Homogenization Limits and Wigner Transforms

Homogenization Limits and Wigner Transforms PATRICK GÉRARD Université de Paris–Sud

PETER A. MARKOWICH NORBERT J. MAUSER TU–Berlin

AND FRÉDÉRIC POUPAUD Université Nice Abstract We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure. The very general theory is illustrated by typical examples like (semi)classical limits of Schrödinger equations (with or without a periodic potential), the homogenization limit of the c 1997 acoustic equation in a periodic medium, and the classical limit of the Dirac equation. John Wiley & Sons, Inc.

0 Introduction We consider the following type of initial value problems: (0.1)

ε∂t uε + P ε uε = 0 ,

uε (t = 0, x) = uεI (x) ,

where ε is a small parameter, uε (t, x) is a vector-valued L2 -function on Rm x , and P ε is an anti-self-adjoint, matrix-valued (pseudo)-differential operator with a Weyl symbol given by P 0 (x, x/ε, εξ) + O(ε). Here P 0 = P 0 (x, y, ξ) is a smooth function that is periodic with respect to y. By ξ we denote the conjugate variable to the position x; that is, ξ = −i∇x . The main assumptions are that the data uεI are bounded in L2 as ε goes to 0 and that uεI oscillates at most at frequency 1/ε; for instance, Z C |∇uεI (x)|2 dx ≤ 2 . ε A more general formulation of the assumptions on uεI is given in definitions (1.26) and (1.27) below. Communications on Pure and Applied Mathematics, Vol. L, 0323–0379 (1997) c 1997 John Wiley & Sons, Inc.

CCC 0010–3640/97/040323-57

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Since P ε is anti-self-adjoint, the following conservation law holds: Z d (0.2) |uε (t, x)|2 dx = 0 . dt Rm In this paper we study the homogenization limit as ε → 0 of the following scalar quantity, which we call energy density: nε (t, x) = |uε (t, x)|2 . According to the physical context of the equation, the function nε may have different interpretations: position density in the case of the Schrödinger equation and energy density in the case of high-frequency wave propagation. The above conservation law shows that nε (t) is uniformly bounded in 1 L (Rm x ), hence we may assume, up to the extraction of a subsequence, that it converges weakly, as ε goes to 0, to a bounded, time-dependent, nonnegative measure n0 (t, x). The main question we address in this paper is to calculate the homogenized energy density n0 (t, x) from quantities related to the data uεI . Clearly, the knowledge of n0I (x) as the weak limit of |uεI (x)|2 is not sufficient, as suggested by a brief inspection of the WKB method (see, for instance, [2]). The main idea is to construct a time-dependent positive measure w0 (t, x, ξ) on the phase space, called the Wigner measure, such that Z n0 (t, x) = (0.3) w0 (t, x, dξ) . ξ∈Rm

The advantage is that w0 can be calculated through transport equations with initial data given by quantities related to uεI . The introduction of such an object in the literature seems to be due to E. Wigner [23] in the context of semiclassical quantum mechanics. Much later on, a similar object was introduced by A. Shnirelman [21] and other authors [25, 3, 12] in spectral theory in order to prove a basic result in what is sometimes called quantum chaos. Finally, a general relevant notion was raised in the late 1980s by L. Tartar [22] and the first author [5, 8] independently, under the names of Hmeasures and microlocal defect measures, respectively. A notable difference of the latter measures with respect to the Wigner measure is that they can be associated to any bounded sequence of L2 , in particular without any assumption about the existence of a typical length or Planck constant ε. On the other hand, this generality forces us to deal with measures on the smaller phase space {(x, ξ) : x ∈ Rm , ξ ∈ Rm , |ξ| = 1}, which is somewhat less precise. Another improvement is that, in the case of vector-valued sequences, matrix-valued

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measures can be defined in the same way to study polarization phenomena, compensated compactness, and so on. More recently Wigner measures were revisited by P. L. Lions and T. Paul [15] and the first author [7]. The motivation of the latter reference was the study of the semiclassical Schrödinger equation with a periodic potential V (x/ε), a well-known problem in solid-state physics. The advantage of Wigner measures in this context is that they allow us to overcome the difficulty caused by crossings of Bloch modes. This work was improved by the three other authors [18], who introduced a third variant of Wigner measures, where the momentum variable ξ is restrained to live in the Brillouin zone, leading to the semiclassical justification of Vlasov equations for semiconductor materials. Our objective in this paper is to give a self-contained survey of Wigner measures, including a treatment of general systems as (0.1), paying special attention to the problems induced by vector-valued solutions. In particular, in the case of slowly varying coefficients, we derive transport equations for the matrix-valued Wigner measures related to each mode of the system, involving zeroth-order terms that describe the evolution of polarization. In the case of oscillating periodic coefficients, we extend the approach of [7] and [18] to a general framework. However, at this time we are not able to deal with symbols depending on both x and x/ε. This difficulty is related to the existence of crossings of Bloch modes, conjugated with the possibility that the integral curves of our transport equations meet the “bad set” where such a crossing occurs. Hence, in particular, we are not able to study the general semiclassical limit of the Schrödinger equation in a crystal under the action of an external, slowly varying potential except in particular cases including one space dimension (see [10]). Finally, our results are illustrated by well-known examples of homogenization problems, including the wave equation and the Schrödinger and Dirac equations. The paper is organized as follows: In Section 1 we present a survey of (matrix-valued) Wigner measures; Section 2 (the constant-coefficient case) is concerned with the case where the symbol of the PDO depends only on the conjugate variable εξ. In Section 4 (the periodic-coefficient case) we admit Weyl symbols with an additional periodic dependence on the “fast-scale” x/ε. Here we use the Wigner series introduced in [18] and give a survey of their general properties. In Section 6 (the slowly variable coefficient case) we consider Weyl symbols that depend on the slow-scale x and the conjugate variable εξ but not on the fast-scale x/ε. The constant-coefficient case, of course, is

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essentially contained in Section 6, but Section 2 was included because the computations are very transparent and require less regularity of the symbol. Sections 3, 5, and 7 contain examples corresponding to their respective preceding sections.

1 A Survey of Wigner Measures In this paper we use the following definition for Fourier transforms on Rm : Z fˆ(ξ) := (Fx→ξ f )(ξ) = (1.1) e−ix·ξ f (x) dx . Rm

If a ∈ S 0 (Rm ), the Fourier multiplier a(D) associated to a is the operator S(Rm ) → S 0 (Rm ) defined by Z a(ξ)eix·ξ fˆ(ξ) dξ . (a(D)f ) (x) = (2π)−m (1.2) Rm

Of course, depending on the regularity of a, a(D) may operate on other function spaces. m For x-dependent symbols a(x, ξ) ∈ S 0 (Rm x ×Rξ ), the “left symbol” Fourier multiplier writes (in the sequel, we assume 0 < ε < ε0 for some ε0 > 0) the following: Z Z 1 (1.3) (a(x, εD)f ) (x) = a(x, εξ)f (y)ei(x−y)·ξ dξ dy . (2π)m Rm m R y ξ The Weyl operator aW (x, εD) is associated to the symbol a(x, ξ) ∈ S 0 × Rm ξ ) by Weyl’s quantization rule [13]

(Rm x

(1.4)


aW (x, εD)f (x)   Z Z 1 x+y = , εξ f (y)ei(x−y)·ξ dξ dy , a (2π)m Rm 2 m Ry ξ

which can be restated as (1.5)

W




a (x, εD)f (x) =

Z

 v  a ˜ x − ε , v f (x − εv) dv 2 Rm v

−1 a)(x, v) denotes the inverse Fourier transform with where a ˜(x, v) := (Fξ→v respect to the second argument. Notice that aW (x, εD) sends S 0 into S if a ∈ S.

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For given f, g ∈ S 0 (Rm ) and ε ∈ ]0, ε0 ], we define the Wigner transform Z  v  iv·ξ v  ε −m e dv . (1.6) w (f, g)(x, ξ) = (2π) f x − ε g¯ x + ε 2 2 Rm For fixed ε, this clearly defines a continuous bilinear mapping from S 0 (Rm ) × S 0 (Rm ) to S 0 (Rm × Rm ). Moreover, we have the following elementary formulae: (1.7) (1.8)

wε (g, f ) = wε (f, g) g , aW (x, εD)f i . hwε (f, g), ai = h¯

In (1.8) we assume a ∈ S(Rm × Rm ) and h·, ·i denotes the duality bracket (linear in both arguments) between S 0 and S on Rm × Rm or Rm , respectively. Of course, formula (1.8) is valid for f , g, and a lying in other spaces, for example, in the dual situation where f, g ∈ S, a(x, ξ) ∈ S 0 . An intermediate situation of current use is where f, g ∈ L2 and the inverse Fourier transform a ˜(x, v) with respect to ξ satisfies Z (1.9) sup |˜ a(x, v)| dv < +∞ Rm

x

so that it can be integrated against the ξ-Fourier transform of wε (f, g) given by  v  v (Fξ→v wε (f, g))(x, v) = f x − ε (1.10) g¯ x + ε . 2 2 In the sequel k · k denotes the L2 -norm on Rm and (· | ·) denotes the L2 inner product on Rm (antilinear in the second argument). Our main interest is in the asymptotic properties of the transformation (1.6) as ε goes to 0. First, we state one of the crucial properties of Wigner transforms. P ROPOSITION 1.1 If f and g lie in a bounded subset of L2 (Rm x ), then m ). More precisely, we have × R wε (f, g) lie in a bounded subset of S 0 (Rm x ξ for the respective Fourier transforms (1.11)

1 m (Fξ→v wε (f, g))(x, v) ∈ C0 (Rm v ; L (Rx )),

(1.12)

1 m (Fx→ζ wε (f, g))(ζ, ξ) ∈ C0 (Rm ζ ; L (Rξ ))

with the respective norms bounded by kf k kgk. Further, we have the following estimate for a, b ∈ S(Rm × Rm ) Z ε ¯ hw (f, g), abi = (1.13) (a(x, εDx )f ) (b(x, εDx )g) dx + rε , Rm

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where |rε | ≤ εC(a, b)kf k kgk. The estimate (1.13) holds analogously for the Weyl operators aW (x, εDx ), W b (x, εDx ) instead of a(x, εDx ), b(x, εDx ) on the right-hand side. P ROOF : In view of (1.10), the estimate (1.11) is immediate for f, g bounded in L2 . This also yields the first S 0 assertion. (1.12) is an immediate consequence of the representation     ξ ζ ¯ ξ ζ 1 ε ˆ (1.14) + gˆ − . f (Fx→ζ w (f, g))(ζ, ξ) = (2πε)m ε 2 ε 2 −1 Using the Plancherel formula and the notation a ˜ = (Fξ→v a), we have

Z hw (f, g), a¯bi = ε

 v  v f x−ε g¯ x + ε a ˜(x, u) ˜b(x, u − v) dx du dv . 2 2 0

If we set v = u − u0 and x = x0 − ε u+u 2 , we obtain hwε (f, g), a¯bi   Z u + u0 = f (x0 − εu) a ˜ x0 − ε , u g¯(x0 − εu0 ) 2   0 u+u 0 , u dx0 du du0 . · ˜b x0 − ε 2 R 0 u+u0 1 u+u0 0 0 Now write k(x0 − ε u+u 2 , ·) = k(x , ·) − ε 2 0 ∂x k(x − sε 2 , ·) ds with k=a ˜, ˜b in the above right-hand side, and the remainder term is easily estimated using a ˜, ˜b ∈ S and the Schwarz inequality in x. 0

u 0 For the case of Weyl operators we write k(x0 − ε u+u 2 , ·) = k(x − ε 2 , ·) − R 0 0 1 ε u2 0 ∂x k(x0 − ε u+su 2 , ·) ds.

R EMARK 1.2 Basic properties of the Wigner transform are Z (1.15) wε (f, g) dξ = f (x) g¯(x) Z (1.16)

Rm ξ

wε (f, g) dx = Rm x

    1 ˆ ξ g¯ˆ ξ f (2πε)m ε ε

The duality in x and ξ can be expressed, for example, as  ·   1 1 ¯ˆ · ε ε ¯ (1.17) w (f, g)(x, ξ) = w , (ξ, x) . gˆ f ε (2πε)m/2 ε (2πε)m/2

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n In general, we can define for f, g in S 0 (Rm x ) an n × n “Wigner matrix”

by 1 (1.18) W (f, g)(x, ξ) = (2π)m

Z

ε

  ε  ε  f ε x − v ⊗ g¯ε x + v eiv·ξ dv , 2 2 Rm v

where ⊗ denotes the tensor product of vectors. As a special case, let (f ε ) be a bounded family in L2 (Rm )n . Then we denote by W ε [f ε ] the n × n matrix with elements ε [f ε ] = wε (fiε , fjε ) . wij

(1.19)

Also, we denote by wε [f ε ] = tr W ε [f ε ] the scalar Wigner transform of f ε . By (1.7), W ε [f ε ] is hermitian, and by Proposition 1.1, there exists a se0) quence (εk ) going to 0 such that W εk (f εk ) has a limit in S 0 . Let W 0 = (wij 0 be such a limit. We claim that W is a nonnegative matrix-valued measure; that is, for any z ∈ Cn , we have X 0 (1.20) wij zi z¯j ≥ 0 i,j

as a measure. Indeed, since every nonnegative function a ∈ C0∞ can be obtained as the limit of |bn |2 for some sequence (bn ) in C0∞ , it is enough to prove X 0 (1.21) hwij , |b|2 izi z¯j ≥ 0 for b ∈ C0∞ . i,j

But, by (1.13) in Proposition 1.1, we have 2 Z X X ε ε ε 2 ε hw (fi , fj ), zi z¯j |b| i = zi b(x, εD)fi dx + O(ε) m R

i,j

i

and (1.20) follows by passing to the limit. R EMARK 1.3 In previous papers [18, 17] the following separable Banach algebra of test functions introduced in [15] has been used:  m 1 m m (1.22) A = ϕ ∈ C0 (Rm x × Rξ ) : (Fξ→v ϕ)(x, v) ∈ L (Rv ; C0 (Rx )) . In view of (1.11) we clearly have (1.23)

kW ε [f ε ]k(An×n )0 ≤ kf ε kL2 (Rm n x )

and the convergence of a subsequence of W ε [f ε ] in (An×n )0 weak-∗ follows.

