in,ǫn ( x ), and which will be made precise later on. Remark 5.1 The external-potential operators U ǫ and U have been defined in (3.6) and (3.8) .The free k · p operator A ( ξ ) and the effective mass operator Λ (2) ( ξ ) (see definitions (4.3) and ) are now re-introduced as operators acting in L . Recalling definition (4.10) , we shallalso consider the diagonal k · p operator (Λ ǫ g ) n ( k ) = 1 ǫ (Λ( ǫk ) g ( k )) n = 1 ǫ λ n ( ǫk ) g n ( k ) . (5.11) The operators A ǫ kp , A ǫ em and Λ ǫ are “fibered” self-adjoint operators in L , with fiber space ℓ .It is well known (see Ref. [18]) that a fibered self-adjoint operator L in L has self-adjointnessdomain D ( L ) = n g ∈ L (cid:12)(cid:12)(cid:12) g ( ξ ) ∈ D ( L ( ξ )) a.e. ξ ∈ R d and Z R d k L ( ξ ) g ( ξ ) k ℓ dξ < ∞ o , where D ( L ( ξ )) is the self-adjointness domain of L ( ξ ) in ℓ . Assuming V ∈ W (Definition 2.7), we know from Lemma 3.2 that U ǫ and U are bounded (and,clearly, symmetric). Therefore, A ǫ kp + U ǫ , A ǫ kp + U and A ǫ em + U ǫ are the generators of theunitary evolution groups G ǫ ( t ) = e − it ( A ǫ kp + U ǫ ) , G ǫ kp ( t ) = e − it ( A ǫ kp + U ) , G ǫ em ( t ) = e − it ( A ǫ em + U ) . Our goal is to compare, in the limit of small ǫ , the three mild solutions of Eqs. (5.1), (5.3) and(5.5), i.e. g ǫ ( t ) = G ǫ ( t ) g in,ǫ , g ǫ kp ( t ) = G ǫ kp ( t ) g in,ǫ , g ǫ em ( t ) = G ǫ em ( t ) g in,ǫ , (5.12) Lemma 5.2 Let g in,ǫ ∈ L µ and V ∈ W µ for some µ ≥ (see Definition 2.7). Then, suitableconstants c ( µ, V ) ≥ and c ( µ, V ) ≥ , independent of ǫ , exists such that k g ǫ kp ( t ) k L µ ≤ e c ( µ,V ) t k g in,ǫ k L µ , k g ǫ em ( t ) k L µ ≤ e c ( µ,V ) t k g in,ǫ k L µ , (5.13) for all t ≥ . Proof We prove the lemma only for g kp , the proof for g em being identical. We also skip the ǫ superscript of g in,ǫ . Let α be a fixed multi-index with | α | ≤ µ . For R > 0, consider the boundedmultiplication operators on L (cid:0) m R g (cid:1) n ( k ) = ( k α g n ( k ) , if | k | ≤ R ,0 , otherwise,Moreover, we denote by m ∞ the (unbounded) limit operator (cid:0) m ∞ g (cid:1) n ( k ) = k α g n ( k ). Since m R (with R < ∞ ) commutes with A ǫ kp on D ( A ǫ kp ), then, by applying standard semigroup techniques,we obtain m R g ǫ kp ( t ) = G ǫ kp ( t ) m R g in + Z t G ǫ kp ( t − s ) (cid:2) m R , U (cid:3) g ǫ kp ( s ) ds and, therefore, k m R g ǫ kp ( t ) k L ≤ k m R g in k L + Z t k (cid:2) m R , U (cid:3) g ǫ kp ( s ) k L ds. (5.14)19sing (3.8) and the identity k α − η α = P β<α (cid:0) αβ (cid:1) ( k − η ) α − β η β , we have (cid:0) (cid:2) m ∞ , U (cid:3) g ǫ kp (cid:1) n ( k ) = X n ′ π ) d/ Z R d ( k α − η α ) ˆ V nn ′ ( k − η ) g ǫ kp ,n ′ ( η ) dη = X n ′ π ) d/ X β<α (cid:18) αβ (cid:19) Z R d ( k − η ) α − β ˆ V nn ′ ( k − η ) η β g ǫ kp ,n ′ ( η ) dη Since V ∈ W µ , the potential U αβ ( x, z ) such that ˆ U αβ ( k, z ) = k α − β ˆ V ( k, z ) belongs to W , with k U αβ k W ≤ k V k W µ , and then, using (3.10), we obtain k (cid:2) m ∞ , U (cid:3) g ǫ kp k L ≤ X β<α (cid:18) αβ (cid:19) k U αβ k W k η β g ǫ kp k L ≤ c ( µ, V ) k g ǫ kp k L µ . (5.15)with c ( µ, V ) = (2 d − k V k W µ . Letting R → + ∞ , it is not difficult to show that the dominatedconvergence theorem applies and yieldslim R → + ∞ k (cid:2) m R , U (cid:3) g ǫ kp k L = k (cid:2) m ∞ , U (cid:3) g ǫ kp k L ≤ c ( µ, V ) k g ǫ kp k L µ . Then, passing to the limit for R → + ∞ in (5.14), we get k g ǫ kp ( t ) k L µ ≤ k g in k L µ + c ( µ, V ) Z t k g ǫ kp ( s ) k L µ ds, and, therefore, Gronwall’s Lemma yields inequality (5.13). (cid:4) Let us begin by comparing the exact dynamics g ǫ ( t ) with the k · p dynamics g ǫ kp ( t ). Theorem 5.3 Let g ǫ ( t ) and g ǫ kp ( t ) be respectively the solution of (5.1) and (5.3) . If g in ,ǫ ∈ L µ and V ∈ W µ , for some µ ≥ , then, for any given τ ≥ , a constant C ( µ, V, τ ) ≥ , independentof ǫ , exists such that k g ǫ ( t ) − g ǫ kp ( t ) k L ≤ ǫ µ C ( µ, V, τ ) k g in k L µ , (5.16) for all ≤ t ≤ τ . Proof The function h ǫ ( t ) = g ǫ ( t ) − g ǫ kp ( t ) satisfies the integral equation h ǫ ( t ) = Z t G ǫ ( t − s ) (cid:0) U − U ǫ (cid:1) g ǫ kp ( s ) ds and, therefore, k h ǫ ( t ) k L ≤ Z t k (cid:0) U − U ǫ (cid:1) g ǫ kp ( s ) k L ds. From Lemma 5.2 we have that g ǫ kp ( t ) belongs to L µ for all t and, therefore, we can apply Theorem3.5, which gives k (cid:0) U − U ǫ (cid:1) g ǫ kp ( s ) k L ≤ ǫ µ c µ k V k W µ k g ǫ kp ( s ) k L µ , for a suitable constant c µ . Then we have k h ǫ ( t ) k L ≤ ǫ µ c µ k V k W µ Z t k g ǫ kp ( s ) k L µ ds and, by (5.13), we have that (5.16) holds with C ( µ, V, τ ) = c µ ( e c ( µ,V ) τ − ) c ( µ,V ) k V k W µ . (cid:4) 20e now compare the k · p dynamics g ǫ kp ( t ) with the effective mass dynamics g ǫ em ( t ) (see definitions(5.12)). Recalling the discussion in Sec. 4, we need, as an intermediate step between g ǫ kp ( t ) and g ǫ em ( t ), the function g ǫ ∗ ( t ) = T ∗ ǫ g ǫ kp ( t ), that is g ǫ ∗ ( t ) = T ∗ ǫ G ǫ kp ( t ) g in ,ǫ = exp (cid:2) − it (cid:0) Λ ǫ + T ∗ ǫ U T ǫ (cid:1)(cid:3) T ∗ ǫ g in ,ǫ , (5.17)representing the diagonalized k · p dynamics (definitions (4.10), (4.11) and (5.11)). Lemma 5.4 Let g ǫ em ( t ) and g ǫ ∗ ( t ) be respectively defined by (5.5) and (5.17) . Let g in ,ǫ ∈ L µ and V ∈ W µ , for some µ > , and assume g in ,ǫ = Π N g in (i.e. g in ,ǫ is concentrated in the first N bands). Then, for any given τ ≥ , a suitable constant C ′ ( µ, N, V, τ ) , independent of ǫ , existssuch that k g ǫ ∗ ( t ) − g ǫ em ( t ) k L ≤ ǫ min { µ/ , } C ′ ( µ, N, V, τ ) k g in ,ǫ k L µ , (5.18) for all ≤ t ≤ τ . Proof Let S ǫ Λ ( t ) = exp( − it Λ ǫ ), S ǫ em ( t ) = exp( − itA ǫ em ) and U ǫT := T ∗ ǫ U T ǫ . Then, g ǫ em ( t ) = S ǫ em ( t ) g in ,ǫ + Z t S ǫ em ( t − s ) U g ǫ em ( s ) ds,g ǫ ∗ ( t ) = S ǫ Λ ( t ) T ∗ ǫ g in ,ǫ + Z t S ǫ Λ ( t − s ) U ǫT g ǫ ∗ ( s ) ds. Putting h ǫ = g ǫ ∗ − g ǫ em , we can write h ǫ ( t ) = ( S ǫ Λ − S ǫ em ) ( t ) g in ,ǫ + S ǫ Λ ( t ) ( T ∗ ǫ − I ) g in ,ǫ + Z t S ǫ Λ ( t − s ) U ǫT h ǫ ( s ) ds + Z t S ǫ Λ ( t − s ) (cid:0) U ǫT − U (cid:1) g ǫ em ( s ) ds + Z t ( S ǫ Λ − S ǫ em ) ( t − s ) U g ǫ em ( s ) ds. (5.19)From the effective mass theorem, Theorem 4.7, a constant C ( µ, N, t ) exists such that k ( S ǫ Λ − S ǫ em ) ( t ) g in ,ǫ k L ≤ ǫ min { µ/ , } C ( µ, N, t ) k g in ,ǫ k L µ . (5.20)Moreover, from Lemma 5.2 we have that both g ǫ em ( t ) and U g ǫ em ( t ) belong to L µ for all t , andthat a constant C ( µ, V, t ) ≥ kU g ǫ em ( t ) k L µ ≤ C ( µ, V, t ) k g in ,ǫ k L µ (5.21)(this stems, in particular, from the commutator inequality (5.15), which still holds for g ǫ em ).This inequality, together with Theorem 4.7, yields k ( S ǫ Λ − S ǫ em ) ( t − s ) U g ǫ em ( s ) k L ≤ ǫ min { µ/ , } C ( µ, N, t − s ) k g in ,ǫ k L µ (5.22)for a suitable constant C ( µ, N, t ) ≥ 0. In order to estimate the last integral in (5.19), let uswrite (cid:0) U ǫT − U (cid:1) g ǫ em ( s ) = ( T ∗ ǫ − I ) U g ǫ em ( s ) + T ∗ ǫ U ( T ǫ − I ) g ǫ em ( s ) . Using inequalities (4.13) and (5.21) we see that another constant C ( µ, N, V, t ) ≥ k (cid:0) U ǫT − U (cid:1) g ǫ em ( s ) k L ≤ ǫ min { µ, } C ( µ, N, V, t ) k g in ,ǫ k L µ . (5.23)21n conclusion, from inequalities (5.20), (5.22) and (5.23), and from Eq. (5.19), we get k h ǫ ( t ) k L ≤ ǫ min { µ/ , } C ( µ, N, V, τ ) k g in ,ǫ k L µ + k V k W Z t k h ǫ ( s ) k L ds, for all 0 ≤ t ≤ τ (here we also used the fact that all the estimation constants introduced sofar are non-decreasing with respect to time). Hence, inequality (5.18), with C ′ ( µ, N, V, τ ) =e τ k V k W C ( µ, N, τ, V ), follows from Gronwall’s Lemma. (cid:4) Theorem 5.5 Let g ǫ kp ( t ) and g ǫ em ( t ) as in (5.12) , and assume g in,ǫ ∈ L µ , with a uniform boundas ǫ tends to zero. Moreover, assume V ∈ W µ for some µ > . Then lim ǫ → k g ǫ kp ( t ) − g ǫ em ( t ) k L =0 , uniformly in bounded time-intervals. If, in addition, g in,ǫ = Π N g in,ǫ , for some N then, forany given τ ≥ , a constant C ′′ ( µ, N, V, τ ) ≥ , independent of ǫ , exists such that k g ǫ kp ( t ) − g ǫ em ( t ) k L ≤ ǫ min { µ/ , } C ′′ ( µ, N, V, τ ) k g in,ǫ k L µ , (5.24) for all ≤ t ≤ τ . Proof We begin by the second statement, assuming g in,ǫ = Π N g in,ǫ . Using inequalities (4.13)and (5.18), and recalling definition (5.17), we can write k g ǫ kp ( t ) − g ǫ em ( t ) k L ≤ k ( T ∗ ǫ − I ) g ǫ kp ( t ) k L + k g ǫ ∗ ( t ) − g ǫ em ( t ) k L ≤ ǫ min { µ, } C ( µ, N ) k g ǫ kp ( t ) k L µ + ǫ min { µ/ , } C ′ ( µ, N, V, τ ) k g in,ǫ k L µ , for 0 ≤ t ≤ τ . Then, using also (5.13), inequality (5.24) follows. If now g in,ǫ simply belongs to L µ , then for any fixed N we can write k ( g ǫ kp − g ǫ em )( t ) k L ≤ k ( G ǫ kp − G ǫ em )( t )Π N g in,ǫ k L + k ( G ǫ kp − G ǫ em )( t )Π cN g in,ǫ k L ≤ ǫ min { µ/ , } C ′′ ( µ, N, τ ) k g in,ǫ k L µ + 2 k Π cN g in,ǫ k L , for all 0 ≤ t ≤ τ . Since Π cN g in,ǫ → L as N → ∞ , then we can fix N large enough and,successively, ǫ small enough (uniformly in 0 ≤ t ≤ τ , by assumption) so that k g ǫ kp ( t ) − g ǫ em ( t ) k L is arbitrarily small, which proves our assertion. (cid:4) The following result is a direct consequence of the above comparisons. Corollary 5.6 Assume that the envelope functions f in,ǫ ( x ) are bounded in H µ for some µ > and that the potential V ( x, z ) belong to W µ . Then we have the local uniform in time convergence lim ǫ → k f ǫ ( t ) − f ǫ em ( t ) k L = 0 , where f ǫ ( t, x ) and f ǫ em ( t, x ) are the respective solutions of (3.4) and (5.7)We are now able to prove the following theorem. Theorem 5.7 Let h ǫ em ( t, x ) and h em ( t, x ) be the mild solutions of, respectively, Eq. (5.8) andEq. (5.10) . Assume lim ǫ → k f in,ǫ − f in k L = 0 and assume that µ > exists such that V ∈ W µ and f in,ǫ is bounded uniformly in H µ . Then lim ǫ → k h ǫ em ( t ) − h em ( t ) k L = 0 , uniformly in bounded time intervals. roof Since the dynamics generated by (5.8) and (5.10) both preserve the L norm, we canassume without loss of generality that the initial condition f in,ǫ and f in are identical and replacethem by the notation h in ∈ H µ . We consider the diagonal operator H in L ( H h ) n ( x ) = 12 div (cid:0) M − n ∇ h n (cid:1) ( x ) + V nn ( x ) h n ( x ) . We recall that the matrix V nn ′ defines a bounded operator on L (that is, the operator U inposition variables, see definition (3.8)). Such operator, as well as its diagonal and off-diagonalparts are bounded operators with bound k V k W (see Lemma 3.2). Then, H is self-adjoint onthe domain D ( H ) = ( h ∈ L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h n ∈ H ( R d ) , X n k div( M − n ∇ h n ) k L ( R d ) < ∞ ) . Let S ( t ) = exp( − itH ) denote the (diagonal) unitary group generated by H . Moreover weconsider the operator R ǫ ( t ) given by( R ǫ ( t ) h ) n ( x ) = X n ′ = n e iω nn ′ t/ǫ V nn ′ ( x ) h