PERIODIC HOMOGENIZATION AND EFFECTIVE MASS THEOREMS FOR THE SCHR - - PowerPoint PPT Presentation

Homogenization of Schr¨

• dinger equation

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• G. Allaire

PERIODIC HOMOGENIZATION AND EFFECTIVE MASS THEOREMS FOR THE SCHR¨ ODINGER EQUATION

Gr´ egoire ALLAIRE, CMAP, Ecole Polytechnique.

• 1. Asymptotic expansions in periodic homogenization
• 2. Two-scale convergence
• 3. Application to homogenization
• 4. Bloch waves
• 5. Schr¨
• dinger equation in periodic media
• 6. Homogenization without drift / with drift
• 7. Localization
• 8. Time oscillating potential

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• I- DEFINITION OF HOMOGENIZATION

☞ Rigorous version of averaging ☞ Process of asymptotic analysis ☞ Extract effective or homogenized parameters for heterogeneous media ☞ Derive simpler macroscopic models from complicated microscopic models ☞ Different methods :

• two-scale asymptotic expansions for periodic media
• two-scale convergence
• H- or G-convergence for general media
• stochastic, or variational methods (Γ-convergence)

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Periodic homogenization

ε

• ✁

Ω

Periodic domain Ω ∈ RN with period ǫ. Rescaled unit cell Y = (0, 1)N. x ∈ Ω, y = x ǫ ∈ Y Example: Composite material with a periodic structure

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Model problem

Conductivity or diffusion equation    − div

• A

x

ǫ

• ∇uǫ
• = f

in Ω uǫ = 0

• n ∂Ω

with a coefficient tensor A(y) which is Y -periodic, uniformly coercive and bounded α|ξ|2 ≤

N

• i,j=1

Aij(y)ξiξj ≤ β|ξ|2, ∀ ξ ∈ RN, ∀ y ∈ Y (β ≥ α > 0).

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Homogenization and asymptotic analysis

➫ Direct solution too costly if ǫ is small ➫ Averaging: replace A(y) by effective homogeneous coefficients ➫ Asymptotic analysis: limit as ǫ → 0 yields a rigorous definition of the homogenized parameters ➫ Error estimates: compare exact and homogenized solutions ➫ Similar to Representative Volume Element method (in mechanics) ➫ Huge literature

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✞ ✝ ☎ ✆

Asymptotic analysis

Rather than considering a single heterogeneous medium with a fixed lengthscale ǫ0, the problem is embedded in a sequence of similar problems parametrized by a lengthscale ǫ. Homogenization amounts to perform an asymptotic analysis when ǫ → 0 lim

ǫ→0 uǫ = u.

The limit u is the solution of an homogenized problem, the conductivity tensor of which is called the effective or homogenized conductivity. This yields a coherent definition of homogenized properties which can be rigorously justified by quantifying the resulting error estimate.

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Two-scale asymptotic expansions

Ansatz for the solution uǫ(x) =

+∞

• i=0

ǫiui

• x, x

ǫ

• ,

with ui(x, y) function of both variables x and y, periodic in y Derivation rule ∇

• ui
• x, x

ǫ

• =
• ǫ−1∇yui + ∇xui

x, x ǫ

• ∇uǫ(x) = ǫ−1∇yu0
• x, x

ǫ

• +

+∞

• i=0

ǫi (∇yui+1 + ∇xui)

• x, x

ǫ

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Cascade of equations

−ǫ−2 [ divyA∇yu0]

• x, x

ǫ

• −ǫ−1 [ divyA(∇xu0 + ∇yu1) + divxA∇yu0]
• x, x

ǫ

• −ǫ0 [ divxA(∇xu0 + ∇yu1) + divyA(∇xu1 + ∇yu2)]
• x, x

ǫ

• −

+∞

• i=1

ǫi [ divxA(∇xui + ∇yui+1) + divyA(∇xui+1 + ∇yui+2)]

• x, x

ǫ

• = f(x).

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✞ ✝ ☎ ✆

ǫ−2 equation

− divy (A(y)∇yu0(x, y)) = 0 in Y where x is just a parameter. Its unique solution does not depend on y u0(x, y) ≡ u(x)

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✞ ✝ ☎ ✆ Technical lemma on cell problems Definition. L2

#(Y ) =

• φ(y) Y -periodic, such that
• Y

φ(y)2dy < +∞

• H1

#(Y ) =

• φ ∈ L2

#(Y ) such that ∇φ ∈ L2 #(Y )N

• Lemma. Let f(y) ∈ L2

#(Y ) be a periodic function. There exists a solution in

H1

#(Y ) (unique up to an additive constant) of

   − div (A(y)∇w(y)) = f in Y y → w(y) Y -periodic, if and only if

• Y f(y)dy = 0 (this is called the Fredholm alternative).

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✞ ✝ ☎ ✆

ǫ−1 equation

− divyA(y)∇yu1(x, y) = divyA(y)∇xu(x) in Y which is an equation for u1. Introducing the cell problem    − divyA(y) (ei + ∇ywi(y)) = 0 in Y y → wi(y) Y -periodic, by linearity we compute u1(x, y) =

N

• i=1

∂u ∂xi (x)wi(y).

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✞ ✝ ☎ ✆

ǫ0 equation

− divyA(y)∇yu2(x, y) = divyA(y)∇xu1 + divxA(y) (∇yu1 + ∇xu) + f(x) which is an equation for u2. Its compatibility condition (Fredholm alternative) is

• Y

( divyA(y)∇xu1 + divxA(y) (∇yu1 + ∇xu) + f(x)) dy = 0. Replacing u1 by its value yields the homogenized equation    − divxA∗∇xu(x) = f(x) in Ω u = 0

• n ∂Ω,

with the constant homogenized tensor A∗

ij =

• Y

[(A(y)∇ywi) · ej + Aij(y)] dy =

• Y

A(y) (ei + ∇ywi) · (ej + ∇wj) dy.

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✞ ✝ ☎ ✆

Comments

➫ The ansatz is not correct after the second term uǫ(x) ≈ u(x) + ǫ

N

• i=1

∂u ∂xi (x)wi x ǫ

• (boundary layers should be included for higher order terms)

➫ Similar ansatz for ∇uǫ, A x

ǫ

• ∇uǫ and A

x

ǫ

• ∇uǫ · ∇uǫ.

➫ Explicit formula for the effective parameters (no longer true for non-periodic problems). ➫ A∗ does not depend on ǫ, f, u or the boundary conditions. ➫ A∗ is positive definite (not necessarily isotropic even if A(y) was so). ➫ Same results for evolution problems. ➫ Very general method, but heuristic and not rigorous.

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✞ ✝ ☎ ✆

Time dependent problems

Schr¨

• dinger equation in a periodic medium:

     i∂uǫ ∂t − div

• A
• x, x

ǫ

• ∇uǫ
• + c
• x, x

ǫ

• uǫ = 0

in RN × R+ uǫ(t = 0, x) = u0(x) in RN, Complex-valued unknown: uǫ(t, x) : R+ × RN → C ☞ y → A(x, y), c(x, y) (0, 1)N-periodic, real, measurable and bounded. ☞ A is a N × N symmetric, uniformly coercive, tensor. ☞ c is a scalar function with no specified sign. ☞ non oscillating initial data u0 ∈ H1(RN).

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✞ ✝ ☎ ✆

Same results !

uǫ(t, x) ≈ u(t, x) + ǫ

N

• i=1

∂u ∂xi (t, x)wi(x ǫ )      i∂u ∂t − div (A∗∇u) + c∗u = 0 in RN × R+ u(t = 0, x) = u0(x) in RN, with c∗(x) =

• Y c(x, y) dy and the homogenized tensor

A∗

ij(x) =

• Y

A(x, y) (ei + ∇ywi) · (ej + ∇wj) dy.    − divyA(x, y) (ei + ∇ywi) = 0 in Y y → wi(x, y) Y -periodic,

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Convergence proof

The goal is to mathematically prove the following

• Theorem. The sequence uǫ(x) of solutions of the model problem converges

weakly in H1

0(Ω) and strongly in L2(Ω), as ǫ goes to 0, to a limit u(x) which is

the unique solution of the homogenized problem. lim

ǫ→0

• Ω

|uǫ − u|2dx = 0, lim

ǫ→0

• Ω

∇(uǫ − u) · φ dx = 0 ∀ φ ∈ L2(Ω)N. Naive idea: pass to the limit in the variational formulation

• Ω

A x ǫ

• ∇uǫ(x) · ∇ϕ(x)dx =
• Ω

f(x)ϕ(x)dx, ∀ ϕ ∈ H1

0(Ω).

impossible because A x

ǫ

• and ∇uǫ converge only weakly (the limit of the product

is not the product of the limits).

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• II- TWO-SCALE CONVERGENCE METHOD

This is a simpler method to prove the convergence theorem.

• Definition. A sequence of functions uǫ in L2(Ω) is said to two-scale converge to

a limit u0(x, y) belonging to L2(Ω × Y ) if, for any Y -periodic smooth function ϕ(x, y), it satisfies lim

ǫ→0

• Ω

uǫ(x)ϕ

• x, x

ǫ

• dx =
• Ω
• Y

u0(x, y)ϕ(x, y)dxdy. Theorem 1. From each bounded sequence uǫ in L2(Ω) one can extract a subsequence, and there exists a limit u0(x, y) ∈ L2(Ω × Y ) such that this subsequence two-scale converges to u0. Theorem 2. Let uǫ be a bounded sequence in H1(Ω). Then, up to a subsequence, uǫ two-scale converges to a limit u0(x, y) ≡ u(x) ∈ H1(Ω), and ∇uǫ two-scale converges to ∇xu(x) + ∇yu1(x, y) with u1 ∈ L2(Ω; H1

#(Y )).

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Lemma 3. For a bounded open set Ω, let B = C(¯ Ω; C#(Y )) be the space of continuous functions ϕ(x, y) on ¯ Ω × Y which are Y -periodic in y. Then, B is a separable Banach space (i.e. it contains a dense countable family), is dense in L2(Ω × Y ), and there exists C > 0 such that

• Ω

|ϕ

• x, x

ǫ

• |2dx ≤ Cϕ2

B,

and lim

ǫ→0

• Ω

|ϕ

• x, x

ǫ

• |2dx =
• Ω
• Y

|ϕ(x, y)|2dxdy, for any ϕ(x, y) ∈ B.

• Remark. The same works with B = L2(Ω; C#(Y )) and Ω not necessarily

bounded.

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ε

• ✁

• ❍

Ω

Proof of Lemma 3. We mesh Ω with cubes of the type (0, ǫ)N Ω = ∪1≤i≤n(ǫ)Y ǫ

i

with Y ǫ

i = xǫ i + (0, ǫ)N.

• Ω

|ϕ

• x, x

ǫ

• |2dx =

n(ǫ)

• i=1
• Y ǫ

i

|ϕ

• x, x

ǫ

• |2dx =

n(ǫ)

• i=1
• Y ǫ

i

|ϕ

• xǫ

i, x

ǫ

• |2dx + O(ǫ)

=

n(ǫ)

• i=1

ǫN

• Y

|ϕ (xǫ

i, y) |2dy + O(ǫ) =

• Ω
• Y

|ϕ (x, y) |2dx dy + O(ǫ)

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Proof of Theorem 1. By Schwarz inequality, we have

• Ω

uǫ(x)ϕ

• x, x

ǫ

• dx
• ≤ C
• Ω

ϕ

• x, x

ǫ

• dx
• 1

2

≤ CϕB. Thus, the l.h.s. is a continuous linear form on B which can be identified to a duality product µǫ, ϕB′,B for some bounded sequence of measures µǫ. Since B is separable, one can extract a subsequence and there exists a limit µ0 such µǫ converges to µ0 in the weak * topology of B′ (the dual of B). On the other hand, Lemma 3 allows us to pass to the limit in the middle term above. It yields |µ0, ϕB′,B| ≤ C

• Ω
• Y

|ϕ(x, y)|2dxdy

• 1

2

. Therefore µ0 is actually a continuous linear form on L2(Ω × Y ), by density of B in this space. Thus, there exists u0(x, y) ∈ L2(Ω × Y ) such that µ0, ϕB′,B =

• Ω
• Y

u0(x, y)ϕ(x, y)dxdy.

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Proof of Theorem 2. Since uǫ and ∇uǫ are bounded in L2(Ω), up to a subsequence, they two-scale converge to limits u0(x, y) ∈ L2(Ω × Y ) and ξ0(x, y) ∈ L2(Ω × Y )N. Thus lim

ǫ→0

• Ω

∇uǫ(x)· ψ

• x, x

ǫ

• dx =
• Ω
• Y

ξ0(x, y)· ψ(x, y)dxdy ∀ ψ ∈ D

• Ω; C∞

# (Y )N

. Integrating by parts the left hand side gives ǫ

• Ω

∇uǫ(x) · ψ

• x, x

ǫ

• dx = −
• Ω

uǫ(x)

• divy

ψ

• x, x

ǫ

• + ǫ divx

ψ

• x, x

ǫ

• dx.

Passing to the limit yields 0 = −

• Ω
• Y

u0(x, y) divy ψ(x, y)dxdy ⇒ u0(x, y) ≡ u(x) ∈ L2(Ω). Next, we choose ψ such that divy ψ(x, y) = 0. We obtain

• Ω

∇uǫ(x) · ψ

• x, x

ǫ

• dx = −
• Ω

uǫ(x) divx ψ

• x, x

ǫ

• dx.

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Passing to the two-scale limit

• Ω
• Y

ξ0(x, y) · ψ(x, y)dxdy = −

• Ω
• Y

u(x) divx ψ(x, y)dxdy. If ψ does not depend on y, it proves that u(x) ∈ H1(Ω). Furthermore,

• Ω
• Y

(ξ0(x, y) − ∇u(x)) · ψ(x, y)dxdy = 0 ∀ ψ with divy ψ = 0. The orthogonal of divergence-free functions are exactly the gradients. Thus, there exists a unique function u1(x, y) in L2(Ω; H1

#(Y )/R) such that

ξ0(x, y) = ∇u(x) + ∇yu1(x, y).

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Theorem 4. Let uǫ ∈ L2(Ω) two-scale converge to u0(x, y) ∈ L2(Ω × Y ).

• 1. Then, uǫ converges weakly in L2(Ω) to u(x) =
• Y u0(x, y)dy, and we have

lim

ǫ→0 uǫ2 L2(Ω) ≥ u02 L2(Ω×Y ) ≥ u2 L2(Ω).

• 2. Assume further that u0(x, y) is smooth and that

lim

ǫ→0 uǫ2 L2(Ω) = u02 L2(Ω×Y ).

Then, we have

• uǫ(x) − u0
• x, x

ǫ

• 2

L2(Ω) → 0.

• Remark. In the last case we say that uǫ two-scale converges strongly to u0.

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Proof of Theorem 4. Take a test function depending only on x lim

ǫ→0

• Ω

uǫ(x)ϕ (x) dx =

• Ω
• Y

u0(x, y)ϕ(x)dxdy =

• Ω

u(x) ϕ(x) dx. Thus, uǫ converges weakly to u in L2(Ω). Then, developing the inequality

• Ω
• uǫ(x) − ϕ
• x, x

ǫ

• 2

dx ≥ 0

• Ω

|uǫ(x)|2dx − 2

• Ω

uǫ(x)ϕ

• x, x

ǫ

• dx +
• Ω

|ϕ

• x, x

ǫ

• |2dx ≥ 0

lim inf

ǫ→0

• Ω

|uǫ(x)|2dx − 2

• Ω
• Y

u0(x, y)ϕ(x, y)dx dy +

• Ω
• Y

|ϕ (x, y) |2dx dy ≥ 0 Take ϕ = u0 to get lim

ǫ→0 uǫ2 L2(Ω) ≥ u02 L2(Ω×Y ).

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Proof of Theorem 4 (continued). If we assume lim

ǫ→0 uǫ2 L2(Ω) = u02 L2(Ω×Y ),

the same computation yields lim inf

ǫ→0

• Ω
• uǫ(x) − ϕ
• x, x

ǫ

• 2

dx =

• Ω
• Y

|u0(x, y) − ϕ (x, y) |2dx dy If u0 is smooth enough to be a test function ϕ (Carath´ eodory function), it gives the desired result

• uǫ(x) − u0
• x, x

ǫ

• 2

L2(Ω) → 0.