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The measure W 0 is called a Wigner measure or semiclassical measure associated to the family (f ε ), where we implicitly assume the choice of a scale ε. If a family (f ε ) admits only one Wigner measure, we shall denote it 0 [f ε ]) and we set w 0 [f ε ] = tr W 0 [f ε ] for the scalar Wigner by W 0 [f ε ] = (wij measure of (f ε ). R EMARK 1.4 The above proof of the positivity of Wigner measures, related to the Bochner-Schwartz theorem, originates from a discussion between L. Tartar and the first author (see also [9]). Other proofs can be found in [7] (using pseudo-differential calculus) and in [11, 15, 16, 18] where a Gaussian regularization (“Husimi function”)   1 |z|2 ε ε ε ε ε ε ε (1.24) wH [f ] := w [f ] ∗x G ∗ξ G , G (z) := exp − ε (πε)m/2 is used, which (as shown by a simple calculation) is a pointwise nonnegative function. Since the accumulation points of wε [f ε ] are also accumulation points ε [f ε ], we conclude that w 0 is a nonnegative measure. of wH The positivity of Wigner measures has many consequences. Here we emphasize one of them, called “orthogonality” by the first author (see [7, 8, 14]). Given a family (f ε ) with a Wigner measure W 0 [f ε ], the Schwarz inequality yields, for any nonnegative a ∈ S, Z 2 Z  Z  0 ε 0 ε 0 ε a dwij [f ] ≤ a dwii [f ] a dwjj [f ] . 0 and w 0 are mutually singular, In particular, if the positive scalar measures wii jj 0 = 0. An equivalent way of stating this property is the following this implies wij result:

P ROPOSITION 1.5 If f ε and g ε have Wigner measures whose traces tr W 0 [f ε ] and tr W 0 [g ε ] are mutually singular, then (1.25)

W 0 [f ε + g ε ] = W 0 [f ε ] + W 0 [g ε ] .

Next we investigate the relationship between the Wigner measure and the weak limits of quadratic forms in f ε . For this purpose we need two definitions. D EFINITION 1.6 A bounded family (f ε ) in L2 is said to be ε-oscillatory as ε goes to 0 if the following property holds for every continuous, compactly supported function ϕ on Rm : Z dε (ξ)|2 dξ → 0 as R goes to +∞. (1.26) lim |ϕf ε→0 |ξ|≥R/ε

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A bounded family (f ε ) in L2 is said to be compact at infinity as ε goes to 0 if Z (1.27) lim |f ε (x)|2 dx → 0 as R goes to +∞. ε→0 |x|≥R

Note that the condition (1.28)

∃κ > 0 such that εκ Dκ f ε is uniformly bounded in L2loc

is sufficient for (1.26). Now we can state the following: P ROPOSITION 1.7 Let (f ε ) be a bounded family in L2 (Rm )n with a Wigner measure W 0 [f ε ]. Then (i) The measure w0 [f ε ] = tr W 0 [f ε ] is bounded on Rm × Rm and, if f ε ⊗ f¯ε → ν as measures on Rm , we have, in the sense of hermitian matrix-valued measures, Z (1.29) W 0 (·, dξ) ≤ ν Rm

with equality if and only if (f ε ) is ε-oscillatory. Z

(ii) We have (1.30)

w (R × R ) ≤ lim 0

m

m

ε→0 Rm

|f ε (x)|2 dx

with equality if and only if (f ε ) is ε-oscillatory and compact at infinity. In this case lim can be replaced by lim in the right-hand side of (1.30). P ROOF : By taking inner products of f ε with fixed vectors in Cn , it is enough to prove the statements for scalar functions f ε . To prove (i), we first observe that, for any uniformly continuous function ϕ on Rm , we have w0 [ϕf ε ] = |ϕ|2 w0 [f ε ] .

(1.31)

This formula is an easy consequence of the definition (1.6) after performing a Fourier transform in ξ. Now choose ψ ∈ C0∞ (Rm ), 0 ≤ ψ ≤ 1, with ψ(ξ) = 1 in a neighborhood of ξ = 0. Applying (1.31) with a continuous compactly supported function ϕ and formula (1.8) with a(x, ξ) = ψ(ξ), we obtain Z (1.32)

ψ(ξ) |ϕ(x)|2 dw0 [f ε ](x, ξ) = lim hwε (ϕf ε , ϕf ε ), ψi ε→0

= lim (ψ(εD)(ϕf ε ) | ϕf ε ) . ε→0

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By the Plancherel formula, the right-hand side of (1.32) is not larger than Z ε 2 lim kϕf k = |ϕ|2 dν . ε→0

Changing ψ(ξ) into ψ(ξ/R) and letting R go to infinity in (1.32), we obtain by Fatou’s lemma the first assertion in (i) and inequality (1.29). In order to investigate equality in (1.29), we come back to the identity (1.32), which we rewrite as follows: Z (1.33)

ψ(ξ/R) |ϕ(x)|2 dw0 [f ε ](x, ξ) Z = |ϕ|2 dν − lim ((1 − ψ(εD/R))(ϕf ε ) | ϕf ε ) ε→0

R Passing to the limit in (1.33) as R goes to infinity, we conclude that w0 [f ε ] (·, dξ) = ν if and only if (f ε ) is ε-oscillatory. Now inequality (1.30) is obtained by integrating (1.29) with respect to x (after possibly selecting a subsequence so that ν exists), and equality in (1.30) holds if and only if equality in (1.29) holds and |f ε |2 converges tightly as a measure on Rm . This completes the proof. Finally, the following proposition shows how the Wigner transform (1.6) translates asymptotically the action of an ε-pseudo-differential operator like (1.4) into a multiplication. The notation {p, q} denotes the Poisson bracket of p = p(x, ξ) and q = q(x, ξ) defined by (1.34) {p, q}(x, ξ) = ∇ξ p(x, ξ) · ∇x q(x, ξ) − ∇x p(x, ξ) · ∇ξ q(x, ξ) . P ROPOSITION 1.8 Let p ∈ C ∞ (Rm × Rm ) satisfy, for some M ≥ 0, (1.35)

α p(x, ξ)| ≤ Cα (1 + |ξ|)M . ∀α ∈ Nm × Nm : |∂x,ξ

Then, because f and g lie in a bounded set of L2 (Rm ), we have the expansion (1.36)


ε wε pW (x, εD)f, g = pwε (f, g) + {p, wε (f, g)} + ε2 rε 2i

where rε is bounded in S 0 (Rm × Rm ) as ε goes to 0. Notice that the right-hand side of (1.36) is well-defined for fixed ε. Indeed, if f ∈ S, integration by parts in ξ shows that formula (1.4) for pW (x, εD)f defines a function ∈ S. Since pW (x, εD)∗ = p¯W (x, εD), it follows that

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pW (x, εD) acts on S 0 by transposition. Hence wε (pW (x, εD)f, g) is welldefined for all f, g ∈ S 0 . In view of formula (1.8), we have, for any a ∈ S, hwε (pW (x, εD)f, g), ai = (aW (x, εD)pW (x, εD)f | g) ; hence the expansion (1.36) can be stated equivalently as an expansion of the symbol (a]p)ε defined by aW (x, εD)pW (x, εD) = (a]p)W ε (x, εD), which is a particular case of the general pseudo-differential Weyl calculus (see, for instance, [13, theorem 18.5.4]). However, for the convenience of the reader, we give a direct proof of Proposition 1.8 in an appendix. It is clear that expansion (1.36) can also be stated for matrix-valued pseudodifferential operators. For the Wigner matrix (1.18) and if P = P (x, ξ) is an n×n matrix-valued function satisfying estimates (1.35), we have the following formulae, which will be useful in Section 6: ε (1.37) W ε (P W (x, εD)f, g) = P W ε (f, g) + {P, W ε (f, g)} + ε2 Rε , 2i ε (1.38) W ε (f, P W (x, εD)g) = W ε (f, g)P ∗ + {W ε (f, g), P ∗ } + ε2 Qε , 2i where Rε and Qε are bounded in S 0 . Notice that on the right-hand sides of these formulae products are matrix products, hence the order is relevant. P ∗ denotes the adjoint P¯ T of a matrix P .

2 Constant Coefficients We consider the initial value problem (2.1a)

εuεt + P (εDx )uε = 0 ,

(2.1b)

uε (t = 0) = uεI

x ∈ Rm x, t∈R on Rm x .

Here ε ∈]0, ε0 ] is a small parameter, uε and uεI are Cn -valued functions, and P (εDx ) is the Fourier multiplier (1.2) associated with the complex n × n– matrix-valued symbol P = P (ξ) on Rm ξ . We impose the following assumptions on P : (A1)

n×n ) (i) P ∈ C 1 (Rm ξ ;C ∗ (ii) ∀ξ ∈ Rm ξ : P (ξ) = −P (ξ).

By Stone’s theorem, − 1ε P (εDx ) is the generator of a unitary, strongly con1 n tinuous group of operators on L2 (Rm x ) . Note that the domain of − ε P (εDx ) n contains H σ (Rm x ) . We assume on the initial datum uεI (see (1.26) and (1.27) in Definition 1.6):

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n (A2) (uεI ) is bounded in L2 (Rm x ) , ε-oscillatory, and compact at infinity.

The “energy density” nε (t, x) := |uε (t, x)|2

(2.2) Z

satisfies (2.3)

Z nε (t, x) dx =

Rm x

Rm x

nεI (x) dx ∀t ∈ R ,

where we set nεI := |uεI |2 . Thus, the total energy is conserved with respect to time t. The main result of this section will be to compute the weak limit of nε (t, ·), under additional assumptions on P and on the data. Up to extraction of a subsequence, we may assume that WI0 = W 0 [uεI ] exists. We impose: (A3) There exists a closed set F in Rm ξ such that (i) For every ξ 6∈ F , the eigenvalues λq = λq (ξ) of −iP (ξ) can be ordered as follows: λ1 (ξ) < · · · < λd (ξ) , where, for 1 ≤ q ≤ d, the multiplicity rq of λq (ξ) does not depend on ξ. (ii) Rm × F is a null set for tr(WI0 ). Notice that, by assumption (A1)(i), the functions λq are C 1 on the open set Rm \ F . For ξ 6∈ F , we denote by Πq (ξ) the orthogonal projection of Cn on the eigenspace corresponding to λq (ξ). Of course, Πq is a C 1 -function on Rm \ F . We can now formulate the following theorem: T HEOREM 2.1 Let (A1), (A2), and (A3) hold, and set, for 1 ≤ q ≤ d, (x, ξ) ∈ Rm × (Rm \ F ), 0 (2.4) (x, ξ) = tr(Πq WI0 )(x, ξ) . wI,q Then, for any t ∈ R, nε (t, ·) converges to the measure n0 (t, ·) given by (2.5)

0

n (t, x) =

d Z X q=1

ξ∈Rm \F

0 wI,q (x − t∇λq (ξ), dξ) .

Moreover, the convergence is uniform for t in bounded intervals.

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Notice that, since WI0 is a positive bounded measure (by Proposition 1.7) 0 on and 0 ≤ Πq ≤ I, (2.4) defines a positive, bounded, scalar measure wI,q m m Rx ×(Rξ \F ), and the right-hand side of (2.5) makes sense as a push-forward sum of partial integrals of bounded measures. P ROOF

OF

T HEOREM 2.1: Fourier transformation of (2.1a) gives εˆ uεt + P (εξ)ˆ uε = 0 ,

(2.6)

ξ ∈ Rm ξ , t ∈ R.

Using the definitions of (A3)(i) we set uε (t, ξ)1εξ6∈F , u ˆεq (t, ξ) := Πq (εξ)ˆ

(2.7)

where 1εξ6∈F is the indicator function, and obtain the IVPs: (2.8a) (2.8b)

ε

∂ ε uεq = 0 , u ˆ + iλq (εξ)ˆ ∂t q u ˆεq (ξ, t = 0) = u ˆεI,q (ξ) ,

ξ ∈ Rm ξ , t∈R ξ ∈ Rm ξ

for 1 ≤ q ≤ d with uεI (ξ)1εξ6∈F . u ˆεI,q (ξ) := Πq (εξ)ˆ

(2.8c)

We now define the scalar Wigner transforms (2.9a)

wqε (t, x, ξ) = wε [uεq (t)](x, ξ) ,

1 ≤ q ≤ d,

(2.9b)

ε (x, ξ) wI,q

1 ≤ q ≤ d.