n ′ ( x ) , which, being unitarily equivalent to the off-diagonal part of U , is again bounded by k V k W (forall t ). The two mild solutions satisfy h ǫ em ( t ) = S ( t ) h in + Z t S ( t − s ) R ǫ ( s ) h ǫ em ( s ) ds, h em ( t ) = S ( t ) h in , and, therefore, what we need to do is proving that h ǫ em ( t ) − h em ( t ) = Z t S ( t − s ) R ǫ ( s ) h ǫ em ( s ) ds goes to zero as ǫ → 0. To this aim we resort to the usual cutoff. For any fixed N ∈ N wedecompose the right hand side of the previous equation h ǫ em ( t ) − h em ( t ) = I N ( t ) + I cN ( t ) , where, using the projection operators Π N and Π cN = I − Π N , introduced in the proof of Theorem4.5, we have put I N ( t ) = Z t S ( t − s )Π N R ǫ ( s )Π N h ǫ em ( s ) ds,I cN ( t ) = Z t S ( t − s ) [Π cN R ǫ ( s )Π N + Π N R ǫ ( s )Π cN + Π cN R ǫ ( s )Π cN ] h ǫ em ( s ) ds. Case of regular data We assume in this part that V ∈ W and that h in ∈ H . We fix a δ > τ . Because R ǫ ( t ) is uniformly bounded and k h ǫ em ( t ) k L = k h in k L then, clearly, a number N ( δ, τ ) (independent of ǫ ) exists such that k I cN ( t ) k L ≤ δ , forall N ≥ N ( δ, τ ) and 0 ≤ t ≤ τ . We now turn our attention to I N ( t ). Using the assumption V ∈ W , it is not difficult to prove the following facts:(i) for every N , if h ∈ H then Π N h ∈ D ( H ), and a constant C N exists such that k H Π N h k L ≤ C N k h k H ; 23ii) for every N a constant C ′ N , independent of t and ǫ , exists such that, if h ∈ H , then k Π N R ǫ ( t )Π N h k H ≤ C ′ N k h k H .Moreover, in a similar way to Lemma 5.2, we can prove the following:(iii) if h in ∈ H , then h ǫ em ( t ) ∈ H for all t and a function C ( t ), bounded on bounded timeintervals and independent of ǫ , exists such that k h ǫ em ( t ) k H ≤ C ( t ) k h in k H .Using (i), (ii) and (iii) we have that Π N R ǫ ( s )Π N h ǫ ( s ) ∈ D ( H ) and, therefore, S ( t − s )Π N R ǫ ( s )Π N h ǫ em ( s )is continuously differentiable in s . This makes possible to perform an integration by parts in theintegral defining I N ( t ). Since R ǫ ( t ) = ǫ Z R ǫω ( t ) dt, where ( R ǫω ( t ) h ) n ( x ) = X n ′ = n iω nn ′ e iω nn ′ t/ǫ V nn ′ ( x ) h n ′ ( x ) , then the integration by parts yields I N ( t ) = ǫ S ( t − s )Π N R ǫω ( s )Π N h ǫ em ( s ) (cid:12)(cid:12)(cid:12) s = ts =0 − ǫ Z t S ( t − s )Π N (cid:20) iH R ǫω ( s )Π N h ǫ em ( s ) + R ǫω ( s )Π N dds h ǫ em ( s ) (cid:21) ds, where, of course, dds h ǫ em ( s ) = H h ǫ em ( s ) + R ǫ ( s ) h ǫ em ( s ) . Since Π N R ǫω ( t ) is uniformly bounded by some constant dependent of N (in particular, suchconstant will depend of 1 / min { ω nn ′ | n ′ = n, n ≤ N } ), then, from (i), (ii) and (iii), and usingΠ N H = H Π N , we obtain that a constant C N ( τ ), independent of ǫ , exists such that k I N ( t ) k L ≤ ǫ C N ( τ ) k h in k H , ≤ t ≤ τ. Thus, fixing N ≥ N ( δ, τ ), a ǫ small enough exists such that k I N ( t ) k L ≤ δ , for all 0 ≤ t ≤ τ .For such N and ǫ we have, therefore, k h ǫ em ( t ) − h em ( t ) k L ≤ k I N ( t ) k L + k I cN ( t ) k L ≤ δ, which proves the theorem in the regular case. Case of general data If µ ≥ 2, then there is nothing to do. Let us assume 0 < µ < δ be a regularizing parameter and let h in δ and V δ be two regularizations of h in and of V suchthat h in δ ∈ H , lim δ → k h in δ − h in k H µ = 0and V δ ∈ W , lim δ → k V δ − V k W µ = 0 . Let h ǫ em ,δ and h em ,δ be the corresponding solutions of (5.8) and (5.10) with the modified initialdata and potential. Then we have k ( h ǫ em − h em )( t ) k L ≤ k ( h ǫ em − h ǫ em ,δ )( t ) k L ++ k ( h ǫ em ,δ − h em ,δ )( t ) k L + k ( h em ,δ − h em )( t ) k L . δ > 0, the second term of theright hand side tends to zero as ǫ tends to zero. Thanks to Theorem 3.5, it is easy to show thatthe third term of the right hand side tends to zero as δ tends to zero and that the first term ofthe right hand also tends to zero as δ tends to zero uniformly in ǫ . (cid:4) In this section, we prove the convergence of the particle density towards the superposition ofthe envelope function densities. Namely, we have the following theorem. Theorem 5.8 Let the initial datum ψ in ,ǫ ∈ L ( R d ) be such that its envelope functions ( f in,ǫn ) form a bounded sequence in H µ which strongly converges in L towards the initial datum f in =( f inn ) , and assume that there exists a positive µ such that V ∈ W µ . Then for any given function θ ∈ L ( R d ) such that b θ ∈ L ( R d ) , the following convergence holds locally uniformly in time: lim ǫ → Z | ψ ǫ ( t, x ) | θ ( x ) dx = X n Z θ ( x ) | h em ,n ( t, x ) | dx, where ψ ǫ is the solution of (2.30) and h em is the solution of (5.10) . Proof let h ǫn ( t, x ) = f ǫn ( t, x )e iE n t/ǫ where f ǫn are the envelope functions of ψ ǫ We deduce fromthe results of the above subsection, in particular from Theorem 5.7, thatlim ǫ → X n k h ǫn ( t ) − h em ,n ( t ) k L ( R d ) = 0 . Let ˜ θ ǫ = T ǫ ( θ ) , e h ǫn = T ǫ ( h ǫn ) , e h ǫ em ,n = T ǫ ( h em ,n )where the truncation operator T γ has been defined in (2.28). Recalling that ψ ǫ ( t, x ) = |C| / X n h ǫn ( t, x )e − iE n t/ǫ v ǫn ( x )let us define ˜ ψ ǫ ( t, x ) = |C| / X n ˜ h ǫn ( t, x )e − iE n t/ǫ v ǫn ( x ) . It is readily seen, in view of (2.29) that k ψ ǫ ( t ) − ˜ ψ ǫ ( t ) k L = X n k h ǫn ( t ) − ˜ h ǫn ( t ) k L ≤ Cǫ µ k h ǫ ( t ) k H µ , where, by Lemma 5.2, k h ǫ ( t ) k H µ remains bounded. It is now clear that (cid:12)(cid:12)(cid:12)(cid:12)Z θ ( x ) | ψ ǫ ( t, x ) | dx − Z θ ( x ) (cid:12)(cid:12)(cid:12) ˜ ψ ǫ ( t, x ) (cid:12)(cid:12)(cid:12) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) k ψ ǫ ( t ) k L − k ˜ ψ ǫ ( t ) k L (cid:12)(cid:12)(cid:12) k θ k L ∞ goes to 0 and, therefore, we can replace ψ ǫ by ˜ ψ ǫ . Now, (cid:12)(cid:12)(cid:12)(cid:12)Z θ ( x ) (cid:12)(cid:12)(cid:12) ˜ ψ ǫ ( t, x ) (cid:12)(cid:12)(cid:12) dx − Z ˜ θ ǫ ( x ) (cid:12)(cid:12)(cid:12) ˜ ψ ǫ ( t, x ) (cid:12)(cid:12)(cid:12) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ˜ ψ ǫ ( t ) k L k ˜ θ ǫ − θ k L ∞ → θ by ˜ θ ǫ . But Z ˜ θ ǫ ( x ) (cid:12)(cid:12)(cid:12) ˜ ψ ǫ ( t, x ) (cid:12)(cid:12)(cid:12) dx = |C| *X n ˜ θ ǫ ( x )˜ h ǫn ( t, x )e − iE n t/ǫ v ǫn ( x ) , X n ˜ h ǫn ( t, x )e − iE n t/ǫ v ǫn ( x ) + and supp( d ˜ θ ǫ ˜ h ǫn ) ⊂ B / ǫ + B / ǫ ⊂ B /ǫ . Therefore the Parseval formula (2.18) shows that Z ˜ θ ǫ ( x ) (cid:12)(cid:12)(cid:12) ˜ ψ ǫ ( t, x ) (cid:12)(cid:12)(cid:12) dx = X n Z e θ ǫ ( x ) (cid:12)(cid:12)(cid:12)e h ǫn ( t, x ) (cid:12)(cid:12)(cid:12) dx → X n Z θ ( x ) | h em ,n ( t, x ) | dx, which completes the proof of the theorem. (cid:4) One of the most restrictive hypotheses that we made in the previous sections is the simplicityof all the eigenvalues of the periodic operator H L . The question of simplicity of the eigenvaluesis central in this problem as already has been noticed in the works of Poupaud and Ringhofer[16] and of Allaire and Piatnistki [3]. In these references, the authors do not assume that allthe eigenvalues are simple but assume that the initial datum is concentrated on finite numberof bands who have multiplicity 1. The difference between our approach and that of thesetwo references is that ours allows for a an infinite number of