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Theorem 5.

• 1. Let uǫ be a bounded sequence in L2(Ω) such that ǫ∇uǫ is also bounded in

L2(Ω)N. Then, there exists a two-scale limit u0(x, y) ∈ L2(Ω; H1

#(Y )/R)

such that, up to a subsequence, uǫ two-scale converges to u0(x, y), and ǫ∇uǫ to ∇yu0(x, y).

• 2. Let uǫ be a bounded sequence in L2(Ω)N such that divuǫ is also bounded in

L2(Ω). Then, there exists a two-scale limit u0(x, y) ∈ L2(Ω × Y )N with divyu0 = 0 and divxu0 ∈ L2(Ω × Y ) such that, up to a subsequence, uǫ two-scale converges to u0(x, y), and divuǫ to divxu0(x, y).

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Proof of Theorem 5 (first part). We have lim

ǫ→0

• Ω

uǫ(x)ϕ

• x, x

ǫ

• dx =
• Ω
• Y

u0(x, y)ϕ(x, y)dxdy lim

ǫ→0

• Ω

ǫ∇uǫ(x) · ψ

• x, x

ǫ

• dx =
• Ω
• Y

ξ0(x, y) · ψ(x, y)dxdy By integration by parts

• Ω

ǫ∇uǫ(x) · ψ

• x, x

ǫ

• dx = −
• Ω

uǫ(x)

• divy

ψ + ǫ divx ψ x, x ǫ

• dx

Passing to the two-scale limit

• Ω
• Y

ξ0(x, y) · ψ(x, y)dxdy = −

• Ω
• Y

u0(x, y) divy ψ(x, y)dxdy which implies that ξ0(x, y) = ∇yu0(x, y).

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• III- APPLICATION TO HOMOGENIZATION

Conductivity or diffusion equation    − div

• A
• x, x

ǫ

• ∇uǫ
• = f

in Ω uǫ = 0

• n ∂Ω

with a coefficient tensor A(x, y) which is Y -periodic, uniformly coercive and bounded α|ξ|2 ≤

N

• i,j=1

Aij(x, y)ξiξj ≤ β|ξ|2, ∀ ξ ∈ RN, ∀ y ∈ Y, ∀ x ∈ Ω (β ≥ α > 0). A priori estimate. If Ω is bounded, then uǫH1(Ω) ≤ CfL2(Ω).

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✞ ✝ ☎ ✆

A priori estimate

Variational formulation

• Ω

A x ǫ

• ∇uǫ(x) · ∇ϕ(x)dx =
• Ω

f(x)ϕ(x)dx, ∀ ϕ ∈ H1

0(Ω).

Take ϕ = uǫ and use coercivity α∇uǫ2

L2(Ω) ≤

• Ω

f(x)uǫ(x)dx ≤ fL2(Ω)uǫL2(Ω) Poincar´ e inequality in Ω uǫL2(Ω) ≤ C(Ω)∇uǫL2(Ω) Thus ∇uǫL2(Ω) ≤ C(Ω)fL2(Ω) α

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✞ ✝ ☎ ✆

Two-scale convergence method

First step. We deduce from the a priori estimates the precise form of the two-scale limit of the sequence uǫ. By application of Theorem 2, there exist two functions, u(x) ∈ H1

0(Ω) and

u1(x, y) ∈ L2(Ω; H1

#(Y )/R), such that, up to a subsequence, uǫ two-scale

converges to u(x), and ∇uǫ two-scale converges to ∇xu(x) + ∇yu1(x, y). In view of these limits, uǫ is expected to behave as u(x) + ǫu1

• x, x

ǫ

• .

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✞ ✝ ☎ ✆

Two-scale convergence method

Second step. We multiply the p.d.e. by a test function similar to the limit of uǫ, namely ϕ(x) + ǫϕ1

• x, x

ǫ

• , where ϕ(x) ∈ D(Ω) and ϕ1(x, y) ∈ D(Ω; C∞

# (Y )).

This yields

• Ω

A

• x, x

ǫ

• ∇uǫ ·
• ∇ϕ(x) + ∇yϕ1
• x, x

ǫ

• + ǫ∇xϕ1
• x, x

ǫ

• dx

=

• Ω

f(x)

• ϕ(x) + ǫϕ1
• x, x

ǫ

• dx.

Regarding At x, x

ǫ

∇ϕ(x) + ∇yϕ1

• x, x

ǫ

• as a test function for the two-scale

convergence, we pass to the two-scale limit

• Ω
• Y

A(x, y) (∇u(x) + ∇yu1(x, y))·(∇ϕ(x) + ∇yϕ1(x, y)) dxdy =

• Ω

f(x)ϕ(x)dx.

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✞ ✝ ☎ ✆

Two-scale convergence method

Third step. We read off a variational formulation for (u, u1). Take (ϕ, ϕ1) in the Hilbert space H1

0(Ω) × L2

Ω; H1

#(Y )/R

• endowed with the norm
• (∇u(x)2

L2(Ω) + ∇yu1(x, y)2 L2(Ω×Y ))

The assumptions of the Lax-Milgram lemma are easily checked. The main point is the coercivity of the bilinear form defined by the left hand side

• Ω
• Y

A(x, y) (∇ϕ(x) + ∇yϕ1(x, y)) · (∇ϕ(x) + ∇yϕ1(x, y)) dxdy ≥ α

• Ω
• Y

|∇ϕ(x)+∇yϕ1(x, y)|2dxdy = α

• Ω

|∇ϕ(x)|2dx+α

• Ω
• Y

|∇yϕ1(x, y)|2dxdy.

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• Ω
• Y

A(x, y) (∇u(x) + ∇yu1(x, y))·(∇ϕ(x) + ∇yϕ1(x, y)) dxdy =

• Ω

f(x)ϕ(x)dx. By application of the Lax-Milgram lemma, there exists a unique solution (u, u1) ∈ H1

0(Ω) × L2

Ω; H1

#(Y )/R

• . Consequently, the entire sequences uǫ and

∇uǫ converge to u(x) and ∇u(x) + ∇yu1(x, y). An easy integration by parts shows that the associated p.d.e.’s are the so-called “two-scale homogenized problem”,              − divy (A(x, y) (∇u(x) + ∇yu1(x, y))) = 0 in Ω × Y − divx

• Y A(x, y) (∇u(x) + ∇yu1(x, y)) dy
• = f(x)

in Ω y → u1(x, y) Y -periodic u = 0

• n ∂Ω.

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• G. Allaire

✞ ✝ ☎ ✆

Two-scale convergence method

Fourth (and optional) step. Eliminate the y variable and the u1 unknown u1(x, y) =

N

• i=1

∂u ∂xi (x)wi(x, y), where wi(x, y) are the unique solutions in H1

#(Y )/R of the cell problems

   − divy (A(x, y) ( ei + ∇ywi(x, y))) = 0 in Y y → wi(x, y) Y -periodic, at each point x ∈ Ω, and    − divx (A∗(x)∇u(x)) = f(x) in Ω u = 0

• n ∂Ω,

with A∗

ij(x) =

• Y A(x, y) (

ei + ∇ywi(x, y)) · ( ej + ∇ywj(x, y)) dy.

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• G. Allaire
• IV- BLOCH WAVES

Theorem 1. For any function u(y) ∈ L2(RN) there exists a unique function ˆ u(y, θ) ∈ L2(Y × Y ) such that u(y) =

• Y

ˆ u(y, θ)e2iπθ·ydθ. The function y → ˆ u(y, θ) is Y -periodic while the function θ → e2iπθ·yˆ u(y, θ) is Y -periodic. Furthermore, the linear map B, called the Bloch transform and defined by Bu = ˆ u, is an isometry from L2(RN) into L2(Y × Y ), i.e. Parseval formula holds for any u, v ∈ L2(RN)

• RN u(y)v(y) dy =
• Y
• Y

ˆ u(y, θ)ˆ v(y, θ) dy dθ.