=

wε [uεI,q ](x, ξ) ,

In view of the min-max principle, λq can be extended as a locally Lipschitz function on the whole of Rm ξ . Then a straightforward calculation gives an evolution equation for the x-Fourier transform w ˆqε (t, ζ, ξ) := (Fx→ζ wqε )(t, ζ, ξ) ε of the Wigner function wq (t, x, ξ): (2.10)

∂ ε w ˆ + iζ ∂t q

Z

−1/2

where (2.11)

1/2

w ˆqε (t, ζ, ξ) =

∇λq (ξ + εsζ) ds w ˆqε = 0 ,

1 ≤ q ≤ d,

    ξ ζ 1 ε ¯ˆε t, ξ − ζ · u t, ˆ + . u q (2π)m εm q ε 2 ε 2

We now prove the following lemma:

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m ε L EMMA 2.2 On Rt × Rm x × (Rξ \ F ), wq converges locally uniformly with m respect to t, to the unique C 0 (Rt , D0 (Rm x × (Rξ \ F ))) solution of the IVP


∂ 0 wq + ∇x · ∇λq (ξ)wq0 = 0 ∂t 0 . wq0 (t = 0) = wI,q

(2.12a) (2.12b)

m ˆqε on Rt × Rm ˆqε is bounded P ROOF : We first study w ζ × (Rξ \ F ). Since w 1 m in L∞ (Rt × Rm ζ , L (Rξ )) in view of (2.11) and since λq is a locally Lipschitz function on the whole of Rm ξ in view of the min-max principle, we con∂w ˆε

m ˆqε is clude from (2.10) that ∂tq is bounded in L∞ (Rt , D0 (Rm ζ × Rξ )), hence w 0 m m equicontinuous in t with values in D (Rζ × Rξ ). If, for some sequence εk → m ∞ m 1 m 0, w ˆqεk → ω(ζ, ξ) in C(Rt , D0 (Rm ζ × (Rξ \ F )), the L (Rt × Rζ , L (Rξ )) ε bound on w ˆq implies that this convergence is also valid in the space of complex m Radon measures on Rt × Rm ζ × (Rξ \ F ). If ζ varies in a compact subset of Rm and ξ varies in a compact subset of m R \ F , then ξ + εsζ belongs to a compact subset of Rm \ F for s ∈ [− 12 , 12 ] and ε small enough. Hence, since ∇λq is continuous on Rm \ F , we can pass to the limit in (2.10) and obtain

∂ω(ζ, ξ) + iζ · ∇λq (ξ)ω(ζ, ξ) = 0 ∂t 0 ω(t = 0) = w ˆI,q

(2.13a) (2.13b)

which clearly admits a unique solution, namely the Fourier transform with respect to x of the unique solution to (2.12). The proof is then completed by using Fourier transform with respect to x. The following result implicitly clarifies the link between W ε (uε )(t, x, ξ) and wqε (t, x, ξ). n ∞ m n×n be L EMMA 2.3 Let (f ε ) be bounded in L2 (Rm x ) and A ∈ L (Rξ ) m continuous on some open subset Ω of Rξ . Then

W ε (A(εD)f ε )(x, ξ) − A(ξ)W ε (f ε )(x, ξ)A(ξ)∗ → 0 in D0 (Rm x × Ωξ ). P ROOF : According to (1.12) the Fourier transform with respect to x of 1 m n×n . It remains to check that W ε (f ε ) is bounded in L∞ (Rm ζ , L (Rξ ))     ζ ∗ ζ ˆε ˆ ε (ζ, ξ)A(ξ)∗ → 0 W (ζ, ξ)A ξ − ε − A(ξ)W A ξ+ε 2 2

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in the space of complex Radon measures on Rm ζ × Ωξ , which is immediate since A is continuous on Ω. Now we can complete the proof of Theorem 2.1. Let us denote the scalar Wigner transform of (uε (t, ·)) by wε (t, x, ξ) = tr W ε (t, x, ξ). Let (tε ) be an arbitrary convergent family of real numbers; denote by t0 its limit. On m Rm x × (Rξ \ F ), we have X X (2.14) wε (tε ) = tr W ε (tε ) = tr(Πq W ε (tε )) = tr(Πq W ε (tε )Πq ) q

q

where the last equality comes from the fact that Π2q = Πq and tr(AB) = tr(BA). Using Lemma 2.3 with f ε (x) = uε (tε , x) and A(ξ) = Πq (ξ), we obtain X X (2.15) wqε (tε ) + o(1) → wq0 (t0 ) wε (tε ) = q

q

m D0 (Rm x × (Rξ \ F )),

in in view of Lemma 2.2. Hence, if µ is a scalar Wigner measure associated to a subsequence of uε (tε ), we get X (2.16) wq0 (t0 , x, ξ)1ξ6∈F . µ(x, ξ) ≥ q

Now, in view of (2.12), the total mass of the right-hand side of (2.16) is XZ Z X 0 0 m wI,q (dx − t0 ∇λq (ξ), dξ) = wI,q (Rm x × (Rξ \ F )). q

m Rm x ×(Rξ \F )

q

Using Lemma 2.3 with f ε = uεI , we have, exactly as in (2.14), X 0 m 0 m m wI,q (Rm x × (Rξ \ F )) = wI (Rx × (Rξ \ F )) q m = wI0 (Rm x × Rξ )

= lim kuεI k2 = lim kuε (tε )k2 ε→0

ε→0

in view of assumptions (A3)(ii), (A2), and conservation of energy. On the other hand, in view of inequality (1.30) in Proposition 1.7(ii), m ε ε 2 µ(Rm x × Rξ ) ≤ lim ku (t )k . ε→0

Hence, we conclude that (2.16) is an equality, and, by Proposition 1.7(ii), that (uε (tε )) is compact at infinity and ε-oscillatory. Using Proposition 1.7(i), this completes the proof of the theorem.

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R EMARK 2.4 The proof of Theorem 2.1 shows that w0 (t, x, ξ) is given by w0 (t, x, ξ) = 1ξ6∈F

(2.17)

d X

0 wI,q (x − t∇λq (ξ), ξ) .

q=1

R EMARK 2.5 We define the energy-flux density J ε (t, x) by its Fourier transε (t, ξ))> : form Jˆε (t, ξ) = (Jˆ1ε (t, ξ), . . . , Jˆm

(2.18)

Jˆlε (t, ξ) = −

i (2π)m

Z Rm ω

¯ˆε (t, ω − ξ)> u

Z

1

0

∂P (ε(ω + (s − 1)ξ)) ∂ξl

ε

ds u ˆ (t, ω) dω . Then the macroscopic conservation law (“continuity equation”) holds: nεt + divx J ε = 0 .

(2.19)

In terms of the Wigner matrix, Jˆlε is given by Jˆlε (t, ξ) (2.20)

= −i

Z

Z ˆ ε (t, ξ, v) tr W Rm v

0

1

∂P ∂ξl

   > ! 1 v+ε θ− dθ dv ξ 2

ˆ ε (t, ξ, v) stands for (Fx→ξ W ε (t, x, v)). Both (2.19) and (2.20) follow where W by direct calculation. Under more stringent assumptions on the symbol and the initial data, for example, (A4)

∃C > 0 , γ ≥ 0 : |∇P (ξ)| ≤ C(1 + |ξ|γ ) ∀ξ ∈ Rm ξ

(A5)

∃C > 0 : εγ kDγ uεI kL2 ≤ C

we can compute the weak limit J 0 of J ε . Similar arguments as in the proof of Theorem 2.1 finally give d Z X 0 0 J (t, x) = (2.21) ∇λq (ξ)wI,q (x − t∇λq (ξ), dξ) . q=1

ξ∈Rm \F

3 Examples with Constant Coefficients The general result Theorem 2.1 applies to a large variety of physically relevant homogenization problems. We give details here for the Schrödinger equation, general first-order hyperbolic systems, and the wave equation. The Maxwell equations in a homogeneous medium (cf. [17]) can be treated analogously to the wave equation.

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3.1 Schrödinger Equation with Vanishing Potential We start with the free-particle transport Schrödinger equation in Rm x for the ε scalar wave function u : (3.1a) (3.1b)

ε2 x ∈ Rm ∆uε = 0 , x , t∈R 2 ε ε m u (t = 0, x) = uI (x) on Rx ,

εuεt − i

where ε is the (scaled) Planck constant. Then we have 1 i P (ξ) = |ξ|2 ⇒ −iP (ξ) = |ξ|2 =: λ(ξ) 2 2 and the x-Fourier-transformed Wigner equation (2.10) reads ˆε = 0 . w ˆtε + iζ · ξ w As is well-known, the Wigner function wε (t, x, ξ) = wε [uε (t)](t, x, ξ) satisfies the free-transport Liouville equation for all ε > 0 (3.2a)

wtε + ξ · ∇x wε = 0 ,

(3.2b)

wε (t = 0) = wε [uεI ](x, ξ) =: wIε

(see, e.g., [23, 19, 7, 15]). The quantity nε = |uε |2 , which is now the position density, is then given by Z nε = (3.3) wε dξ . Rm ξ

The assumptions (A1) (with σ = 2), and, trivially, (A3), are automatically satisfied. We assume the initial datum uεI to be ε-oscillatory. Note that compactness at infinity is not necessary to carry out the homogenization of nε (since the set F of assumption (A3)(i) is empty). Then the homogenization gives Z ε ε→0 0 wI0 (x − ξt, dξ) , n −→ n = Rm ξ

where wI0 is a Wigner measure of uεI . We remark that WKB-initial data   i SI (x) , uεI (x) = aI (x) exp ε

x ∈ Rm x ,

1,1 with aI ∈ L2 (Rm ), SI ∈ Wloc (Rm ) and real-valued satisfy assumption (A2).

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A simple calculation gives wI0 (x, ξ) = |aI |2 (x)δ(ξ − ∇SI (x)) and w0 (t, x, ξ) = |aI |2 (x − ξt)δ(ξ − ∇SI (x − ξt)) . Note that because of the low regularity of aI , SI , a WKB method cannot be applied.

3.2 First-Order Symmetric, Strictly Hyperbolic Systems Let A1 , . . . , Am be real, symmetric, n × n matrices. We consider ∂uε X ∂uε Al = 0, + ∂t ∂xl m

(3.4a) (3.4b)

l=1 ε

u (t = 0) = uεI

x ∈ Rm x , t∈R on Rm x

where uε (t, x), uεI (x) ∈ Cn . The energy density is given by nε = |uε |2 . We calculate m X P (ξ) = i Al ξl , ξ = (ξ1 , . . . , ξm )> . l=1

We assume that (3.4a) is strictly hyperbolic, which isP equivalent to assuming that the eigenvalues λ1 (ξ), . . . , λn (ξ) of the matrix m l=1 Al ξl (= −iP (ξ)) are simple for ξ 6= 0. By homogeneity we have λq (αξ) = αλq (ξ), ∀α ∈ R, and in particular (3.5)

λq (ξ) = |ξ|λq (ω) ,

ω=

ξ ∈ S m−1 . |ξ|

Since λq is smooth on S m−1 (by strict hyperbolicity) we have F = {0} in Theorem 2.1. P Let a1 (ω), . . . , an (ω) be eigenvectors of nl=1 Al ξl corresponding to λ1 (ξ), . . . , λn (ξ), respectively, with |al (ω)| = 1 for l = 1, . . . , n. Then the transport equations read ∂ 0 1 ≤ q ≤ n, w + ∇λq (v) · ∇x wq0 = 0 , ∂t q 0 wq0 (t = 0) = wI,q (3.6b) , 1 ≤ q ≤ n,   0 (x, ξ) = tr a ( ξ ) ⊗ a ( ξ )W 0 (x, ξ) and W 0 = W 0 [uε ]. where wI,q q |ξ| q |ξ| I I I

(3.6a)

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We assume that Rm x × {0} is a null set of the scalar Wigner measure 0 ε ε tr W [uI ] and that uI is uniformly bounded in L2 , ε-oscillatory, and compact at infinity (cf. [22, 8]). Then the homogenization limit of nε = |uε |2 is constructed according to Theorem 2.1, that is, by summing the zeroth-order ξ-moments of the solutions of (3.6). R EMARK 3.1 Assume that the scalar Wigner measure tr W 0 [f ε ] of a sequence f ε ∈ L2 (Rm )n does not charge ξ = 0. Then Z ∞ 0 ε m−1 n×n W 0 [f ε ](x, rω)rm−1 dr ∈ M+ (Rm ) H [f ](x, ω) := x × Sω 0

is an H-measure matrix of the sequence f ε and h0 := tr H 0 [f ε ](x, ω) is a scalar H-measure (cf. [22, 5, 4, 11]). Because of (3.5) and the assumption that the initial Wigner measure does not charge ξ = 0, we can obtain an H-measure h0 (t) of the system (3.4) from ∂ m−1 hq + ∇λq (ω) · ∇x hq = 0 , x ∈ Rm , t∈R x , ω ∈ Sω ∂t   Z ∞ 0 m−1 WI (x, rω)r dr hq (t = 0, x, ω) = tr aq (ω) ⊗ aq (ω) and h0 [uε (t)](x, ω) =

0

Pn

q=1 hq (t, x, ω).

3.3 Wave Equation in Rm We consider the wave equation (3.7a) (3.7b)

uεtt − ∆uε = 0 ,

x ∈ Rm x , t ∈ R,

uε (t = 0) = uεI , uεt (t = 0) = sεI

on Rm x

for the scalar, complex-valued function uε . The energy density is given by nε = |uεt |2 + |∇uε |2 . Therefore we introduce the new dependent variables sε = uεt , and obtain the system 

sε rε



 − t

rε = ∇uε ,

0 divx ∇x 0



sε rε

 =0

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sε (t = 0) = sεI , rε (t = 0) = ∇uεI =: rIε . Thus   0 ξ> P (ξ) = i ξ 0 and λ1 (ξ) = −|ξ| ,

λ2 (ξ) = 0 ,

λ3 (ξ) = |ξ| ,

Rm ξ ,

and we have F = {0} under follows. The multiplicity of λ2 is m − 1 on assumption (A3)(i). We assume that the initial data fulfill (A2) and  ε  sI m 0 . Rx × {0} is a null set of w rIε The limiting transport equations read ξ ∂ 0 w − w0 = 0 , ∂t 1 |ξ| 1 ∂ 0 w = 0, ∂t 2 ξ ∂ 0 w3 + w30 = 0 , ∂t |ξ|

(3.8a) (3.8b) (3.8c)

0 w10 (t = 0) = wI,1 , 0 w20 (t = 0) = wI,2 , 0 w30 (t = 0) = wI,3 ,

where we obtain after the straightforward computation of the projections Π1 (ξ), Π2 (ξ), and Π3 (ξ) 0 (3.9a) wI,1 (x, ξ) =

1 0 ε w [sI ](x, ξ) 2   1 ξ 0 ε ε · w (rI , sI )(x, ξ) + w0 [rIε ](x, ξ) , − Re |ξ| 2

0 (3.9b) wI,2 (x, ξ) = 0 , 0 (3.9c) wI,3 (x, ξ) =

1 0 ε w [sI ](x, ξ) 2   1 ξ · w0 (rIε , sεI )(x, ξ) + w0 [rIε ](x, ξ) . + Re |ξ| 2

Here w0 (rIε , sεI ) is a measure-valued vector whose components are limits of wε (∂xl uεI , sεI ). P ROOF tion (3.10)

OF

(3.8)–(3.9)::

In computing formulae (3.9), we used the equa-

W 0 [rIε ] =

ξ⊗ξ 0 ε w [rI ] |ξ|2

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(note that w0 [rIε ] does not charge ξ = 0 by assumption). The proof of (3.10) is based on the fact that rIε is a gradient, which implies ε ε − εDk rI,j = 0, εDj rI,k ε = ∂ uε . An application of (1.36) with p = ξ and where we denoted rI,l xl I j ε 2 p = ξk , respectively, gives for any bounded sequence f in L ε ε ξj w0 [rI,k , f ε ] − ξk w0 [rI,j , f ε] = 0

and, analogously, ε ε ] − ξl w0 [f ε , rI,j ] = 0. ξj w0 [f ε , rI,l ε in the first formula, f ε = r ε in the second formula, mulWe set f ε = rI,l I,j tiply both formulae by ξj , and sum over j. The claim then follows from straightforward linear algebra.