envelope functions. Besides, thehypothesis of simplicity of all the eigenvalues at k = 0 can be removed and replaced by the factthat the initial datum envelope functions corresponding to multiple eigenvalues are vanishing.The proof has however to be reshuffled and we have chosen to stick to the restrictive hypothesisof simple eigenvalues. Let us however briefly explain how we can deal with this problem. Oneimportant step is the diagonalization of the k · p Hamiltonian which gives rise to the equation(5.17). In this formula the operator Λ ǫ is diagonal in the n index while T ∗ ǫ U T ǫ is not (theexistence of the unitary transformation is still valid even in the case of multiple eigenvalues; itis continuous, but not regular for eigenvalues with multiplicity larger than one). Because of theseparation of the eigenvalues, it is easy to show that the eigenspaces with different energies aredecoupled from each other (adiabatic decoupling) and we can replace T ∗ ǫ U T ǫ by U nn δ nn ′ . Ifthe initial data are only concentrated on modes with multiplicity one, then the solution itselfis almost concentrated on these modes and for these modes, we can make the expansion ofeigenvalues and obtain the effective mass equation (5.10). Let us also mention a recent work byF. Fendt-Delebecque and F. M´ehats [10] where the effective mass approximation is performedfor the Schr¨odinger equation with large magnetic field and which relies on large time averagingof almost periodic functions. This approach might be of help for analyzing the limit for multipleeigenvalues.One final question which has not been addressed so far is the relationship between the regularityof function ψ and that of its corresponding sequence f ǫ of envelope functions. In particular, onemay look for sufficient conditions on ψ so that f ǫ ∈ H µ . Since the envelope function is a Fourierlike expansion of the function ψ on the basis v n , then their decay as n becomes bigger dependsnot only on the regularity of ψ but also on that of the basis ( v n ) which itself will depend on26he regularity of the potential W L . We show in the following subsection some results in thisdirection. In this section we study the asymptotic behavior as ǫ tends to zero of the scaled envelopefunctions relative to the basis ( v n ) defined in (2.25).From (2.19), it is readily seen that the limit as ǫ tends to zero of the envelope function is givenby lim ǫ → ˆ f ǫn ( k ) = lim ǫ → |B | − / Z R d B /ǫ ( k ) e − ik · x v n (cid:16) xǫ (cid:17) ψ ( x ) dx = |B | − / |C| − h v n , i Z R d e − ik · x ψ ( x ) dx = |C| − / h v n , i b ψ ( k ) . Therefore lim ǫ → π ǫn ( ψ ) = |C| − / h v n , i ψ. (6.1)The following Proposition, shows that the regularity of the crystal potential leads to decayproperties on the coefficients h v n , i = R C v n ( x ) dx . Proposition 6.1 Let W L be in C ∞ . Then for any integer p , the coefficients h v n , i satisfy theinequality |h v n , i| ≤ C p E pn , where C p is a constant only depending on k W L k W p, ∞ . Proof We first remark that E pn h v n , i = (cid:10) H p L v n , (cid:11) = (cid:10) v n , H p L (cid:11) (where H p L denotes the p -th power of H L , not to be confused with the notation H ǫ L introducedin Sec. 2). Now it is readily seen that if W L ∈ W p, ∞ , then H p L ∈ L ∞ with k H p L k L ∞ ≤ C k W L k W p, ∞ , for a suitable constant C ≥ 0. Then E pn |h v n , i| ≤ k v n k L k H p L k L ≤ C p , with C p only depending on k W L k W p, ∞ , which ends the proof. (cid:4) We also have the following property. Lemma 6.2 Let λ and λ ′ two elements of the reciprocal lattice L ∗ . Assume that W L ∈ C ∞ .Then, for any integers k, p , we have the estimate (cid:12)(cid:12)(cid:12)D H k L e iλ · x , H k L e iλ ′ · x E(cid:12)(cid:12)(cid:12) ≤ C k,p (1 + | λ | k | λ ′ | k )1 + | λ − λ ′ | p , for a suitable constant C k,p ≥ . Proof It is clear that H k L e iλ · x = P k | α | =0 λ α V α ( x )e iλ · x , where V α contains products of W L andits derivatives up to order 2 k − | α | . Therefore D H k L e iλ · x , H k L e iλ ′ · x E = k X | α | , | β | =0 λ α ( λ ′ ) β Z C V α ( x ) V β ( x ) e i ( λ − λ ′ ) · x dx. p times. (cid:4) The estimate of Lemma 6.2 is not optimal and can certainly be refined, but this is not the scopeof our paper. Next proposition follows from the previous result. Proposition 6.3 Assume W L ∈ L ∞ and let f ǫn = π ǫn ( ψ ) be the envelope functions of ψ . Thenthe following estimate holds for any µ ≥ : k f ǫ k L µ = X n k (1 + | k | ) µ/ c f ǫn ( k ) k L ≤ C µ k ψ k H µ . (6.2) Let now W L be in C ∞ , then the following estimate holds for any integer s X n E sn k f ǫn k L ≤ C s ( k ψ k L + ǫ s k ψ k H s ) . (6.3) Proof Let us first prove (6.2). Using the identity c f ǫn ( k ) = |B | − / Z R d B /ǫ ( k ) e − ik · x v n (cid:16) xǫ (cid:17) ψ ( x ) dx, as well as the decomposition v n ( x ) = 1 |C| / X λ ∈L ∗ v n,λ e iλ · x where v n,λ = D v n , e iλ · x |C| / E , we obtain, X n k (1 + | k | ) µ/ c f ǫn ( k ) k L = X n X λ,λ ′ Z B /ǫ (1 + | k | ) µ v n,λ v n,λ ′ b ψ (cid:0) k − λǫ (cid:1) b ψ (cid:16) k − λ ′ ǫ (cid:17) dk. Summing first with respect to n and using the identity X n v n,λ v n,λ ′ = 1 |C| h e iλ · x , e iλ ′ · x i = δ λ,λ ′ , the right hand side of the above identity takes the simple form X n k (1 + | k | ) µ/ c f ǫn ( k ) k L = X λ ∈L ∗ Z B /ǫ (1 + | k | ) µ (cid:12)(cid:12)(cid:12) b ψ (cid:0) k − λǫ (cid:1)(cid:12)(cid:12)(cid:12) dk. It is now readily seen that there exists a constant c ≥ 1, only depending on the fundamental cell C , such that for all k ∈ B and for all λ ∈ L ∗ , we have the estimate | k | ≤ c | k − λ | , so that X n k (1 + | k | ) µ/ c f ǫn ( k ) k L ≤ c µ X λ ∈L ∗ Z B /ǫ (cid:16) (cid:12)(cid:12) k − λǫ (cid:12)(cid:12) (cid:17) µ (cid:12)(cid:12)(cid:12) b ψ (cid:0) k − λǫ (cid:1)(cid:12)(cid:12)(cid:12) dk = c µ Z R d (1 + | k | ) µ (cid:12)(cid:12)(cid:12) b ψ ( k ) (cid:12)(cid:12)(cid:12) dk. C µ exists such that (6.2) holds. Let us now prove (6.3).We proceed analogously and find X n E sn k f ǫn k L = 1(2 π ) d X n X λ,λ ′ Z B /ǫ E sn v n,λ v n,λ ′ b ψ (cid:0) k − λǫ (cid:1) b ψ (cid:16) k − λ ′ ǫ (cid:17) dk. As above, we first make the sum over the index n and, therefore, we need to evaluate X n E sn v n,λ v n,λ ′ . We first remark that E sn v n,λ = D H s L v n , e iλ · x |C| / E = D v n , H s L e iλ · x |C| / E . Therefore X n E sn v n,λ v n,λ ′ = 1 |C| D H s L e iλ · x , e iλ ′ · x E . Contrary to the proof of (6.2), the obtained formula is not diagonal in ( λ, λ ′ ) but Lemma 6.2leads to the following estimate, which holds for large enough integers p : X n E sn k f ǫn k L ≤ C s,p X λ,λ ′ Z B /ǫ | λ | s | λ − λ ′ | p (cid:12)(cid:12)(cid:12) b ψ (cid:0) k − λǫ (cid:1) b ψ (cid:16) k − λ ′ ǫ (cid:17)(cid:12)(cid:12)(cid:12) dk ≤ C s,p X λ,λ ′ Z B /ǫ | λ | s | λ − λ ′ | p (cid:20)(cid:12)(cid:12)(cid:12) b ψ (cid:0) k − λǫ (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) b ψ (cid:16) k − λ ′ ǫ (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) dk ≤ C s,p X λ ∈L ∗ Z B /ǫ (1 + | λ | s ) (cid:12)(cid:12)(cid:12) b ψ (cid:0) k − λǫ (cid:1)(cid:12)(cid:12)(cid:12) dk. Note that we used the fact that, for large enough p , the following estimates