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• G. Allaire
• Proof. Let u(y) be a smooth compactly supported function in RN. Define

ˆ u(y, θ) =

• k∈ZN

u(y + k)e−2iπθ·(y+k). This sum is well defined because it has a finite number of terms since u has compact support. It is also clearly a Y -periodic function of y. On the other hand, for j ∈ ZN, we have ˆ u(y, θ + j) = e−2iπj·y

k∈ZN

u(y + k)e−2iπθ·(y+k) = e−2iπj·yˆ u(y, θ). Thus, θ → e2iπθ·yˆ u(y, θ) is Y -periodic. Next, we compute

• TN ˆ

u(y, θ)e2iπθ·ydθ =

• k∈ZN

u(y + k)

• TN e−2iπθ·kdθ = u(y)

since all integrals vanish except for k = 0. This proves the result for smooth compactly supported functions.

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• G. Allaire

In particular, it shows that the Bloch transform B is a linear map, well defined

• n C∞

c (RN) and bounded on L2(RN).

Since C∞

c (RN) is dense in L2(RN), B can be extended by continuity and the

result holds true in L2(RN).

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• G. Allaire
• Lemma. Let a(y) ∈ L∞(TN) be a periodic function.

For any u(y) ∈ L2(RN), we have B(au) = aB(u) ≡ a(y)ˆ u(y, θ).

• Remark. TN is the flat unit torus, i.e. TN ≡ Y + periodic B.C.
• Lemma. Let u(y) ∈ H1(RN). The Bloch transform of its gradient is

B(∇yu) = (∇y + 2iπθ)B(u) ≡ ∇yˆ u(y, θ) + 2iπθˆ u(y, θ) and y → B(u) ≡ ˆ u(y, θ) belongs to H1(TN).

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• G. Allaire

✞ ✝ ☎ ✆ Application Find u ∈ H1(RN) solution of − divy

• A(y)∇yu
• + c(y)u = f

in RN, with f ∈ L2(RN). We assume that A(y) and c(y) belong to L∞(TN) and ∃ ν > 0 such that for a.e. y ∈ TN A(y)ξ · ξ ≥ ν|ξ|2 for any ξ ∈ RN and c(y) ≥ c0 > 0. The variational formulation is to find u ∈ H1(RN) such that

• RN
• A(y)∇yu · ∇yφ + c(y)uφ
• dy =
• RN fφ dy

∀φ ∈ H1(RN).

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• G. Allaire
• Proposition. The p.d.e. in RN is equivalent to the family of p.d.e.’s, indexed by

θ ∈ TN, −( divy + 2iπθ)

• A(y)(∇y + 2iπθ)Bu
• + c(y)Bu = Bf

in TN, which admits a unique solution y → (Bu)(y, θ) ∈ H1(TN) for any θ ∈ TN.

• Proof. We apply the previous Lemmas
• RN
• A(y)∇yu · ∇yφ + c(y)uφ
• dy =
• RN fφ dy

∀φ ∈ H1(RN)

• TN
• Y
• A(y)(∇y + 2iπθ)ˆ

u · (∇y + 2iπθ)ˆ φ + c(y)ˆ uˆ φ

• dy dθ =
• TN
• Y

ˆ f ˆ φ dy dθ which is a variational formulation in y ∈ TN integrated with respect to θ which is just a parameter.

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• G. Allaire

For a given θ ∈ Y , consider the Green operator Gθ    L2(TN) → L2(TN) g(y) → Gθg(y) = v(y) where v ∈ H1(TN) is the unique solution of −( divy + 2iπθ)

• A(y)(∇y + 2iπθ)v
• + c(y)v = g

in TN. One can check that Gθ is a self-adjoint compact complex-valued linear operator acting on L2(TN). As such it admits a countable sequence of real increasing eigenvalues (λn)n≥1 (repeated with their multiplicity) and normalized eigenfunctions (ψn)n≥1 with ψnL2(TN ) = 1. The eigenvalues and eigenfunctions depend on the dual parameter or Bloch frequency θ which runs in the dual cell of Y , which is again Y .

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• G. Allaire

In other words, the eigenvalues and eigenfunctions satisfy the so-called Bloch (or shifted) spectral cell equation −( divy + 2iπθ)

• A(y)(∇y + 2iπθ)ψn
• + c(y)ψn = λn(θ)ψn

in TN. θ ∈ Y is the Bloch frequency (or quasi momentum). A Bloch wave is φn(y) = ψn(θ, y)e2iπθ·y which satisfies − divy (A(y)∇yφn) + c(y)φn = λ(θ)φn in RN.

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• G. Allaire

Theorem 2. For any u(y) ∈ L2(RN), ∃ ˆ un(θ) ∈ L2(TN), n ≥ 1, such that u(y) =

• n≥1
• TN ˆ

un(θ)ψn(y, θ)e2iπθ·ydθ. Furthermore, the linear map B, called the Bloch transform and defined by Bu = (ˆ un)n≥1, is an isometry from L2(RN) into ℓ2

• L2(TN)
• , i.e. Parseval

formula holds for any u, v ∈ L2(RN)

• RN u(y)v(y) dy =
• n≥1
• Y

ˆ un(θ)ˆ vn(θ) dθ.

• Proof. We decompose each ˆ

u(y, θ) on the corresponding eigenbasis ˆ u(y, θ) =

• n≥1

ˆ un(θ)ψn(y, θ) with ˆ un(θ) =

• TN ˆ

u(y, θ)ψn(y, θ)dy. Commuting the sum with respect to n and the integral with respect to θ is a standard Fubini type result.

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• G. Allaire

✞ ✝ ☎ ✆ Application − divy

• A(y)∇yu
• + c(y)u = f

in RN, with f ∈ L2(RN). We obtain an explicit algebraic formula for the solution ˆ un(θ) = ˆ fn(θ) λn(θ) ∀n ≥ 1, ∀θ ∈ Y, which is a generalization of a similar formula, using Fourier transform, for a constant coefficient p.d.e..

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• G. Allaire

✞ ✝ ☎ ✆ Smoothness of the Bloch eigenvalues

• Lemma. θ → λn(θ) is Lipschitz.

Remark: for multiple eigenvalues θ → λn(θ) is usually not smoother. Lemma (Kato, Rellich). If an eigenvalue λn(θ) is simple at the value θ = θn, then it remains simple in a small neighborhood of θn and the n-th eigencouple is analytic in this neighborhood of θn.

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• G. Allaire

Assumption: λn(θn) is a simple eigenvalue. Operator An(θ)ψ = −( divy + 2iπθ)

• A(y)(∇y + 2iπθ)ψ
• + c(y)ψ − λn(θ)ψ,

Then, in a neighborhood of θn, one can differentiate An(θ)∂ψn ∂θk = 2iπekA(y)(∇y +2iπθ)ψn +( divy +2iπθ) (A(y)2iπekψn)+ ∂λn ∂θk (θ)ψn, and the second derivative is An(θ) ∂2ψn ∂θk∂θl = 2iπekA(y)(∇y + 2iπθ)∂ψn ∂θl + ( divy + 2iπθ)

• A(y)2iπek

∂ψn ∂θl

• +2iπelA(y)(∇y + 2iπθ)∂ψn

∂θk + ( divy + 2iπθ)

• A(y)2iπel

∂ψn ∂θk

• +∂λn

∂θk (θ)∂ψn ∂θl + ∂λn ∂θl (θ)∂ψn ∂θk −4π2ekA(y)elψn − 4π2elA(y)ekψn + ∂2λn ∂θl∂θk (θ)ψn

Homogenization of Schr¨

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• G. Allaire
• V- SCHR¨

ODINGER EQUATION

Schr¨

• dinger equation in a periodic medium:

     i∂uǫ ∂t − div

• A

x ǫ

• ∇uǫ
• +
• ǫ−2c

x ǫ

• + d
• x, x

ǫ

• uǫ = 0

in RN × R+ uǫ(t = 0, x) = u0

ǫ(x)

in RN, Complex-valued unknown: uǫ(t, x) : R+ × RN → C ☞ y → A(y), c(y), d(x, y) (0, 1)N-periodic, real, measurable and bounded. ☞ A is a N × N symmetric, uniformly coercive, tensor. ☞ c and d are scalar functions with no sign. ☞ d(x, y) is a Carath´ eodory function, u0

ǫ ∈ H1(RN).