For an H-measure-based analysis of the homogenization of the energy density of the wave equation, we refer to [4, 22, 5].

4 Periodic Coefficients We now deal with the case where the pseudo-differential operator P depends on the position x in an oscillating way with period of order ε. A problem of this kind is the semiclassical limit ε → 0 for the Schrödinger equation in a crystal (cf. [7, 18]). For analytical convenience we use the Weyl operator formulation (cf. (1.4)). Hence we consider the following IVP for the Cn -valued function uε (t, x): (4.1a) (4.1b)

εuεt + (P ε )W (x, Dx )uε = 0 , uε (t = 0) = uεI ,

x ∈ Rm x , t ∈ R, on Rm x ,

where the complex n × n–matrix-valued symbol P ε is given by x  (4.1c) , εξ . P ε (x, ξ) = P ε We require the following periodicity of P : (4.2)

P (x, ξ) = P (x + γ, ξ) ,

ξ ∈ Rm ξ , γ ∈L

where γ ∈ L is a lattice vector of the lattice L ⊂ Rm defined as follows: (4.3)

L = {a(1) j1 + · · · + a(m) jm | j1 , . . . , jm ∈ Z}

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and a(1) , . . . , a(m) form a basis of Rm . The dual basis vectors a(1) , . . . , a(m) are determined by the equation a(`) a(k) = 2πδ`k ,

(4.4)

`, k = 1, . . . , m,

∗

and the dual lattice L ⊂ Rm ξ (“reciprocal lattice”) reads n o ∗ (4.5) L = a(1) j1 + . . . + a(m) jm | j1 , . . . , jm ∈ Z . The basic period cell of the lattice L is denoted by ) (m X (4.6) ti a(i) 0 < t1 , . . . , tm < 1 . C := i=1

By B we denote the Brillouin zone, that is, a distinct fundamental domain of L∗ : ∗

(4.7) B := {k ∈ Rm ξ | k is closer to zero than to any other point of L } . ∗

Note that we do not define B as the torus Rm ξ /L , since we will need the periodic extensions of functions of k ∈ B (in which case we will consistently use ξ ∈ Rm ξ instead of k ∈ B as the argument). In addition to (4.2), we impose the following assumptions on P : (B1)

(i) P (x, ξ)∗ = −P (x, ξ). (ii) P (x, ξ) is a polynomial of degree σ in ξ and C ∞ in x. (iii) If u and P W (x, Dx )u ∈ L2 (Rm )n (respectively, L2loc (Rm )n ), then σ/2 u ∈ H σ/2 (Rm )n ) (respectively, Hloc (Rm )n ).

R EMARK 4.1 Assumption (B1)(iii) is crucial for obtaining the properties (P1) below, which are the basis of the generalized Bloch decomposition. The ellipticity condition (4.8)

| det P ε (x, ξ)| ≥ Cε |ξ|nσ

for |ξ| large enough

is sufficient for (B1)(iii), but, for example, the acoustic equation in Section 5.2 shows that (4.8) can be too stringent. In view of (B1), the operator −iP W (x, Dx ) is essentially self-adjoint on Indeed, the domain of P W (x, Dx )∗ can be easily identified as the set of u ∈ L2 (Rm )n such that P W (x, Dx )u ∈ L2 (Rm )n in the distributional sense.

L2 (Rm )n .

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Note that only for the Weyl formalism the formal self-adjointness of the operator is equivalent to the real-valuedness of a scalar symbol or to the pointwise self-adjointness of a symbol matrix. The assumptions on the initial datum uεI are again (B2) uεI is ε-oscillatory and compact at infinity, that is, (1.26) and (1.27) hold. Note that we require the period of the oscillations of initial data to be of the same order of magnitude as the period of the coefficients of the pseudodifferential operator in the evolution equation. This is the “crossover case” of the more general two-scale situation. In this section we replace the Wigner function (1.6) by the Wigner series introduced in [18] obtained by replacing the Fourier integral in (1.6) by Fourier series on the lattice L. This is historically motivated by the semiclassical equations in solid state physics (cf. [1, 18]) where k is called “crystal momentum.” The obvious advantage of these Wigner series is the “reduction” of the “whole space variable” ξ ∈ Rm ξ to the variable k in the bounded domain B. For f, g ∈ S 0 (Rm ) x we define wsε (f, g)(x, k) := 1 X  ε   ε  (4.9) f x − γ g x + γ eik·γ , |B| 2 2

x ∈ Rm , k ∈ B .

γ∈L

For fixed ε, this defines a continuous bilinear mapping from S 0 (Rm ) × S 0 (Rm ) to C 0 where C is defined as the space of ξ-periodic test functions a(x, ξ): m m C := {a ∈ C ∞ (Rm x × Rξ ) | ∀α ∈ N , ∀β, γ ∈ N :

(4.10)

m (1 + |x|2 )α/2 ∂xβ ∂ξγ a(x, ξ) ∈ L∞ (Rm x × Rξ )

and a(x, ξ) is L∗ -periodic in ξ} . In analogy to (1.7) and (1.8) we have

ε L EMMA 4.2 Let f, g ∈ L2 (Rm x ) and a(x, ξ) ∈ C. Then ws lie in a bounded 0 set in C and

(4.11)

wsε (g, f ) = wsε (f, g)

(4.12)

hwsε (f, g), aiC 0 ,C = aW (x, εD)f | g




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P ROOF : For f, g ∈ S(Rm x ) we have hwsε (f, g), aiC 0 ,C  X 1 Z ε   ε  = f x − γ g x + γ a(x, k)eik·γ dx dk |B| Rm 2 2 x ×B γ∈L Z X Z   ε 1 = f (x − εγ) a x − γ, k eik·γ dk g(x) dx |B| B 2 Rm x γ∈L Z Aε (x) g(x) dx . =: Rm x

We compute Aε (x): Aε (x) =

  X 1 Z Z Z ε ei(y−γ)·ξ f (x−εy) a x − y, k eik·γ dy dξ dk . m |B| B Rm 2 (2π) m R y ξ

γ∈L

By rearrangement for using the L∗ -periodic δ-distribution, X 1 X i(k−ξ)·γ e = δ(k − ξ − γ ∗ ) |B| ∗ ∗

(4.13)

γ ∈L

γ∈L

and by the L∗ -periodicity of a(x, k) in k we have Z   X 1  ε ε a x − y, k ei(k−ξ)·γ dk = a x − y, ξ 2 |B| 2 B γ and Aε (x) =

1 (2π)m

Z Rm y

Z

 y  f (x − εy) a x − ε , ξ eiy·ξ dy dξ 2 Rm ξ

= aW (x, εD)f .

In the sequel we use the notation fˇ(γ) for the Fourier coefficients with respect to k: Z X 1 ik·γ ˇ ˇ f (k) = (4.14) , f (γ) = f (k)e−ik·γ dk . f (γ)e |B| γ∈L

B

In analogy to Proposition 1.1, we have the following:

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347

ε P ROPOSITION 4.3 If f, g lie in a bounded subset of L2 (Rm x ), then ws (f, g) belongs to a bounded set in C 0 . Moreover, if a(x, k), b(x, k) ∈ C we have

(4.15)

hwε (f, g), abiC 0 ,C = (a(x, εD)f | b(x, εD)g) + rε ,

where |rε | ≤ C(a, b)ε kf k kgk. P ROOF : We take a(x, k) and b(x, k) in C as defined in (4.10) and denote by a ˇ(x, γ) and ˇb(x, γ) the Fourier coefficients (4.14) of a, b. By the Plancherel formula we obtain Z X  ε   ε  ∨ ε f x − γ g x + γ (ab) (x, −γ) dx hws (f, g)), abiC 0 ,C = 2 2 Rm x γ∈L Z XX  ε   ε  f x− γ g x+ γ a = ˇ(x, µ) 2 2 Rm x γ∈L µ∈L

· ˇb(x, µ + γ) dx . Using the change of variables γ → γ − µ and x → x + 2ε (γ + µ), this can be written as Z XX
ε hws (f, g), abiC 0 ,C = a x + 2ε (γ + µ), µ f (x + εµ) Rm x γ∈L µ∈L


· ˇb x + 2ε (γ + µ), γ g(x + εγ) dx . For a function a(x, ξ) that is L∗ -periodic in ξ, we define (4.16)

a(x, εD)f :=

X

a ˇ(x, γ)f (x + εγ)

γ∈L

and a calculation similar to the proof of Lemma 4.2 shows that this is indeed the Fourier multiplier (1.3) for a periodic symbol. The regularity of a, b allows us to conclude (4.15). R EMARK 4.4 The relation between Wigner series (4.9) and Wigner functions (1.6) follows directly using the reciprocal Poisson summation formula of (4.13) and the relation |C||B| = (2π)m for the basic cells (4.17)

wsε (f, g)(x, k) =

X γ ∗ ∈L∗

wε (f, g)(x, k + γ ∗ ) .

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n For a uniformly bounded sequence f ε in L2 (Rm x ) we use (in analogy to the Wigner functions in Chapter 2) the notations wsε (t, x, ξ) = wsε [f ε (t)](x, ξ) and Wsε (t, x, ξ) = Wsε [f ε (t)](x, ξ) for the n × n “Wigner series matrix” assigned to an n-vector sequence f ε , with the relation wsε (t, x, ξ) = tr Wsε (t, x, ξ). As a consequence of the above general properties of Wigner series, Wsε [f ε ] has accumulation points Ws0 [f ε ] which are nonnegative, definite, matrix-valued measures. The Wigner series are less general than the Wigner transforms of Section 1 in the sense that their application makes sense only in the case of periodic PDOs (i.e., finite difference operators in the sense of (4.16)). However, they have better convergence properties (cf. the proposition below with Proposition 1.7). n P ROPOSITION 4.5 Let f ε be a bounded sequence in L2 (Rm x ) such that n×n . Then Wsε [f ε ] → Ws0 [f ε ] in (C 0 )n×n and f ε ⊗ f ε * ν in M(Rm x ) Z (4.18) Ws0 [f ε ](·, dk) = ν . B

Z

Moreover, tr(Ws0 [f ε ])(Rm ξ × B) = lim

ε→0 Rm x

|f ε (x)|2 dx ,

if f ε is compact at infinity. P ROOF : The proof is based on (4.12) with a independent of ξ. The following proposition is analogous to Proposition 1.8 and is useful for the proof of the evolution equation of the limiting Wigner series measures. P ROPOSITION 4.6 Let λ be a bounded Γ∗ -periodic function such that λ ∈ n C 1 (Ω), Ω ⊂ B. Let f ε , g ε be bounded sequences in L2 (Rm x ) such that ε ε ε 0 ε ε 0 Ws (f , g ) → Ws (f , g ) in C . Then (4.19)

Wsε (λ(εD)f ε , g ε ) = λ(k)Ws0 (f ε , g ε ) +

ε ∇λ(k) · ∇x Ws0 (f ε , g ε ) + εrε 2i

where rε → 0 in D0 (Rm x × Ω) as ε goes to 0. P ROOF : We first remark that the Fourier transform with respect to x gives ε ε ε d W s (f , g )(ζ, k)     X k γ∗ ζ k γ∗ ζ 1 (4.20) ε c bε = + + ⊗ + − g f (2πε)m ∗ ∗ ε ε 2 ε ε 2 γ ∈L

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349

∞ m 1 ∞ m ε dε ˆ0 d therefore W s is bounded in L (Rζ ; L (B)) and Ws → Ws in L (Rζ ; 0 0 C (B) ) weak-∗. We have  ε  dε ε ε ε (λ(εD)f ε , g ε ) = λ k + ζ W d (4.21) W s s (f , g ) 2

since λ is Γ∗ -periodic. Also, λ is continuous on Ω, and for (k, ζ) in a compact ε subset of Ω × Rm ζ we have k + 2 ζ ∈ Ω for ε small enough, which yields ε ε 0 m d0 ε ε ε d W s (λ(εD)f , g ) → λ(k)Ws (f , g ) in D (Ω × Rζ ) .

dε is bounded in D0 (Ω, S 0 (Rm )), which leads to the conclusion. The But W s ζ final form (4.19) follows from a Taylor expansion in (4.21). For the Wigner series of the solution uε (t) of (4.1), that is, for wsε (t, x, ξ) := = tr Wsε [uε (t)](x, ξ), we have the following important result:

wsε [uε (t)](x, ξ)

P ROPOSITION 4.7 Assume (B1) and (B2) hold. Then the sequence (wsε ) in C(Rt , C 0 ) is equicontinuous with respect to t as ε → 0. P ROOF : We first prove that we may assume that (P ε )W uεI belongs to a bounded subset of L2 . Indeed, assume the proposition is proved for such data. ε by Given (uεI ) satisfying only (B2) and R > 0, introduce vI,R ε (ξ) = 1|ξ|≤ R u ˆεI (ξ) . vˆI,R ε

ε belongs to a bounded subset of L2 . Indeed, it We claim that f ε = (P ε )W vI,R is enough to prove that we can estimate independently of ε the L2 -norm of   ε (εx) . εm/2 f ε (εx) = P W (x, Dx ) εm/2 vI,R ε (εx) is given by The Fourier transform of εm/2 vI,R

ˆεI 1|ξ|≤R ε−m/2 u

  ξ ; ε

hence this function is bounded in any H S -space, and so is εm/2 f ε (εx). On the other hand, since uεI is ε-oscillatory and compact at infinity, we have R→∞ ε lim kuεI − vI,R kL2 = α(R) −→ 0 . ε→0

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ε ) to (4.1a) with initial Hence, by the conservation of L2 -norm, the solution (vR ε satisfies datum vI,R R→∞

ε lim kuε (t) − vR (t)k = α(R) −→ 0 .