hold with constants C and C only depending on s and p X λ ∈L ∗ | λ | s | λ − λ ′ | p ≤ C (1 + (cid:12)(cid:12) λ ′ (cid:12)(cid:12) s ) , X λ ′ ∈L ∗ | λ | s | λ − λ ′ | p ≤ C (1 + | λ | s ) . Now, for λ = 0 and ǫk ∈ B it is readily seen that | λ | ≤ c | λ − ǫk | , where c is a positive constantindependent of λ and k . Therefore, X λ ∈L ∗ Z B /ǫ (1 + | λ | s ) (cid:12)(cid:12)(cid:12) b ψ (cid:0) k − λǫ (cid:1)(cid:12)(cid:12)(cid:12) dk ≤ k ψ k L + ǫ s c s k ψ k H s , which implies that a suitable constant C s exists such that (6.3) holds. (cid:4) This section is devoted to the proofs of some results stated in the beginning of the paper. For any Schwartz function ψ we can write ψ ( x ) = (2 π ) − d/ Z R d ˆ ψ ( k ) e ik · x dk = X η ∈L ∗ (2 π ) − d/ Z B + η ˆ ψ ( k ) e ik · x dk X η ∈L ∗ (2 π ) − d/ e iη · x Z B ˆ ψ ( ξ + η ) e iξ · x dξ = X η ∈L ∗ e iη · x G η ( x ) , where G η ( x ) = (2 π ) − d/ Z B ˆ ψ ( ξ + η ) e iξ · x dξ clearly belongs to F ∗ L B ( R d ). Moreover, we have X η ∈L ∗ k G η k L = X η ∈L ∗ k ˆ G η k L = X η ∈L ∗ Z R d (cid:12)(cid:12)(cid:12) ˆ ψ ( ξ + η ) B ( ξ ) (cid:12)(cid:12)(cid:12) dk = X η ∈L ∗ Z B + η (cid:12)(cid:12)(cid:12) ˆ ψ ( ξ ) (cid:12)(cid:12)(cid:12) dk = k ψ k L . Thus, defining F ( x, y ) = X η ∈L ∗ e iη · x G η ( y ) , ( x, y ) ∈ C × R d , (7.1)we have that F ∈ L ( C × R d ) and |C| − k F k L ( C× R d ) = X η ∈L ∗ k G η k L ( R d ) = k ψ k L ( R d ) (where we used the fact that {|C| − / e iη · x | η ∈ L ∗ } is a orthonormal basis of L ( C )). Since { v n | n ∈ N } is another orthonormal basis of L ( C ), then we can also write F ( x, y ) = |C| / X n f n ( y ) v n ( x ) , where f n ( y ) = |C| − / h F ( · , y ) , v n i L ( C ) . (7.2)Note that ˆ f n ∈ L B ( R d ) for every n and that k ψ k L ( R d ) = |C| − k F k L ( C× R d ) = X n k f n k L ( R d ) . For y = x , (7.1) yields (2.14), at least for Schwartz functions. However, it can be easily provedthat the mapping ψ ( f , f , . . . ) can be uniquely extended to an isometry between L ( R d )and ℓ ( N , F ∗ L B ( R d )), with the properties (2.14) and (2.15). Recalling definition (2.28), let ˜ θ ǫ = T ǫ ( θ ) , ˜ f ǫn = T ǫ ( f ǫn ) , and define ˜ ψ ǫ ( x ) = |C| / X n ˜ f ǫn ( x ) v ǫn ( x ) . Z R d θ ( x ) h | ψ ( x ) | − X n | f ǫn ( x ) | i dx = Z R d θ ( x ) h | ψ ( x ) | − | ˜ ψ ǫ ( x ) | i dx + Z R d h θ ( x ) − ˜ θ ǫ ( x ) i | ˜ ψ ǫ ( x ) | dx + Z R d ˜ θ ǫ ( x ) h | ˜ ψ ǫ ( x ) | − X n | ˜ f ǫn ( x ) | i dx + Z R d ˜ θ ǫ ( x ) X n h | ˜ f ǫn ( x ) | − | f ǫn ( x ) | i dx + Z R d h ˜ θ ǫ ( x ) − θ ( x ) i X n | f ǫn ( x ) | dx = I + I + I + I + I . Since supp( d ˜ θ ǫ ˜ f ǫn ) ⊂ B / ǫ + B / ǫ ⊂ B /ǫ , then ˜ θ ǫ ˜ f ǫn are the envelope functions of ˜ θ ǫ ˜ ψ ǫ and theParseval identity (2.18) can be applied to the functions ˜ ψ ǫ and ˜ θ ǫ ˜ ψ ǫ , which yields I = 0.As far as the terms I and I are concerned, we have | I | ≤ k θ − ˜ θ ǫ k L ∞ k ˜ ψ ǫ k L = k θ − ˜ θ ǫ k L ∞ X n k ˜ f ǫn k L ≤ k θ − ˜ θ ǫ k L k ψ k L and, therefore, I → ǫ → 0. Similarly we can prove that I → R is the radius of a ball contained in B , we have | I | ≤ k θ k L ∞ (cid:16) k ψ k L − k ˜ ψ ǫ k L (cid:17) = k θ k L ∞ X n (cid:16) k f ǫn k L − k ˜ f ǫn k L (cid:17) ≤ k θ k L ∞ X n Z | k | > R ǫ (cid:12)(cid:12)(cid:12) ˆ f ǫn ( k ) (cid:12)(cid:12)(cid:12) dk. The last integral goes to 0 as ǫ → 0, because P n R R d | ˆ f ǫn ( k ) | dk = k ψ k L and the dominatedconvergence theorem applies. Thus I → I → Z R d θ ( x ) h | ψ ( x ) | − X n | f ǫn ( x ) | i dx → ǫ → 0, which proves the theorem. A cknowledgements. N. 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