Homogenization of Schr¨

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Physical motivation: ☞ Solid state physics: one-electron model in a periodic crystal. ☞ c(y) is the crystal potential and d(x, y) the (small) exterior potential. Other possible equation:      i∂uǫ ∂t − ( div + iǫAǫ) (∇ + iǫAǫ) uǫ + ǫ−2c x ǫ

• uǫ = 0

in RN × R+ uǫ(t = 0, x) = u0

ǫ(x)

in RN, where Aǫ = A

• t, x, x

ǫ

• is the electromagnetic vector potential such that the

electric field E and magnetic field B are E = −∂Aǫ ∂t and B = curlAǫ. ✗ Periodic homogenization: ǫ → 0 is the period. ✗ Singular perturbation: potential of order ǫ−2.

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✞ ✝ ☎ ✆

SCALING

✗ This is not the scaling of semi-classical analysis: i ǫ ∂uǫ ∂t − div

• A

x ǫ

• ∇uǫ
• +
• ǫ−2c

x ǫ

• + d
• x, x

ǫ

• uǫ = 0

✗ This is a parabolic scaling (for longer times). ✗ Scaling of a long time asymptotic: change of variables y = x/ǫ and τ = t/ǫ2 ⇒ cells of size 1, no more oscillations i∂uǫ ∂τ − divy (A(y)∇yuǫ) +

• c(y) + ǫ2d(ǫy, y)
• uǫ = 0

⇒ small and slowly varying exterior potential. Goal: find homogenized models, justify the notion of electron effective mass.

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✞ ✝ ☎ ✆

A priori estimates

For any final time T > 0, there exists a constant C > 0 that does not depend on ǫ such that uǫL∞((0,T );L2(RN )) = u0

ǫL2(RN),

ǫ∇uǫL∞((0,T );L2(RN)N ) ≤ C

• u0

ǫL2(RN) + ǫ∇u0 ǫL2(RN )N

• .
• Proof. Multiply the equation by uǫ and by ∂uǫ

∂t , integrate by parts and take the

real part. No strong convergence a priori.

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• G. Allaire

A FIRST SIMPLE RESULT

We use a trick of Vanninathan (81’), Kozlov (84’), A.-Malige (97’). Introduce the spectral cell problem for θ = 0 − divy (A(y)∇yψn) + c(y)ψn = λnψn in the unit torus TN. First eigenvalue λ1 is simple and first eigenfunction ψ1(y) > 0 is positive by Krein-Rutman theorem (maximum principle). Interpretation of ψ1: periodic ground state. Change of unknowns: vǫ(t, x) = e−i λ1t

ǫ2 uǫ(t, x)

ψ1( x

ǫ )

Algebraic trick: ψ1 div (A∇(ψ1v)) = ψ1v div (A∇ψ1) + div

• ψ2

1A∇v

Homogenization of Schr¨

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• G. Allaire

A simpler equation for vǫ:      i|ψ1|2 x ǫ ∂vǫ ∂t − div

• (|ψ1|2A)

x ǫ

• ∇vǫ
• + (|ψ1|2d)
• x, x

ǫ

• vǫ = 0

in RN × R+ vǫ(t = 0, x) =

u0

ǫ(x)

ψ1( x

ǫ )

in RN.

• Theorem. (Standard periodic homogenization)

If u0

ǫ(x) = v0(x)ψ1

x

ǫ

• , then vǫ, converges weakly in L2

(0, T); H1(RN)

• to the

solution v of the homogenized problem      i∂v ∂t − div (A∗∇v) + d∗(x) v = 0 in RN × (0, T) v(t = 0, x) = v0(x) in RN, where d∗(x) =

• TN |ψ1|2(y)d(x, y) dy and A∗ is the “usual” homogenized tensor

for |ψ1(y)|2A(y).

Homogenization of Schr¨

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✞ ✝ ☎ ✆

INTERPRETATION

Convergence result: uǫ(t, x) ≈ ei λ1t

ǫ2 ψ1

x ǫ

• v(t, x)

☞ It is crucial that ψ1(y) > 0 (maximum principle). ☞ This trick is called factorization principle. ☞ Equivalent results for parabolic or hyperbolic equations, transport equation (A.-Bal, A.-Capdeboscq ...). ☞ Still true if ψ1 x

ǫ

• u0

ǫ(x) converges weakly to v0(x) in L2(RN).

What happens if the weak limit is v0 ≡ 0 ?

Homogenization of Schr¨

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• G. Allaire

WKB METHOD

A different kind of ansatz. Geometric optics or WKB method (Wentzel, Kramers, Brillouin).    i ǫ ∂uǫ ∂t − div

• A
• x, x

ǫ

• ∇uǫ
• + 1

ǫ2 c

• x, x

ǫ

• uǫ = 0

in RN × (0, T), uǫ(t = 0, x) = u0

ǫ(x)

in RN ☞ This is precisely the scaling of semi-classical analysis. ☞ Short time scaling. ☞ A is symmetric and uniformly coercive. ☞ Formal method. For rigorous analysis, see: Buslaev, Guillot-Ralston, G´ erard-Martinez-Sjostrand, G´ erard-Markowich-Mauser-Poupaud, Panati-Sohn-Teufel...

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☞ Introduce the operator A(x, θ)ψ ≡ −( divy + 2iπθ)

• A(x, y)(∇y + 2iπθ)ψ
• + c(x, y)ψ

☞ Eigenvalues λn(x, θ) and eigenfunctions ψn(x, y, θ). ☞ Choose an energy level n such that λn(x, θ) is simple ∀θ ∈ TN, ∀x ∈ RN. ☞ Replace θ par ∇S0 and consider an initial data with oscillating phase u0

ǫ(x) = e2iπ S0(x)

ǫ

u0(x)ψn

• x, x

ǫ , ∇S0(x)

Homogenization of Schr¨

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• G. Allaire

✞ ✝ ☎ ✆ High frequency asymptotic expansion or WKB ansatz uǫ(t, x) = e2iπ S(t,x)

ǫ

• v
• t, x, x

ǫ

• + ǫv1
• t, x, x

ǫ

• + ...
• The first derivatives are

ǫ∂uǫ ∂t = e2iπ S(t,x)

ǫ

• 2iπ(v + ǫv1)∂S

∂t + ǫ∂v ∂t + O(ǫ2)

• ǫ∇uǫ = e2iπ S(t,x)

ǫ

• 2iπ(v + ǫv1)∇S + ∇yv + ǫ(∇xv + ∇yv1) + O(ǫ2)
• We compute also the second derivatives, plug the ansatz in the p.d.e. and deduce

a cascade of equations.

Homogenization of Schr¨

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• G. Allaire

✞ ✝ ☎ ✆ Cascade of equations Order ǫ−2: A(x, ∇S)v = 2π ∂S ∂t v in TN. This is a spectral problem, y being the space variable and (t, x) being fixed

• parameters. Thus, we deduce

2π ∂S ∂t = λn(x, ∇S) and by simplicity of the eigenvalue v(t, x, y) = u(t, x)ψn(x, y, ∇S(t, x)). We have obtained an eikonal equation (Hamilton-Jacobi) to compute the phase with the initial data S(0, x) = S0(x).

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• G. Allaire

Order ǫ−1: A(x, ∇S)v1 = λn(x, ∇S)v1 + f in TN, with f = −i∂v ∂t +

• divy + 2iπ∇S
• (A∇xv) + divx
• A(∇y + 2iπ∇S)v
• .

To solve for v1 the Fredholm alternative requires that

• TN f(t, x, y)ψn(x, y, ∇S) dy = 0.