ε→0

In particular, the Wigner series of uε (t) is uniformly approximated by the ε (t) as R → +∞; hence it is equicontinuous if the latter Wigner series of vR is. Now we assume that (P ε )W uεI belongs to a bounded subset of L2 . Since (P ε )W uε solves equation (4.1a), we conclude that

ε W ε

(P ) u (t) 2 ≤ C ; L hence, by setting u ˜ε (t, x) = εm/2 uε (t, εx),

W

P (x, Dx )(˜ uε (t)) L2 ≤ C . Using assumption (B1)(iii), we conclude that u ˜ε (t) is uniformly bounded in σ/2 H , which implies the following estimate: εσ/2 kuε (t)kH σ/2 ≤ C .

(4.22)

To complete the proof of the proposition, we need the following generalization of the conservation of energy: L EMMA 4.8 Given a ∈ C, we have  ε  h X Z 1/2 Z ∂ws = tr W ε (t, x, ξ)ζ ,a ∂t 3m C 0 ,C γ∈L −1/2 R i x γ (4.23) + , ξ + εζθ · ∇ξ P ε 2 dζ ˇ ·a ˆ(ζ, γ)ei(xζ+γξ) dx dξ dθ (2π)m P ROOF : Using equation (4.1a), we have    ε 
1 ε ε W ε ε ∂ws ε ε ε W ε w [(P ) u , u ] + ws [u , (P ) u ] , a . (4.24) ,a = − ∂t ε s Now we have the following formula, which is the Wigner series version of the lemma in the appendix,

ε W
 ws P (x, εD)f, g , a C 0 ,C Z h i = tr W ε (f, g)(x, ξ)(a]P )ε (x, ξ) dx dξ R2m

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where (a]P )ε (x, ξ) = (2π)−m

XZ

ˇˆ(ζ, γ)ei(xζ+γξ) P a

m γ∈L R

  γ ζ x +ε ,ξ −ε dζ . 2 2

Observe that this formula is an easy consequence of the lemma in the appendix and of the relation (4.17) between Wigner series and Wigner transforms. Plugging this formula into (4.24) and using P ε (x, ξ)∗ = −P ε (x, ξ), we get 

     XZ ∂wsε γ 1 ζ ε ε tr W (t, x, ξ) ,a = − P x+ ε ,ξ −ε ∂t ε 2 2 3m γ∈L R   γ dζ ζ ˇ − Pε x − ε ,ξ + ε . a ˆ(ζ, γ)ei(xζ+γξ) dx dξ 2 2 (2π)m

Using P ε (x, η) = P ( xε , η) and the L-periodicity with respect to xε , we finally get 

     XZ ∂wsε 1 x γ ζ ε tr W (t, x, ξ) · ,a = − P + ,ξ −ε ∂t ε ε 2 2 3m γ∈L R   x γ dζ ξ ˇ −P , + ,ξ +ε a ˆ(ζ, γ)ei(xζ+γξ) dx dξ ε 2 2 (2π)m

which leads to the claimed formula. ε

s We now complete the proof of the proposition by showing that ∂w ∂t is bounded in L∞ (Rt , C 0 ). By expanding ∇ξ P (·, ξ + εζθ) in Fourier series in the right-hand side of (4.23), we get



   X X Z 1 Z  2 ∂wsε γ∗ ε tr Fx W t, − − ζ, ξ ζ ,a = 1 ∂t ε 2m γ∈L γ ∗ ∈L∗ − 2 R  dζ ∗ ˇ · ∇ξ Pˇ (γ ∗ , ξ + εζθ) a ˆ(ζ, γ) eiγ(ξ+γ /2) dξ dθ (2π)m    X Z 12 Z γ∗ ε = tr Fx W t, − − ζ, ξ ζ 1 ε 2m γ ∗ ∈L∗ − 2 R    dζ γ∗ ∗ ˇ dξ · ∇ξ P (γ , ξ + εζθ) a ˆ ζ, ξ + dθ . 2 (2π)m

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Using the formula (1.14), this leads to X Z

=

1 2

− 12

γ ∗ ∈L∗

Z



∗

R2m

ζ∇ξ Pˇ (γ , εξ + εζθ)ˆ u

ε

γ∗ ζ t, ξ − − 2ε 2



    γ∗ ζ dζ γ∗ dξ > ε ¯ ·u ˆ t, ξ + dθ + a ˆ ζ, εξ + m ε 2 2 (2π) (2π)m 1

Using the inequalities (with the notation hξi = (1 + ξ 2 ) 2 ) |∇ξ Pˇ (γ ∗ , η)| ≤ Chγ ∗ i−N hηiσ−1 |ˆ a(ζ, k)| ≤ Chζi−N for N as large as we like, we finally get  ε  Z X ∂ws ∗ −N hγ i hζi−m hεξi(σ−1) |ˆ uε (t, ξ)|2 dξ dζ ∂t , a ≤ C 2m R ∗ ∗ γ ∈L

which is bounded in view of (4.22). The periodicity (4.2) of the symbol matrix P allows the use of the so-called Bloch decomposition of L2 (Rm )n , thus making the periodic-coefficient case similar to the constant-coefficient case of Section 2. We set Z ⊕ dk 2 (4.25) L2 (εC)n L],ε := |B| B (see [20] for the constant-fiber direct integral of a Hilbert space). L2],ε is equipped with the norm  ku] k],ε :=

(4.26)

1 |B|

1

Z

2

|u] (x, k)| dx dk 2

.

εC×B

The following proposition is essentially given in [20]: P ROPOSITION 4.9 The map I ε from L2 (Rm )n to L2],ε defined by Iε

u(x) −→ u] (x, k) :=

(4.27)

X

u(x − εγ)eik·γ

γ∈L

is an isometry. Its inverse is (4.28)

ε −1

(I )




1 u] (x + εγ) = |B|

Z u] (x, k)eik·γ dk , B

x ∈ εC , γ ∈ L .

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353

Now the operator (P ε )W on L2 (Rm )n is transformed into pε = I ε (P ε )W (I ε )−1 on L2],ε . From a calculation based on (4.27) and (4.28) and the periodicity (4.2) of the symbol P ε , we conclude that (P ε )W commutes with I ε and thus (4.29)

pε u] (x, k) = (P ε )W (x, D)u] (x, k) ,

x ∈ εC , k ∈ B

(note that (P ε )W acts locally since its Weyl symbol is a polynomial in ξ). Here k plays the role of a parameter. Thus we are led to define the operator pε (k) : D(pε (k)) → L2 (εC)n for fixed k ∈ B in the following way: (4.30a)

pε (k)u(x) = (P ε )W (x, D)u(x) ,

x ∈ εC .

Its domain D(pε (k)) is the set of restrictions to εC of functions in {u ∈ L2],ε (k) | (P ε )W (·, D)u ∈ L2loc (Rm )n }, where L2],ε (k) is the space of locally square-integrable k-quasi-periodic functions:

(4.30b)

n n L2],ε (k) = u ∈ L2loc (Rm x ) |

o ∀γ ∈ L : u(x + εγ) = u(x)eik·γ a.e. in Rm x .

−ipε (k) is an essentially self-adjoint operator on L2 (εC)m . We thus obtain (cf. [20]) the direct-fiber decomposition of (P ε )W in operators acting on kquasi-periodic functions: Z ⊕ dk ε ε W ε −1 (4.31) pε (k) (P ) = (I ) I . |B| B For ε = 1 we set up the following spectral problem: (4.32)

−ip1 (x, Dx )u] (x, k) = λ(k)u] (x, k)

In view of assumption (B1)(iii), the resolvent of −ie−ik·x p1 (k)(eik·x .) belongs to the Schatten class Lp (L2 (C)n ), p > 2m σ . Hence the arguments of Wilcox in [24] (see also [6]) for the case of the Schrödinger equation can be carried out in this general framework in order to obtain the properties (P1) stated below. The basic idea is to show that the eigenvalue problem for −ip1 (k) is equivalent (with multiplicities) to an analytic equation D(λ, k) = 0 and then use general properties of such equations. (P1)

(i) For any k, the spectrum of −ip1 (k) is an infinite discrete set {λl (k), l ∈ D} of real eigenvalues with finite multiplicities. Here

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D denotes either Z, Z+ , or Z− depending on whether P is not semibounded or semibounded from below or above. The functions λl are L∗ -periodic and the labeling above is chosen so that if j < l, then λj (k) < λl (k) for all k. Notice that this definition implies that functions λl may be discontinuous (at crossing points). (ii) |λl (k)| → +∞ as |l| → +∞, l ∈ D, uniformly for k ∈ B. (iii) For any positive integer N , there exists a closed subset FN ⊂ B of Lebesgue measure 0 (in fact, a closed analytic set) such that, for any l ∈ D such that |l| < N , the function λl is analytic on ΩN := B \ FN and the multiplicity of λl (k) as an eigenvalue of −ip1 (k) is constant on the components of ΩN . 1 (iv) Let Sl1 (k) ⊆ L2 (C)n denote the eigenspace of λl (k) and L by Πl1 (k) 2 n 1 the projector L (C) l∈Z Sl (k) L → Sl (k). For any k ∈ ΩN , 2 n = L (C) , where l denotes the direct sum of closed subspaces of a Hilbert space. Moreover, Π1l (k) is analytic in k ∈ ΩN for N >l.

Note that the eigenspaces Slε (k) (and their projectors Πεl (k) of −ipε (k)) are obtained from Sl1 (k) by the rescaling x → xε . The eigenvalues λ(k) are invariant under this rescaling. In generalization of [20, 2, 18, 17], we use the above eigenvalue problems on spaces of quasi-periodic functions to construct invariant spectral subspaces of L2 (Rm )n . The ε-dependent “Bloch decomposition” of L2 (Rm )n is obtained from the direct fibers Slε (k) introduced in property (P1)(iv). We denote by Slε the subspace of L2 (Rm )n corresponding to λl

(4.33)

Slε

Z

⊕ dk := (I ) Slε (k) |B| B  2 m n = u(x) ∈ L (R ) | (I ε u)(·, k) ∈ Slε (k) a.e. in B ε −1

where I ε is the isometry (4.27). The projection L2 (Rm )n → Slε is denoted by Πεl . It is obtained from (4.34)

Πεl = (I ε )−1

Z

⊕

B

Πεl (k)

dk ε I . |B|

By construction the subspaces Slε are mutually orthogonal, and due to Proposition 4.9 and property (P1)(iv) we have the following:

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355

P ROPOSITION 4.10 Let Slε be defined by (4.33). Then we have for ε ∈ (0, ε0 ] L2 (Rm )n =

(4.35)

M

Slε .

l∈Z

The action of (P ε )W in the subspaces is given by the following: P ROPOSITION 4.11 Let f (x) ∈ Slε . Then X
λˇl (γ)f (x + εγ) = ((iλl (εDx )f )) (x) ∈ Slε (4.36) (P ε )W f (x) = i γ∈L

where λˇl (γ), γ ∈ L, are the Fourier coefficients (4.14) of λl (k), k ∈ B. P ROOF : The proof is by direct calculation using (4.33), Proposition 4.9. The Hilbert space L2 (Rm )n is thus decomposed (by Bloch decomposition) into a direct sum of countably many “Band spaces” (Floquet spaces) Slε , which are invariant under the action of (P ε )W (i.e., the operator (P ε )W commutes with Πεl , ∀l ∈ D). Hence the IVP (4.1) can be replaced by denumerably many decoupled IVPs: C OROLLARY 4.12 The solution u(t, x) of (4.1a) and (4.1b) is given by X uε (t, x) = (4.37) uεl (t, x) l∈Z

where uεl (t, x) is obtained from (4.38a) ε

X ∂ ε ul + i λˇl (γ)uεl (x + εγ) = 0 , ∂t

x ∈ Rm x , t ∈ R,

γ∈L

uεl (x, t = 0) = uεI,l (x) ,

(4.38b) for l ∈ Z, with (4.38c)

x ∈ Rm x

uεI,l (x) := Πεl uεI (x) .

Note that in analogy to (2.8), the time evolution of the Fourier transforms uˆl (t, ξ) is governed by (4.39)

ε

∂ ε uεl = 0 , u ˆ + iλl (εξ)ˆ ∂t l

ξ ∈ Rm ξ , t ∈ R,

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for l ∈ Z, with the Fourier transform of uεI,l (x) in (4.38c) as initial datum. We define the lth Band-Wigner function (i.e., the Wigner series (4.9) of the th l component in the decomposition (4.37)) ε (4.40) wlε (t, x, k) := wsε [uεl (t)](x, k) , wI,l (x, k) := wsε [uεI,l ](x, k) ,

l ∈ Z,

which have uniform C 0 -bounds according to Proposition 4.3. For the zerothorder k-moment Z ε nl (t, x) := (4.41) wlε (t, x, k) dk = |uεl (t, x)|2 , B

we again have the energy conservation Z Z ε (4.42) nl (t, x) dx = |uεI,l (x)|2 dx , Rm x

Rm x

and due to Proposition 4.10 Z XZ ε (4.43) n (t, x) dx = Rm x

l∈Z

Rm x

nεl (t, x) dx ,

∀t ∈ Rt ,

∀t ∈ Rt .

The general “orthogonality result” Proposition 1.5 also holds for Wigner series measures Ws0 , where the proof is again a consequence of the positivity of Ws0 and the Schwarz inequality. For the Band-Wigner measures wl0 as the limits of (4.40), that is, the Wigner series measures of components (4.37) of uε (t, x), we have another result of that type: L EMMA 4.13 Let j 6= l and ΩN as in (P1)(iii) with N > |j|, |l|. Then ws0 [uεj + uεl ] = ws0 [uεj ] + ws0 [uεl ] in C 0 (Rt ; Cc0 (ΩN )0 ) . P ROOF : It is sufficient to prove that ws0 (uεj , uεl ) = 0 in D0 (Rt × ΩN ). We have ε∂t wsε (uεj , uεl ) + iwsε (λj (εD)uεj , uεl ) + iwsε (uεj , λl (εD)uεl ) = 0 . Passing to the limit with the help of Proposition 4.6 yields (λj (k) − λl (k)) ws0 (uεj , uεl ) = 0 in D0 (Rt × ΩN ) .

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357

Using (4.20) with f ε = g ε = uεl and the periodicity of λl we obtain from (4.39) an evolution equation for the x-Fourier transform w ˆlε (t, ξ, k) (4.44)

∂ ε w ˆ + iξ · ∂t l

Z

1 2

− 12

∇λl (k + εsξ) ds w ˆlε = 0 ,

l ∈ Z.

1 Note that w ˆlε (t) is uniformly bounded in L∞ (Rm ξ ; L (B)). As in Section 2 we have the desired result on ΩN as defined in property (P1)(iii) for ε → 0 (cf. Proposition 4.6).

˜ N , with Ω ˜ N = Rm ×(ΩN +L∗ ) and with N > |l|, wε L EMMA 4.14 On Rt × Ω x l ˜ N )) converges locally uniformly with respect to t to the unique C 0 (Rt ; M(Ω solution of the IVP:

(4.45b)

∂ 0 w + ∇λl (k) · ∇x wl0 = 0 ∂t l wl0 is periodic in k

(4.45c)

0 . wl0 (t = 0) = wI,l

(4.45a)

Of course, now we have to relate the Wigner series measure w0 (t) of to the Band-Wigner measures wl0 (t) of uεl (t) and thus reconstruct the “total” limiting energy density n0 from the countably infinite Band densities n0l . Defining A as the intersection of all the open sets ΩN \ (4.46) A := ΩN uε (t)

N ∈N

we have the following: L EMMA 4.15 Let ws0 (t) be the Wigner series measure of the solution uε (t) of the IVP (4.1) and wl0 (t), l ∈ Z, be given by Lemma 4.14. Then we have for ws0 (t), which, due to Proposition 4.7, depends continuously on t as a measure on the whole of Rm x × B, X 1ξ∈A ws0 (t) = 1ξ∈A (4.47) wl0 (t) l∈Z

in the strong topology of measures, ∀t ∈ Rt . P ROOF : We define for N ∈ N X sεN (t, x) := (4.48) uεl (t, x) , |l|≤N

ε rN (t, x) := uε − sεN (t, x) .

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csε (ζ, k): For f, g in L2 (Rm )n , we obtain by direct estimation for w (4.49)

ε ε [ [ kw ≤ kf − gk(kf k + kgk) 1 s [f ] − ws [g]kL∞ (Rm ζ ;L (B))

which gives, using the k-periodic extensions ws0 (x, ξ), ε \ \ 0 ε 0 ε m 0 ≤ C sup kr kL2 (Rm )n . (4.50) kw 0 N s [u ](t) − ws [sN ](t)kL∞ (Rm ζ ;Cc (Rξ ) ) ε>0

In the proof of Proposition 4.7 it was shown how to obtain k(P ε )W (·, Dx )uεI kL2 (Rm n ≤ C x )

(4.51)

from the ε-oscillatory property (1.26) by uniform approximation. From the Plancherel theorem and (4.39), we hence obtain k(P ε )W uεl kL2 (Rm uεl kL2 (Rm n = kiλl (ε·)ˆ n x ) ξ ) and a calculation using the orthogonality and invariance of the subspaces Slε as well as the conservation k(P ε )W uε (t)kL2 (Rm )n = k(P ε )W uεI kL2 (Rm )n gives ε 2 krN k ≤

X

1 ε W ε 2 2 k(P ) ul k min |λl (k)|

|l|>N k∈B

(4.52)

≤ max |l|>N

1 k(P ε )W uεI k2 . min |λl (k)|2 k∈B

Thus we can conclude ε kL2 (Rm )n = 0 . lim sup sup krN

N →∞ ε>0 t∈Rt

For the finite sum, application of Lemma 4.13 gives X 1ξ∈A ws0 [sεN ](t) = 1ξ∈A ws0 [uεl ](t) , ∀t ∈ Rt . |l|≤N

By observing that ∀α > 0: ws0 [uε1

+ uε2 ]

≤

(1 + α)ws0 [uε1 ] +

  1 ws0 [uε2 ] 1+ α

we get that the total mass of ws0 [uε ] − ws0 [sεN ] goes to 0 and we conclude (4.47).

HOMOGENIZATION LIMITS AND WIGNER TRANSFORMS

359

With the additional assumption on the initial data that wI0 does not charge on the “bad set,” that is, the complement of A (cf. (P1)(iii) and (4.46)): c 0 (B3) Rm x × A is a null set of the limiting initial Wigner series measure wI ,

we obtain the final result as follows: T HEOREM 4.16 Let (B1)–(B3) hold. Then, up to subsequences, we have for the Wigner series measures ws0 [uε ](t) =

X

ws0 [uεI,l ] (x − ∇λl (k)t, k) ∈ Cb (Rt ; M(Rm x × B)) ,

l∈Z

and the energy density nε converges locally uniformly in time ε ε→0

Z

n −→

ws0 [uε ](t, x, dk) B

and

Z lim

ε→0 Rm x

nε (t, x) dx = ws0 [uε ](t)(Rm x × B) .

P ROOF : We first apply Lemma 4.15 for t = 0 and compare with the Parseval formula X kuεI k2 = kuεl,I k2 . l∈D 0 is supported in A. This shows, by using assumption (B3), that wl,I Now we use the explicit formula for wl0 (t) in A given by Lemma 4.14 and compare with the energy conservation of uεl (t). This shows that wl0 (t) is supported in A. Finally, we use Lemma 4.15 and compare with the energy conservation for uε (t). This shows that ws0 (t) is supported in A, which gives the result.

5 Examples with Periodic Coefficients Below we present two typical examples for the general theory of Section 4, namely, the Schrödinger equation in a crystal and the acoustic equation in a periodic medium. The case of Maxwell equations in a periodic medium is again somewhat similar to the acoustic equation; for details we refer to [17].

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P. GÉRARD ET AL.

5.1 Schrödinger Equation in a Crystal A single electron in a crystal is described by the Schrödinger equation in Rm x for the scalar uε (t, x) (cf. [20, 7, 18]): x ε2 ∆uε + iVp uε = 0 , x ∈ Rm x , t ∈ R, 2 ε
with the initial condition (3.1b). Vp xε is the properly rescaled, periodic, real potential of the crystal ions and obeys (cf. (4.2)) Vp (x + µ) = Vp (x), ∀µ ∈ L. The symbol is hence given by (5.1)

εut − i

i P (x, ξ) = |ξ|2 − iVp (x) 2

(5.2)

which fulfills (B1) if we require the potential to be in C ∞ (Rm x ). (In [18] it ) is enough.) was shown that actually Vp (x) ∈ L∞ (Rm x Note that for a symbol P ε (x, ξ) that is additive in x and ξ the Fourier multiplier (“left symbol”) P ε(x, Dx ) coincides with the Weyl operator (P ε )W(x, Dx ) and our results of Chapter 4 apply directly. The cell problem (4.32) is given by a stationary Schrödinger equation subject to the k-quasi-periodicity condition, leading to the well-known energy bands (eigenvalues) El (k) and Bloch functions (eigenfunctions) Ψl (x, k), l ∈ N (see, e.g., [24, 18]). The Band-Wigner functions wlε (t, x, k), l ∈ N, obey [18] (5.3)

∂ ε w + Θε [El ]wlε = 0 , ∂t l

x ∈ Rm x , k ∈ B, t > 0,

with the finite difference (pseudo-differential) operator (Θε [El ]wlε ) (t, x, k) ε ε ε ε X ˆl (γ)eik·γ wl (t, x + 2 γ, k) − wl (t, x − 2 γ, k) = 0 . =i E ε γ∈L

In the limit we recover the free-streaming, semiclassical equations (5.4)

∂ 0 w (t, x, k) + vl (k)∇x wl0 (t, x, k) = 0 , ∂t l

with the “band velocities” vl (k) := ∇k El (k) ,

k ∈B.

l ∈ N,

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Assuming (B3), that is, that the limiting initial Wigner series measure does not charge the null sets where the energy bands cross, the limiting position density is given by 1 X n −→ n0 = |B| ε ε→0

Z 0 wI,l (x − vl (k) · t, k) dk .

B

l∈N

It is also possible to perform the homogenization limit for other physically interesting densities. With some additional assumptions (on the energy bands) we have (cf. [18]), for example, for the “current density” J ε (t, x) = ε Im[uε ∇uε ] 1 X J −→ |B| ε ε→0

l∈N

Z B

vl (k)wl0 (t, x, k) dk .

5.2 The Acoustic Equation in a Periodic Medium We now consider the homogenization of the acoustic equation in a periodic medium (cf. [2, 4]): x ε




uεtt = div s

(5.5)

r

(5.6)

uε (t = 0) = uεI ,







x ε ε ∇u uεt (t = 0)

in Rm x × Rt on Rm x

= p˜εI

for the scalar real-valued function uε . The energy density is given by nε := r

(5.7)

x ε

(uεt )2 + s

x ε

|∇uε |2 .

We assume that the functions r and s are L-periodic and C ∞ and that there is α > 0 such that r, s ≥ α > 0 on Rm . As for the wave equation in Section 3.3, we can rewrite (5.5) in the general form (4.1a) by introducing the new variables r   x ε u , p = r ε t

r   x ∇uε , q := s ε

ε

ε

which yields the (m + 1)-dimensional IVP  (5.8a)

 0  pε   − p x qε t s( ) ∇ √ · ε

r( xε )



p x
 s( ε ) ·  ε   p  ε =0 q 0

√ 1 x div r( ε )

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r   x ε p (t = 0) = r p˜ =: pεI , ε I r   x ε ∇uεI =: qIε , q (t = 0) = s ε ε

(5.8b)

The energy density (5.7) now reads nε = |(pε , q ε )|2 . The spatial operator defining (5.8a) is not elliptic for m > 1 since its Weyl symbol matrix has rank 2. In fact, it does not allow control on all first derivatives of the vector q ε but only on its divergence. As a remedy we set up a “penalized” version of (5.8). We define A(ξ) = |ξ|2 Id − ξ ⊗ ξ such that ~ + ∇x (div) . A(D) = −∆ The “penalization” now reads  ε  ε p ε p ε +L (5.9) =0 qε t qε where



0

  Lε = −  p x ε s( ε ) ∇x √ ·

r( xε )

p x
 s( ε ) · r( ε )    . ε2 · √ √ i x A(D) x √ ε x div s( ε )

s( ε )

m+1 . Clearly, −iLε is self-adjoint on L2 (Rm x ) Let us show for L1 that the crucial assumption (B1)(iii) is satisfied. Assume

√
1 √ div s q ∈ L2 (Rm ) r   i q + √ A(D) √ ∈ L2 (Rm )m . s s √ Multiplying the second relation by s we obtain    p ∈ H −1 (Rm ) div s∇x √ r

and

p s(x) ∇x



p √ r



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by taking the divergence. This implies p ∈ H 1 (Rm ). Moreover, the first relation leads to     √ 1 √ 1 q = ∇x √ sq + div( sq) ∈ L2 (Rm ) div √ s s s √ ~ and, coming back to the second relation, ∆(q/ s) ∈ H −1 (Rm )m , which implies q ∈ H 1 (Rm )m . Hence (B1)(iii) holds. Observe that, since A(D)∇x = 0, the IVP (5.9) and (5.8b) has the same solution as the IVP (5.8). The spectral problem for −ip1 (k) is given by √ 1 (5.10a) i √ div( sq) = λ(k)p r     √ 1 q p i s∇x √ (5.10b) − √ A(D) √ = λ(k)q r s s (5.10c)

∀γ ∈ L : p(x + γ) = eikγ p(x) , q(x + γ) = eikγ q(x) .

The property (P1) holds for the eigenvalues and eigenprojections and the theory of Section 4 can be applied. However, the problem (5.10) can be substantially simplified. First of all, a simple calculation shows that λ = 0 is an eigenvalue only for k = 0. The (m + 1)-dimensional null space of −ip1 (0) is given by      √ α ∈ C, β ∈ Cm , ϕ ∈ L2],1 (0) α r p √ √ = , β s + s∇ϕ q solves div(s∇ϕ) = −β · ∇s with L2],1 (0) as defined in (4.30b). The nonzero eigenvalues split into two p groups. The first group satisfies λ(k) = ± µ(k), where µ(k) are the eigenvalues of    1 p (5.11a) = µ(k)p − √ div s∇ √ r r ∀γ ∈ L : p(x + γ) = eik·γ p(x) .   p The corresponding eigenfunctions are where p satisfies (5.11) and q±   i √ p (5.12) q± = ± p s∇ √ . r µ(k) (5.11b)

The second group of eigenvalues λ(k) of −ip1 (k) is obtained by solving the spectral problem   1 q (5.13a) = λ(k)q − √ A(D) √ s s √ div( sq) = 0 , q ∈ L2],1 (k) . (5.13b)

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  0 The corresponding eigenfunctions are , where q satisfies (5.13). q qε

Since √Is is a gradient, the Floquet projections corresponding to this second group of eigenvalues do not contribute to the Wigner series measure. Thus it is sufficient to solve the spectral problem (5.11).