Recall that v(t, x, y) = u(t, x)ψn(x, y, ∇S) and Fredholm alternative for ∇θψn −∇xu·∇θλn = 2iπ

• TN [( divy + 2iπθ) (A(y)∇xuψn) + A(y)(∇y + 2iπθ)ψn · ∇xu] dy

We deduce an homogenized transport equation ∂u ∂t − ∇θλn(∇S) 2π · ∇xu + b∗u = 0

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• G. Allaire

✄ ✂

• ✁

Reminder on Bloch waves ✖ Bloch eigenvalue λn(x, θ) and eigenvector ψn(x, y, θ) ✖ Assumption: λn(θ) is a simple eigenvalue. ✖ Operator An(x, θ)ψ = −( divy + 2iπθ)

• A(x, y)(∇y + 2iπθ)ψ
• + c(x, y)ψ − λn(x, θ)ψ,

✖ Bloch spectral problem An(x, θ)ψn = 0 ✖ We differentiate with respect to θ An(x, θ)∂ψn ∂θk = 2iπekA(∇y + 2iπθ)ψn + ( divy + 2iπθ) (A2iπekψn) + ∂λn ∂θk ψn, ✖ Fredholm alternative: the r.h.s. is orthogonal to ψn.

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✘ Group velocity V = ∇θλn(∇S) 2π ✘ If we write v(t, x, y) = u(t, x, θ)ψn(x, y, θ) with θ = ∇S, then we deduce an homogenized Liouville equation in the phase space ∂|u|2 ∂t − V · ∇x|u|2 + ∇xλn · ∇θ|u|2 = 0

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✄ ✂

• ✁

Conclusion uǫ(t, x) ≈ e2iπ S(t,x)

ǫ

ψn

• x, x

ǫ , ∇S(t, x)

• u(t, x, ∇S(t, x))

The semi-classical limit is given by the dynamic of the following Hamiltonian system in the phase space (x, θ) ∈ RN × TN    ˙ x =

1 2π∇θλn(x, θ)

˙ θ = −∇xλn(x, θ) ☞ Phase S solution of an eikonal equation. ☞ Amplitude |u|2 solution of a transport equation in the phase space. ☞ Valid up to caustics. ☞ Rigorous version by using semi-classical or Wigner measures. ☞ WKB ansatz are useful for computations too.

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✞ ✝ ☎ ✆ Special case Monochromatic initial data. Periodic coefficients. Assume S0(x) = θ · x and A(x, y) ≡ A(y), c(x, y) ≡ c(y). Then the explicit solution of the eikonal equation is S(t, x) = θ · x + 2πλn(θ) t Furthermore V is constant, b∗ = 0 and ∇xλn(θ) = 0, thus u(t, x) = u0(x + Vt) In such a simpler case we can find a better ”long time” ansatz of the solution.

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• VI- HOMOGENIZATION

We consider a monochromatic wave packet as initial data u0

ǫ(x) = ψn

x ǫ , θn e2iπ θn·x

ǫ v0(x)

with v0 ∈ H1(RN). Assumption: the eigenvalue λn(θn) is simple. ✗ We replace the maximum principle by Bloch wave theory. ✗ Simplicity is a generic assumption. ✗ Other initial data are possible (up to extracting a weakly two-scale converging subsequence).

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For θ ∈ Y the Bloch (or shifted) spectral cell equation is −( divy + 2iπθ)

• A(y)(∇y + 2iπθ)ψn
• + c(y)ψn = λn(θ)ψn

in TN Schr¨

• dinger equation in a periodic medium:

     i∂uǫ ∂t − div

• A

x ǫ

• ∇uǫ
• +
• ǫ−2c

x ǫ

• + d
• x, x

ǫ

• uǫ = 0

in RN × R+ uǫ(t = 0, x) = u0

ǫ(x)

in RN,

Homogenization of Schr¨

• dinger equation

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• G. Allaire

Theorem 1. Assume further that the group velocity vanishes ∇θλn(θn) = 0. Then uǫ(t, x) = ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ ψn

x ǫ , θn v(t, x) + rǫ(t, x) with lim

ǫ→0 sup t∈[0,T ]

• RN |rǫ(t, x)|2 dx = 0,

and v ∈ C

• (0, T); L2(RN)
• is the unique solution of the homogenized

Schr¨

• dinger equation

     i∂v ∂t − div (A∗

n∇v) + d∗ n(x) v = 0

in RN × (0, T) v(t = 0, x) = v0(x) in RN, with A∗

n =

1 8π2 ∇θ∇θλn(θn) and d∗

n(x) =

• TN d(x, y)|ψn(y)|2 dy.

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✞ ✝ ☎ ✆

REMARKS

☞ The inverse of A∗

n is called the effective mass (a well-known concept in solid

state physics). ☞ The effective mass can be negative or infinite ! ☞ Previous work on effective mass: Bensoussan-Lions-Papanicolaou, Poupaud-Ringhofer. ☞ The homogenized coefficients do depend on the initial data ! It does not fit in the framework of G- or H-convergence.

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Theorem 2. Assume lim

|x|→+∞ d(x, y) = d∞(y)

uniformly in TN. Define the group velocity or drift V = 1 2π ∇θλn(θn) ∈ RN. Then uǫ(t, x) = ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ ψn

x ǫ , θn v

• t, x + V

ǫ t

• + rǫ(t, x) with

lim

ǫ→0

T

• RN |rǫ(t, x)|2 dxdt = 0, where v ∈ C
• (0, T); L2(RN)
• is the unique

solution of the homogenized Schr¨

• dinger equation

     i∂v ∂t − div (A∗

n∇v) + d∗ n v = 0

in RN × (0, T) v(t = 0, x) = v0(x) in RN, with A∗

n =

1 8π2 ∇θ∇θλn(θn) and d∗

n =

• TN d∞(y)|ψn(y)|2 dy.

Homogenization of Schr¨

• dinger equation

68

• G. Allaire

✞ ✝ ☎ ✆ Generalization for a multiple eigenvalue What happens if λn(θn) is not simple ? In general, if λn(θn) is of multiplicity p > 1, we expect an homogenized system of p equations. Assume that λn(θn) = λn+1(θn) is of multiplicity 2 and that, up to a convenient relabelling, λn(θ) and λn+1(θ) are smooth branches of eigenvalues (and eigenfunctions). (Very strong assumption !) We can prove that the homogenized system is              i∂v1 ∂t − div (A∗

n∇v1) + d∗ 11(x) v1 + d∗ 12(x) v2 = 0

in RN × R+ i∂v2 ∂t − div

• A∗

n+1∇v2

• + d∗

21(x) v1 + d∗ 22(x) v2 = 0

in RN × R+ (v1, v2)(t = 0, x) = (v0

1, v0 2)(x)

in RN.

Homogenization of Schr¨

• dinger equation

69

• G. Allaire

✞ ✝ ☎ ✆

Formal proof of Theorem 1

We make an ansatz for the solution uǫ(t, x) = ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ

+∞

• i=0

ǫiui

• t, x, x

ǫ

• where the oscillating phase is deduced from WKB.

We recall the operator An(θ)ψ ≡ −( divy + 2iπθ)

• A(y)(∇y + 2iπθ)ψ
• + c(y)ψ − λn(θ)ψ

We plug the ansatz in the equation and get a cascade of equations.

Homogenization of Schr¨

• dinger equation

70

• G. Allaire

✞ ✝ ☎ ✆ Cascade of equations Order ǫ−2: An(θn)u0(t, x, y) = 0 from which we deduce, by simplicity of λn(θn), that u0(t, x, y) ≡ v(t, x)ψn(y, θn). Order ǫ−1: An(θn)u1(t, x, y) = divx (A(y)(∇y + 2iπθn)ψnv) + ( divy + 2iπθn) (A(y)∇xvψn) from which we deduce (up to the addition of a multiple of ψn) u1(t, x, y) ≡ 1 2iπ

N

• k=1

∂v ∂xk (t, x)∂ψn ∂θk (y, θn)

Homogenization of Schr¨

• dinger equation

71

• G. Allaire

Order ǫ0: An(θn)u2(t, x, y) = f from which we deduce that u2(t, x, y) ≡ −1 4π2

N

• k,l=1

∂2v ∂xk∂xl (t, x) ∂2ψn ∂θk∂θl and, more important, by the Fredholm alternative,

• TN f(t, x, y)ψn(t, x, y) dy = 0.

This last condition yields the homogenized equation.

Homogenization of Schr¨

• dinger equation

72

• G. Allaire

✞ ✝ ☎ ✆

Proof of Theorem 1

It is made of 3 steps

• 1. Deriving the spectral cell problem.
• 2. Deriving the homogenized equation.
• 3. Strong convergence.