6 Slowly Variable Coefficients In this section we consider the Cauchy problem (6.1a)

εuεt + (P ε )W (x, εDx )uε = 0 ,

(6.1b)

uε (t = 0) = uεI

x ∈ Rm x , t ∈ R, on Rm x

for the pseudo-differential operator (P ε )W (x, εDx ) associated by Weyl’s quantization to a complex n × n–matrix-valued, ε-dependent symbol P ε (x, ξ) on m ε Rm x × Rξ . We impose the following assumptions on the symbol P : (C1)

(i) ∃σ ∈ R : P ε ∈ S σ (Rm )n×n uniformly for ε ∈ (0, ε0 ], n (ii) i(P ε )W (·, εDx ) is essentially self-adjoint on L2 (Rm x ) , ∞ (Rm × (iii) P ε (x, ξ) = P 0 (x, ξ) + εQ0 (x, ξ) + o(ε) uniformly in Cloc x m n×n . Rξ )

The hypothesis (C1)(i) means that P ε is of order σ uniformly as ε → 0; that is, for all α, β ∈ N0 there exists Cα,β > 0 such that for all l, k ∈ {1, . . . , m} and for all ε ∈ (0, ε0 ] we have ∂ α+β (6.2) α β P ε (x, ξ) ≤ Cα,β (1 + |ξ|)σ−β ∂x ∂ξ k

l

m σ m n for all (x, ξ) ∈ Rm x × Rξ . In particular, this implies that H (Rx ) is in the ε W domain of (P ) (x, εD) (see [13]). (C1)(ii) implies (6.3) P ε (x, ξ)∗ = −P ε (x, ξ) .

The condition (6.3), although necessary for the self-adjointness of the PDO −i(P ε )W (·, εDx ), is by no means sufficient. Examples of sufficient conditions are 1. σ = 1 2. P ε is elliptic; that is, (4.8) holds.

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On the initial function uεI we assume n (C2) uεI is bounded in L2 (Rm x ) , ε-oscillatory, and compact at infinity.

Obviously, − 1ε (P ε )W (x, εD) generates a strongly continuous group of unin tary operators on L2 (Rm x ) , and we have conservation of the total energy Z Z ε (6.4) n (t, x) dx = nεI (x) dx ∀t ∈ R. Rm x

Rm x

Again, we use the notation nε = |uε |2 and nεI = |uεI |2 for the energy density. Our task in this section will be to calculate the Wigner matrix W 0 of (uε (t, ·)). For this, we assume that (uεI ) admits a Wigner matrix WI0 , and the following: (C3)

m (i) There exists a closed subset E of Rm x × Rξ such that, for every (x, ξ) 6∈ E, the eigenvalues of −iP 0 (x, ξ) can be ordered as follows: λ1 (x, ξ) < · · · < λd (x, ξ) ,

where, for 1 ≤ q ≤ d, the multiplicity of λq (x, ξ) does not depend on (x, ξ). (ii) For 1 ≤ q ≤ d, the Hamiltonian flow of λq leaves invariant the set m Ω = (Rm x × Rξ ) \ E .

(iii) E is a null set of the measure wI0 = tr(WI0 ). For 1 ≤ q ≤ d, (x, ξ) ∈ Ω, we denote by Πq (x, ξ) the orthogonal projection of Cn on the eigenspace associated to λq (x, ξ). Of course, Πq ∈ C ∞ (Ω)n×n . We can now formulate the following theorem: T HEOREM 6.1 Let (C1), (C2), and (C3) hold and let {·} denote the Poisson bracket (1.34). (i) For 1 ≤ q ≤ d, we denote by wq0 (t) the continuously t-dependent positive m scalar measure on Rm x × Rξ defined by (6.5a) (6.5b) (6.5c)

∂ 0 w + {λq , wq0 } = 0 on Rt × Ω ∂t q wq0 (t = 0) = tr(Πq WI0 ) on Ω wq0 (t, E) = 0 ,

t ∈ R.

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Then the scalar Wigner transform wε (t, x, ξ) of uε (t) converges locally uniformly in t to w0 (t, x, ξ) =

d X

wq0 (t, x, ξ) ,

q=1

and

nε (t, x)

converges locally uniformly in t to Z 0 w0 (t, x, dξ) . n (t, x) = Rm ξ

(ii) For 1 ≤ q ≤ d, we set Fq = [Πq , {λq , Πq }] + Πq Q0 Πq with Q0 given by (C1)(iii); denote by Wq0 the continuously t-dependent, positive-matrixm valued measure on Rm x × Rξ defined by ∂ 0 W + {λq , Wq0 } = [Wq0 , Fq ] on Rt × Ω ∂t q

(6.6a)

Wq0 (t = 0) = Πq WI0 Πq

(6.6b)

t ∈ R.

Wq0 (t, E) = 0 ,

(6.6c)

on Ω

Then the Wigner transform W ε (t, x, ξ) of uε (t) converges in L∞ (Rt , S 0 ) weak-∗ to d X W0 = Wq0 , q=1

and

uε

⊗u ¯ε

converges in

L∞ (Rt , S 0 ) Z

N0 = Rm ξ

weak-∗ to

W 0 (·, ·, dξ).

R EMARK 6.2 (1) Unlike (i), the convergence in (ii) may not be uniform and not even almost everywhere in t (a counterexample is given in Section 7). ˜ q0 the solution to (2) Denote by W (6.7a) (6.7b) (6.7c)

˜ q0 ∂W ˜ 0 } = 0 on Rt × Ω + {λq , W q ∂t ˜ q0 (t = 0) = Πq WI0 Πq on Ω W ˜ q0 (t, E) = 0 , W

t ∈ R.

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˜ 0 (t), it Then, since Fq∗ = −Fq , Wq0 (t) remains unitarily equivalent to W q is actually positive. The precise formula is ˜ 0 (t)Uq (t)∗ Wq0 (t) = Uq (t)W q

(6.8) where

∂Uq + {λq , Uq } = Fq Uq ∂t Uq (t = 0) = I .

(6.9a) (6.9b)

Moreover, in view of the expression of Fq , we have {λq , Πq } = [Πq , Fq ], hence Wq0 = Πq Wq0 Πq . Then an elementary computation shows that equation (6.6a) can be rewritten as    ∂Wq0 + Πq λq , Wq0 Πq = Wq0 , Πq Q0 Πq . ∂t

(6.10)

(3) Observe that Πq {λq , W }Πq can be seen as the covariant derivative of the matrix-valued distribution W along the Hamiltonian vector field of λq associated to the induced connection on the subbundle defined by the equations W = Πq W Πq . P ROOF (6.11) ε

OF

∂W ε

T HEOREM 6.1: Using equation (6.1a), we have



= −W ε (P ε )W (x, εD)uε , uε − W ε uε , (P ε )W (x, εD)uε .

∂t In view of (1.37), (1.38), (C1)(i), and (6.3), we can rewrite (6.11) as (6.12)

∂W ε W εP ε − P εW ε 1 = + ({W ε , P ε } − {P ε , W ε }) + Rε , ∂t ε 2i

m where Rε → 0 in L∞ (Rt , S 0 (Rm x × Rξ )). From (6.12), we first conclude two facts: ∞ 1. Taking the trace of (6.12), we observe that ∂w ∂t is bounded in L (Rt , 0 m m S (Rx ×Rξ )). This implies the uniform convergence stated in Theorem 6.1(i). ε

2. Passing to the limit in (6.12), we observe that every limit W 0 of a subsequence of (W ε ) in L∞ (Rt , S 0 ) weak-∗ has to satisfy (6.13)

W 0P 0 = P 0W 0 .

In particular, on Rt × Ω, for 1 ≤ q ≤ d, (6.14)

W 0 Πq = Πq W 0 .

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Now we fix W 0 to be the limit in L∞ (Rt , S 0 ) weak-∗ of some subsequence of (W ε ), and we study the propagation identity (6.12) on the open set Rt × Ω. Multiplying by Πq on the left and right and then passing to the limit, we obtain

(6.15)

∂ Πq W 0 Πq = Πq W 0 Q0 Πq − Πq Q0 W 0 Πq ∂t
1 + Πq {W 0 , P 0 } − {P 0 , W 0 } Πq . 2i

Set Wq0 = Πq W 0 Πq on Rt × Ω. In view of (6.14), we have  1  
∂ 0  0 Wq = Wq , Πq Q0 Πq + Πq W 0 , P 0 − P 0 , W 0 Πq . ∂t 2i P On Ω, we have P 0 = i dl=1 λl Πl . Hence (6.16)

d X  0 0  0 W , P Πq = i W , λl Πl Πq

(6.17) =i

l=1 d X

d X  0  W , λl Πl Πq + i λl W 0 , Πl Πq .

l=1

l=1

Notice that Πl Πq = 0 if l 6= q. Hence, plugging (6.17) and its adjoint in (6.16), we obtain

(6.18)

 ∂ 0  0 Wq = Wq , Πq Q0 Πq − Πq {λq , W 0 }Πq ∂t d
1X − λl Πq {Πl , W 0 } − {W 0 , Πl } Πq 2 l=1

We claim that the third term in the right-hand side cancels. This is a consequence of the following lemma: L EMMA 6.3 Let Π1 and Π2 be two projector-valued C ∞ -functions and W m be a matrix-valued distribution on Ω ⊂ Rm x × Rξ . Assume that Π1 Π2 = Π2 Π1 = 0 , Πj W = W Πj , Then (6.19)

j = 1, 2.

Π1 ({Π2 , W } − {W, Π2 }) Π1 = 0 .

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P ROOF : We shall make systematic use of the elementary identity (6.20)

A{B, C} − {A, B}C = {AB, C} − {A, BC}

Particular cases of (6.20) are Π1 {Π2 , W } − {Π1 , Π2 }W = −{Π1 , Π2 W } −{W, Π2 }Π1 + W {Π2 , Π1 } = {W Π2 , Π1 }. Hence Π1 ({Π2 , W } − {W, Π2 }) Π1 (6.21)

= {Π1 , Π2 }W Π1 − Π1 W {Π2 , Π1 } − {Π1 , Π2 W }Π1 + Π1 {W Π2 , Π1 } .

Since Π2 W = W Π2 , the two last terms on the right-hand side of (6.21) give, in view of (6.20), −{Π1 , Π2 W }Π1 + Π1 {W Π2 , Π1 } = −{Π1 , Π2 W Π1 } + {Π1 W Π2 , Π1 } =0 since Π1 W Π2 = W Π1 Π2 = 0 and similarly Π2 W Π1 = 0. Coming back to (6.21) and using W Π1 = Π1 W , we have Π1 ({Π2 , W } − {W, Π2 }) Π1 = {Π1 , Π2 }Π1 W − W Π1 {Π2 , Π1 } . But {Π1 , Π2 } = {Π1 , Π22 } = {Π1 , Π2 }Π2 in view of (6.20). Hence {Π1 , Π2 } Π1 = 0 and similarly Π1 {Π2 , Π1 } = 0. This completes the proof. Now we come back to (6.18). For q 6= l, we have  
Πq Πl , W 0 − W 0 , Πl Πq = 0 because of the above lemma. This is also true for q = l, since     Πq , W 0 − W 0 , Πq = W 0 , Id − Πq − Id − Πq , W 0 . Finally, (6.18) becomes (6.22)

  ∂ 0  0 Wq = Wq , Πq Q0 Πq − Πq λq , W 0 Πq . ∂t

It remains to observe that   λq , Wq0 = λq , Πq W 0 Πq  = Πq λq , W 0 Πq + Wq0 {λq , Πq } + {λq , Πq } Wq0

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and using Wq0 = Wq0 Πq = Πq Wq0 with {λq , Πq } Πq = (Id − Πq ) {λq , Πq } because (Πq )2 = Πq , we have Πq {λq , Πq } Πq = 0, hence   Wq0 {λq , Πq } = Wq0 , Πq {λq , Πq }

  {λq , Πq } Wq0 = {λq , Πq } Πq Wq0 = Wq0 , − {λq , Πq } Πq

so that h i   λq , Wq0 = Πq λq , W 0 Πq + Wq0 , [Πq , {λq , Πq }] Plugging this relation in (6.22), we obtain precisely (6.6a). It is now easy to complete the proof of Theorem 6.1. On Rt × Ω, we have X X Πq W 0 = Wq0 , W0 = q

q

m where the last equality comes from (6.14). This implies that, on Rt ×Rm x ×Rξ , we have X (6.23) Wq0 1Ω W0 ≥ q

where 1Ω is the indicator function on Ω. Let us compare the total masses of both sides of (6.23). Observe that, by (6.6a), tr(Wq0 ) = wq0 solves (6.5a), hence tr(Wq0 )1Ω is the solution to (6.5), so its total mass equals tr(WI0 Πq )(Ω). Hence the total mass of the left-hand side equals X m tr(WI0 Πq )(Ω) = tr(WI0 )(Ω) = tr(WI0 )(Rm x × Rξ ) q

= lim kuεI k2 by assumptions (C3)(iii) and (C2). Because of the conservation of energy, we have, for every t, m lim kuεI k2 = lim kuε (t)k2 ≥ (tr W 0 )(Rm x × Rξ , t)

where the last inequality comes from Proposition 1.7(ii). We conclude that (6.23) is an equality. Repeating the same argument as in the proof of Theε Rorem 2.1, 0we also conclude that n (t, ·) converges locally uniformly in t to Rm (tr W )(t, x, dξ). Finally, the last statement comes from Proposition 1.7(i) ξ

and the fact that ∀z ∈ Cn and ∀t ∈ R, (z · uε ) is ε-oscillatory. Thus Theorem 6.1 is completely proved.

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7 Examples with Slowly Varying Coefficients 7.1 Schrödinger Equation with a Potential A single electron under the influence of an arbitrary (given) potential V (x) ε is described by the Schrödinger equation in Rm x for the scalar u (t, x) (cf. [16, 15]): (7.1)

εut − i

ε2 ∆uε + iV (x)uε = 0 , 2

x ∈ Rm x , t ∈ R,

with the initial condition (3.1b). In this case the symbol is given by i P (x, ξ) = |ξ|2 + iV (x) , 2

(7.2)

which fulfills (C1)(i) if we require V (x) ∈ C ∞ (Rm x ). Note that there is no εdependence, hence P = P 0 and (C1)(iii) is trivially satisfied. Since the “left symbol” (1.2) coincides with the Weyl operator (P ε )W and V (·) is assumed to be real, (C1)(ii) also holds. Clearly, for the scalar symbol, (C3) is fulfilled with λ(x, ξ) = 12 |ξ|2 + V (x) and empty set E. The Wigner function wε (t, x, ξ) = wε [uε (t)] (x, ξ) obeys the so-called Wigner equation [19] (7.3)

∂ ε w + ξ · ∇x wε + Θε [V ] wε = 0 , ∂t

m x ∈ Rm x ξ ∈ Rξ , t > 0 ,

with the pseudo-differential operator 1 Θ [V ] w = (2π)m ε

(7.4)

Z

Z

ε

i Rm ξ0

Rm v

V (x + 2ε v) − V (x − 2ε v) ε

0

· w(t, x, ξ 0 )ei(ξ−ξ )·v dv dξ 0 . The position density nε = |uε |2 is again given by (3.3). Assuming (C2) on the initial data, our theory of Chapter 6 applies. In the classical limit ε → 0 we recover the Vlasov equation (7.5)

∂ 0 w (t, x, ξ) + ξ · ∇x w0 (t, x, ξ) − ∇x V (x) · ∇ξ w0 (t, x, ξ) = 0 ∂t

with the initial datum wI0 , that is, the initial Wigner measure (3.2b) of uεI . The position density nε (t, x) converges (locally) uniformly in t: Z ε→0 0 ε n (t, x) −→ n (t, x) = w0 (t, x, dξ) . Rm ξ

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R EMARK 7.1 (1) The assumptions on V (x), of course, can be significantly weakened. Detailed results are given in [15] (and partially in [16]), where the case of a self-consistent potential is also dealt with. (2) Again, it is possible to homogenize other quantities like the “current density” (see [16]).