Main tools: Bloch wave decomposition, two-scale convergence.

Homogenization of Schr¨

• dinger equation

73

• G. Allaire

✞ ✝ ☎ ✆

STEP 1. SPECTRAL CELL PROBLEM

We apply two-scale convergence to the bounded sequence vǫ defined by vǫ(t, x) = uǫ(t, x)e−i λn(θn)t

ǫ2

e−2iπ θn·x

ǫ ,

which admits a two-scale limit v∗(t, x, y) ∈ L2 (0, T) × RN; H1(TN)

• .

We multiply the Schr¨

• dinger equation by the complex conjugate of

ϕǫ ≡ ǫ2φ(t, x, x ǫ )ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ

Homogenization of Schr¨

• dinger equation

74

• G. Allaire

Variational formulation T

• RN
• i∂uǫ

∂t ϕǫ + Aǫ∇uǫ · ∇ϕǫ + (ǫ−2cǫ + dǫ)uǫϕǫ

• dt dx = 0.

Replace uǫ by vǫ and ϕǫ by ǫ2φǫ T

• RN
• Aǫ(ǫ∇ + 2iπθn)vǫ · (ǫ∇ − 2iπθn)φ

ǫ + (cǫ − λn(θn))vǫφ ǫ

dt dx = O(ǫ2). Passing to the two-scale limit yields the variational formulation of −( divy + 2iπθn)

• A(y)(∇y + 2iπθn)v∗

+ c(y)v∗ = λn(θn)v∗ in TN. Since λn(θn) is simple, there exists a scalar function v(t, x) ∈ L2 ((0, T) × Ω) such that v∗(t, x, y) = v(t, x)ψn(y, θn).

Homogenization of Schr¨

• dinger equation

75

• G. Allaire

✞ ✝ ☎ ✆

STEP 2. HOMOGENIZED PROBLEM

We multiply the Schr¨

• dinger equation by the complex conjugate of

Ψǫ = ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ

• ψn(x

ǫ , θn)φ(t, x) + ǫ

N

• k=1

∂φ ∂xk (t, x)ζk(x ǫ )

• where ζk(y) is the solution of

A(θn)ζk = ekA(y)(∇y + 2iπθn)ψn + ( divy + 2iπθn) (A(y)ekψn) in TN with the operator A(θ) defined on L2(TN) by A(θn)ψ = −( divy + 2iπθn)

• A(y)(∇y + 2iπθn)ψ
• + c(y)ψ − λn(θn)ψ

The existence of ζk is guaranteed because ∂ψn ∂θk = 2iπζk since ∇θλn(θn) = 0

Homogenization of Schr¨

• dinger equation

76

• G. Allaire

✞ ✝ ☎ ✆ Everything you always wanted to know about the proof but were afraid to ask ☞ We integrate by parts and put all derivatives on the test function. ☞ The ǫ−2 terms cancel out because of the equation for ψn. ☞ The ǫ−1 terms cancel out because of the equation for ζk. ☞ We pass to the two-scale limit in the ǫ0 terms. ☞ We obtain a very weak form of the homogenized equation. ☞ The homogenized tensor A∗

n is equal to 1 8π2 ∇θ∇θλn(θn) because of the

compatibility condition (Fredholm alternative) for

∂2ψn ∂θk∂θl .

Homogenization of Schr¨

• dinger equation

77

• G. Allaire
• RN Aǫ∇uǫ · ∇Ψǫdx =
• RN Aǫ(∇ + 2iπ θn

ǫ )(φvǫ) · (∇ − 2iπ θn ǫ )ψ

ǫ n

+ǫ

• RN Aǫ(∇ + 2iπ θn

ǫ )( ∂φ ∂xk vǫ) · (∇ − 2iπ θn ǫ )ζ

ǫ k

−

• RN Aǫek

∂φ ∂xk vǫ · (∇ − 2iπ θn ǫ )ψ

ǫ n

+

• RN Aǫ(∇ + 2iπ θn

ǫ )( ∂φ ∂xk vǫ) · ekψ

ǫ n

−

• RN Aǫvǫ∇ ∂φ

∂xk · ekψ

ǫ n −

• RN Aǫvǫ∇ ∂φ

∂xk · (ǫ∇ − 2iπθn)ζ

ǫ k

+

• RN Aǫζ

ǫ k(ǫ∇ + 2iπθn)vǫ · ∇ ∂φ

∂xk

Homogenization of Schr¨

• dinger equation

78

• G. Allaire

Variational formulation for ψn x

ǫ

• RN Aǫ(∇ + 2iπ θn

ǫ )ψǫ

n · (∇ − 2iπ θn

ǫ )Φ + 1 ǫ2

• RN (cǫ − λn(θn))ψǫ

nΦ = 0

Take Φ = φvǫ ⇒ cancellation of the ǫ−2 terms

Homogenization of Schr¨

• dinger equation

79

• G. Allaire

Variational formulation for ζk x

ǫ

• RN Aǫ(∇ + 2iπ θn

ǫ )ζǫ

k · (∇ − 2iπ θn

ǫ )Φ + 1 ǫ2

• RN (cǫ − λn(θn))ζǫ

kΦ =

ǫ−1

• RN Aǫ(∇ + 2iπ θn

ǫ )ψǫ

n · ekΦ − ǫ−1

• RN Aǫekψǫ

n · (∇ − 2iπ θn

ǫ )Φ. Take Φ =

∂φ ∂xk vǫ ⇒ cancellation of the ǫ−1 terms

Homogenization of Schr¨

• dinger equation

80

• G. Allaire

✞ ✝ ☎ ✆

STEP 3. STRONG CONVERGENCE

This is a consequence of the energy conservation vǫ(t)L2(RN) = uǫ(t)L2(RN) = u0

ǫL2(RN) → ψnv0L2(RN ×TN) = v0L2(RN)

and of the notion of strong two-scale convergence. Recall that u0

ǫ(x) = ψn

x ǫ , θn e2iπ θn·x

ǫ v0(x)

Homogenization of Schr¨

• dinger equation

81

• G. Allaire

✞ ✝ ☎ ✆

Proof of Theorem 2

New ingredient: two-scale convergence with drift. Proposition (Marusic-Paloka, Piatnitski). Let V ∈ RN be a given drift

• velocity. Let uǫ be a bounded sequence in L2((0, T) × RN). Up to a subsequence,

there exist a limit u0(t, x, y) ∈ L2((0, T) × RN × TN) such that uǫ two-scale converges with drift weakly to u0 in the sense that lim

ǫ→0

T

• RN uǫ(t, x)φ
• t, x + V

ǫ t, x ǫ

• dt dx =

T

• RN
• TN u0(t, x, y)φ(t, x, y) dt dx dy

for all functions φ(t, x, y) ∈ L2 (0, T) × RN; C(TN)

• .

Homogenization of Schr¨

• dinger equation

82

• G. Allaire

Lemma (Marusic-Paloka, Piatnitski). Let φ(t, x, y) ∈ L2 (0, T) × RN; C(TN)

• . Then

lim

ǫ→0

T

• RN
• φ
• t, x + V

ǫ t, x ǫ

• 2

dt dx = T

• RN
• TN |φ(t, x, y)|2dt dx dy.
• Proof. Change of variables x′ = x + V

ǫ t

T

• RN
• φ
• t, x + V

ǫ t, x ǫ

• 2

dt dx = T

• RN
• φ
• t, x′, x′

ǫ − V ǫ2 t

• 2

dt dx′ We mesh RN with cubes of size ǫ, RN = ∪i∈ZY ǫ

i with Y ǫ i = xǫ i + (0, ǫ)N

• RN
• φ
• t, x′, x′

ǫ − V ǫ2 t

• 2

dx =

• i
• Y ǫ

i

• φ
• t, xǫ

i, x′

ǫ − V ǫ2 t

• 2

dx + O(ǫ) =

• i∈Z

ǫN

• TN |φ (xǫ

i, y)|2 dy + O(ǫ) =

• Ω
• Y

|φ (x, y)|2 dx dy + O(ǫ)

Homogenization of Schr¨

• dinger equation

83

• G. Allaire
• VII- LOCALIZATION

We come back to the semi-classical scaling and to locally periodic coefficients i ǫ ∂uǫ ∂t − div

• A
• x, x

ǫ

• ∇uǫ
• + ǫ−2c
• x, x

ǫ

• uǫ = 0

We choose well-prepared initial data, Bloch wave packets.