7.2 The Dirac Equation We now consider the semiclassical limit of the Dirac equation. This relativistic version of the Schrödinger equation describes very fast electrons in an electromagnetic field. The equation reads 3 X

(7.6)

(iγ µ ∂µ − M + gγ µ Aµ ) Ψ = 0 .

µ=0

Here Ψ = Ψ(t, x) ∈ C4 , t ≡ x0 ∈ R, and x = (x1 , x2 , x3 ) ∈ R3 , the “Spinorfield,” is the unknown playing the role of uε (t, x) in Section 6. ∂µ ∂ stands for ∂x∂ µ , that is, ∂0 = ∂t , ∂k = ∂x∂ k , where we adopt the notation that the Greek letter µ denotes 0, 1, 2, 3 and k denotes the three spatial-dimension indices 1, 2, 3. γ µ ∈ C4×4 , µ = 0, . . . , 3, are the 4 × 4 Dirac matrices, which are closely related to the 2 × 2 Pauli matrices. Their elements are 0, 1, i, and they satisfy ∗

∗

(7.7)

γ 0 = γ 0 , γ k = −γ k ,

(7.8)

γµγν + γν γµ = 0 ,

k = 1, 2, 3,

µ 6= ν ,

(γ 0 γ k )∗ = γ 0 γ k ,

2

2

γ 0 = Id , γ k = −Id .

The relativistic current density J is a 4-vector with elements Jµ given by (7.9)

Jµ (t, x) = γ 0 γ µ Ψ(t, x) · Ψ(t, x) ,

and using (7.8) the relativistic position density n(t, x) is given by (7.10)

n(t, x) = J0 (t, x) =

3 X

Ψµ (t, x) · Ψµ (t, x) .

µ=0

Aµ (x) ∈ R, µ = 0, . . . , 3, are the components of the electromagnetic potential; in particular, A0 is the electric potential and A = (A1 , A2 , A3 )> is the magnetic potential vector. Hence the electric field is given by E = ∇x A0 and the magnetic field B = curlx A.

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373

Finally, we have the physical constants M = m~0 c and g = ~e , where m0 is the electron’s rest mass, c is the velocity of light, ~ is the Planck constant, and e is the unit charge. There are two physically meaningful limits: the relativistic limit c → ∞ and the classical limit ~ → 0. Here we shall be concerned with the classical limit only and therefore identify ~ ≡ ε as our small parameter. In order to obtain the form (6.1a) we rewrite (7.6) as (7.11)

εΨεt + P (x, εDx )Ψε = 0 ,

x ∈ R3x , t ∈ R ,

Ψε (t = 0) = ΨεI ∈ L2 (R3x )4

(7.12)

with the symbol matrix P = P 0 (7.13) P (x, ξ) = i

3 X

! γ 0 γ k (ξk − eAk (x)) + m0 cγ 0 − eA0 (x)Id

k=1

Notice that here the Weyl symbol (P )W coincides with the left symbol (1.3). From (7.7) and (7.8) we immediately see that iP is hermitian. We set Q :=

3 X

γ 0 γ k (ξk − eAk (x)) + m0 cγ 0 .

k=1

A straightforward computation using (7.7) and (7.8) leads to ! 3 X |ξk − eAk |2 + m20 c2 Id QQ∗ = Q2 = k=1

and we remark that if y is an eigenvector of Q corresponding to the eigenvalue λ, then (γ 0 − mλ0 c )y is an eigenvector corresponding to −λ. Hence the 4 × 4 matrix −iP (x, ξ) has two eigenvalues of multiplicity 2, which are given by v u 3 uX (7.14) |ξk − eAk (x)|2 + m20 c2 − eA0 (x) . λ± (x, ξ) = ±t k=1

Therefore Theorem 6.1 applies with the empty set E in assumption (C3) and we have Z Z ε→0 ε 0 0 (7.15) w+ (t, x, dξ) + w− (t, x, dξ) n (t, x) −→ R3ξ

R3ξ

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0 and w 0 are given by where the positive measures w+ −

(7.16)

0 + ∇ λ± · ∇ w 0 − ∇ λ± · ∇ w 0 = 0 ∂t w± x ± x ξ ξ ±

(7.17)

0 (t = 0) = tr(Π W 0 ) . w± ± I

As usual, WI0 denotes a Wigner measure matrix of ΨεI . The projections are easily calculated. We have (7.18)

1 1 Π+ = (Id + Q) , 2 σ

1 1 Π− = (Id − Q) , 2 σ

P with σ = ( 3k=1 |ξk − eAk |2 + m20 c2 )1/2 . We remark that (7.16) is not yet the Liouville equation for transport with a Lorentz force. With the definition (7.19)

0 f± (t, x, ξ) = w± (t, x, eA ± ξ)

and the change of variable to a relativistic velocity (7.20)

ξ v(ξ) = p , 2 |ξ| + m20 c2

we finally obtain the Liouville equations (7.21) ∂t f± + v(ξ) · ∇x f± ∓ e (∇x A0 + v(ξ) ∧ curl A) · ∇ξ f± = 0 R and n0 (t, x) = R3 f+ (t, x, dξ) + f− (t, x, dξ). Note that f− represents the ξ contribution of positrons and hence the term of the Lorentz force changes sign in comparison to the electrons, which correspond to f+ . The homogenization limit of (the other components of) the relativistic current density vector can also be easily computed. For its components Jkε , given by (7.9), we have Z   ε Jk (t, x) = tr γ 0 γ k W ε [Ψε (t)] (x, ξ) dξ , R3ξ

and from Theorem 6.1(ii) we obtain Jkε → Jk0 with Z (7.22)

Jk0 (t, x)

tr (γ 0 γ k W 0 (t, x, dξ)) .

= R3ξ

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375

Here W 0 (t) is the Wigner measure of the solution Ψε (t) of (7.11)–(7.13). The final expression for the current (7.22) using (7.20) is given by Z 0 (7.23) vk (ξ) (f+ (t, x, dξ) + f− (t, x, dξ)) Jk (t, x) = R3ξ

which is consistent with equation (7.21) and the continuity equation ∂µ J µ = 0. P ROOF

OF

(7.23):: Observe that Qγ 0 γ k + γ 0 γ k Q = 2(ξk − Ak ) .

Using the expression (7.18) for Π± , we obtain γ 0 γ k Π+ = Π− γ 0 γ k + vk I ; hence, since W 0 = Π+ W 0 Π+ + Π− W 0 Π− because of (6.14), we have γ 0 γ k W 0 = Π− γ 0 γ k W 0 Π+ + vk W 0 Π+ + Π+ γ 0 γ k W 0 Π− − vk W 0 Π− . Then take the trace of both sides and use Π+ Π− = 0 to obtain 0 0 tr(γ 0 γ k W 0 ) = vk (w+ − w− ).

By integrating with respect to ξ and using the change of variables given by (7.20), we obtain equation (7.23).

7.3 A Simple Counterexample to Uniform Convergence As stated in Theorem 6.1(ii), the convergence of the Wigner matrix is in general not uniform in time, as the following simple counterexample  ε shows: ε Let us consider the following 2 × 2 system for u = fgε ∂t f ε + ∂x f ε = 0 ,

∂t g ε − ∂x g ε = 0 ,

with initial data fIε (x) = a(x + 1) ei(x+1)/ε ,

gIε (x) = b(x − 1) ei(x−1)/ε ,

where a, b are two functions in L2 (R). An elementary calculation yields f ε (t, x) = a(x + 1 − t) ei(x+1−t)/ε ,

g ε (t, x) = b(x − 1 + t) ei(x−1+t)/ε ;

hence wε (uε (t))(x, ξ) = e2i(1−t)/ε wε (a(· + 1), b(· − 1))(x, ξ − 1) which goes to 0 weakly in t, while w0 [uε (1)] (x, ξ) = a(x + 1) ¯b(x − 1) δ(ξ − 1) . Thus the convergence is not uniform in t ∈ [0, 1].

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Appendix: Proof of Proposition 1.8 Let f, g ∈ L2 (Rm ) and let p be as in Proposition 1.8. The proof is based on the following lemma: L EMMA A.1 Given a ∈ S(Rm × Rm ), we have

ε W
 w p (x, εD)f, g , a = hwε (f, g), (a]p)ε i , (A.1) where −2m

Z

(a]p)ε (x, ξ) = (2π)

i(xζ+ξz)

a ˆ(ζ, z) e

  ζ z dz dζ . p x +ε ,ξ −ε 2 2

(A.2) P ROOF : It is enough to prove the lemma with f, g ∈ S, so we only have to make formal calculations. First of all, we observe the following formula, which follows from a simple change of variable in (1.6): ε wε (f, g)(x, ξ) = (f | Mx,ξ g)

(A.3)



where (A.4)

ε Mx,ξ g(y) =

1 πε

m

e−2i(x−y)·ξ/ε g(2x − y) .

Applying (A.3) and (1.8), we obtain




 ε ε wε pW (x, εD)f, g (x, ξ) = pW (x, εD)f | Mx,ξ g = wε f, Mx,ξ g ,p . Hence, by Fubini’s theorem, we have, for a ∈ S,

ε W
 w p (x, εD)f, g , a = hwε (f, G), pi (A.5) where

Z ε Mx,ξ g(y)¯ a(x, ξ) dx dξ Z = (πε)−m g(2x − y) e−2i(x−y)ξ/ε a ¯(x, ξ) dx dξ Z ¯ = (2π)−2m g(y − εz)e−iζ(y−εz/2) a ˆ(ζ, z) dz dζ .

G(y) =

An elementary calculation then gives Z ε −2m ε −i(tζ+τ z) ¯ a ˆ(ζ, z) dz dζ . Mt−εz/2,τ Mt,τ G(y) = (2π) +εζ/2 g(y) e

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Hence, using (A.3) again, wε (f, G)(t, τ ) −2m

Z



z ζ w (f, g) t − ε , τ + ε 2 2 ε

= (2π)

 ˆ(ζ, z) dz dζ . ei(tζ+τ z) a

Coming back to (A.5) and setting t = x + εz/2 and τ = ξ − εζ/2 in the integral, we obtain (A.1) and (A.2). We can now easily prove Proposition 1.8. If f, g vary in a bounded subset of L2 and ε ∈ (0, ε0 ], we know that wε (f, g) remains bounded in S 0 ; hence it is enough to prove an expansion of (a]p)ε in S. Let us first prove that (a]p)ε is bounded in S. Indeed, by integration by parts, we have, for any multi-indices α, β ∈ Nm × Nm , β (a]p)ε (x, ξ) (x, ξ)α ∂x,ξ   Z ζ z β dz dζ . = (2π)−2m ei(xζ+ξz) (i∂ζ,z )α a ˆ(ζ, z)∂x,ξ p x +ε ,ξ −ε 2 2

Using assumption (1.35) on p and a ˆ ∈ S, we easily obtain β (x, ξ)α ∂x,ξ (a]p)ε (x, ξ) ≤ Cα,β (1 + |x| + |ξ|)M , which proves that (a]p)ε is bounded in S. Now, by performing a Taylor expansion of p and using the Fourier inversion formula for the first terms, we obtain ε (a]p)ε (x, ξ) = a(x, ξ) p(x, ξ) + (∇ξ a(x, ξ) · ∇x p(x, ξ) 2i − ∇x a(x, ξ).∇ξ p(x, ξ)) + ε2 rε (x, ξ) , with −2m

Z

1Z i(xζ+ξz)

e

rε (x, ξ) = (2π)

0

  ζ z a ˆ(ζ, z) p x + tε , ξ − tε 2 2 00

(z, ζ)2 dz dζ (1 − t) dt . 2 It remains to prove that rε is bounded in S, which can be done along the same lines as we did before for (a]p)ε . This completes the proof. ·

R EMARK A.2 The proof above shows that assumption (1.35) on p can be relaxed to α p(x, ξ) ≤ Cα (1 + |ξ|)M+ρ|α| ∀α ∈ Nm × Nm : ∂x,ξ for some ρ < 1.

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Acknowledgement. The last three authors acknowledge financial support from the HCM Network ERBCHRXCT 930413 and from the DAADPROCOPE. The second and third author also acknowledge support from DFG Project MA 1662/1-1.

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PATRICK G ÉRARD Université de Paris–Sud Mathématiques Bâtiment 425 F-91405 Orsay FRANCE E-mail: Patrick.Gerard@ math.u-psud.fr

P ETER A. M ARKOWICH Fachbereich Mathematik TU-Berlin Straße des 17. Juni 136 D-10623 Berlin GERMANY E-mail: markowic@ math.tu-berlin.de

N ORBERT J. M AUSER Fachbereich Mathematik TU-Berlin Straße des 17. Juni 136 D-10623 Berlin GERMANY E-mail: mauser@ math.tu-berlin.de

F REDERIC P OUPAUD Université de Nice Lab. J. A. Dieudonne URA 168 du CNRS—UNSA Parc Valrose F-06108 Nice FRANCE E-mail: poupaud@ math.unice.fr

Received March 1996.