Homogenization of Schr¨

• dinger equation

84

• G. Allaire

✞ ✝ ☎ ✆

ASSUMPTIONS

We choose a point (xn, θn) ∈ RN × TN in the phase space such that λn(xn, θn) is a simple eigenvalue and ∇θλn(xn, θn) = ∇xλn(xn, θn) = 0 We consider well-prepared initial data u0

ǫ(x) = ψn

• xn, x

ǫ , θn e2iπ θn·x

ǫ v0x − xn

√ǫ

• with v0 ∈ H1(RN) (degenerate case for WKB !)
• Notations. New scale z = x−xn

√ǫ

Homogenization of Schr¨

• dinger equation

85

• G. Allaire

✞ ✝ ☎ ✆

Main result

Theorem (A.-Palombaro). uǫ(t, x) = ei λn(xn,θn)t

ǫ

e2iπ θn·x

ǫ ψn

• xn, x

ǫ , θn v

• t, x − xn

√ǫ

• + rǫ
• t, x − xn

√ǫ

• with lim

ǫ→0

• RN |rǫ(t, z)|2 dz = 0, uniformly in time, and v(t, z) is the unique

solution of the homogenized Schr¨

• dinger equation

     i∂v ∂t − divz (A∗∇zv) + divz(vB∗z) + c∗v + vD∗z · z = 0 in RN × R+ v(0, z) = v0(z) in RN where c∗ is a constant coefficient and A∗, B∗, D∗ are constant matrices defined by A∗ = 1 8π2 ∇θ∇θλn(xn, θn) , B∗ = 1 2iπ ∇θ∇xλn(xn, θn) , D∗ = 1 2∇x∇xλn(xn, θn) .

Homogenization of Schr¨

• dinger equation

86

• G. Allaire

✞ ✝ ☎ ✆

Homogenized problem

     i∂v ∂t − divz (A∗∇zv) + divz(vB∗z) + c∗v + vD∗z · z = 0 in RN × R+ v(0, z) = v0(z) in RN

• Lemma. The operator A∗ : L2(RN) → L2(RN) defined by

A∗ϕ = − div (A∗∇ϕ) + div(ϕB∗z) + c∗ϕ + ϕD∗z · z is self-adjoint.

• Corollary. The homogenized Schr¨
• dinger equation is well-posed in

C(R+; L2(RN)) and satisfies the energy conservation ||v(t, ·)||L2(RN) = ||v0||L2(RN ) ∀ t ∈ R+ .

Homogenization of Schr¨

• dinger equation

87

• G. Allaire

Proof of the corollary. By self-adjointness

• RN A∗vv dz = A∗v, v = v, A∗v = A∗v, v ∈ R.

i

• RN

∂v ∂t v dz +

• RN A∗vv dz = 0

Take the imaginary part to get 1 2 d dt

• RN |v|2dz = 0

Homogenization of Schr¨

• dinger equation

88

• G. Allaire

Proof of the lemma. By integration by parts and since B∗ ∈ iR, B∗ = −B∗

• RN
• div(vB∗z)w − 1

2tr B∗vw

• dz =
• RN
• div(wB∗z)v − 1

2tr B∗wv

• dz

Thus

• RN A∗vw dz =
• RN
• A∗∇v · ∇w + D∗z · z vw + div(vB∗z)w − 1

2tr B∗vw

• dz

+

• RN

1 2tr B∗ + c∗ vw dz which is symmetric (obvious for the blue terms) since 1 2itr B∗ + Imc∗ = 0 as a consequence of the Fredholm alternative for

∂2ψn ∂zk∂θl .

Homogenization of Schr¨

• dinger equation

89

• G. Allaire

✞ ✝ ☎ ✆

Compactness and localization

∇∇λn =   ∇x∇xλn ∇θ∇xλn ∇θ∇xλn ∇θ∇θλn   (xn, θn) .

• Lemma. If the matrix ∇∇λn is positive definite, then there exists an
• rthonormal basis {ϕn}n≥1 in L2(RN) of eigenfunctions of the homogenized

problem. Moreover for each n there exists a real constant γn > 0 such that eγn|z|ϕn , eγn|z|∇ϕn ∈ L2(RN) . This is localization ! (cf. Anderson in a stochastic framework)

Homogenization of Schr¨

• dinger equation

90

• G. Allaire
• Proof. For simplicity assume that Re(c∗) = 0.

A∗v, v =

• RN A∗vv dz =
• RN
• A∗∇v · ∇v + D∗z · z|v|2 − iB∗Im(vz · ∇v)
• dz

Recall that A∗ = 1 8π2 ∇θ∇θλn , B∗ = 1 2iπ ∇θ∇xλn , D∗ = 1 2∇x∇xλn . Define Φ(z) =

• 2iπv(z) z , ∇v(z)

t . Then A∗v, v =

• RN

1 8π2 ∇∇λnΦ · Φ dz ≥ C

• ||∇v||2

L2(RN) + ||z v||2 L2(RN )

• The space {v ∈ H1(RN) s.t. (zv) ∈ L2(RN)} is compactly embedd in L2(RN).

Thus (A∗)−1 is compact and admits an hilbertian basis of eigenfunctions. It is classical to show that eigenfunctions decay exponentially at infinity.

Homogenization of Schr¨

• dinger equation

91

• G. Allaire

✞ ✝ ☎ ✆

Proof of the homogenization theorem

Variational proof with an oscillating test function.

• 1. Change of variables z = x−xn

√ǫ

• 2. 2nd order Taylor expansion around z = 0 for the macroscopic variable
• 3. Two-scale convergence at the scale √ǫ
• 4. Bloch wave decomposition as in the purely periodic case

No need of WKB or semiclassical arguments.

Homogenization of Schr¨

• dinger equation

92

• G. Allaire

Change of variables: z = x − xn √ǫ , Change of unknowns: wǫ(t, z) := e2iπ θn·z

√ǫ vǫ(t, z) = e−i λn(θn)t ǫ

uǫ(t, x) . New equation:      i∂wǫ ∂t − div[A √ǫz, z/√ǫ

• ∇wǫ] + 1

ǫ [c(√ǫz, z/√ǫ) − λn(θn)]wǫ = 0 wǫ(0, z) = u0

ǫ(√ǫz)

The oscillations are at scale √ǫ: this is the parabolic scaling !

Homogenization of Schr¨

• dinger equation

93

• G. Allaire

Taylor expansion around zero for the macroscopic variable c √ǫz, z √ǫ

• = c
• 0, z

√ǫ

• +√ǫz·∇xc
• 0, z

√ǫ

• +1

2ǫ∇x∇xc

• 0, z

√ǫ

• z·z+O(√ǫ|z|)3.

We obtain 1 ǫ

• c(√ǫz, z/√ǫ) − λn(θn)
• =

1 ǫ

• c(0, z/√ǫ) − λn(θn)
• + 1

√ǫz · ∇xc(0, z/√ǫ) + 1 2∇x∇xc

• 0, z

√ǫ

• z · z + O(√ǫ)

as usual + new singular term to cancel + bounded harmonic potential New test function: Ψǫ(t, z) = e2iπθn· z

√ǫ

• ψǫ

nφ(t, z) + √ǫ N

• k=1

1 2iπ ∂ψǫ

n

∂θk ∂φ ∂zk (t, z) + zk ∂ψǫ

n

∂xk φ(t, z)

Homogenization of Schr¨

• dinger equation

94

• G. Allaire
• VIII- TIME OSCILLATING POTENTIAL

     i∂uǫ ∂t − ∆uǫ +

• ǫ−2c

x ǫ

• + dǫ(t, x)
• uǫ = 0

in RN × (0, T) uǫ(t = 0, x) = u0

ǫ(x)

in RN, Goal: transfer some energy from the initial data to some target final data u0

ǫ(x) = ψn

x ǫ , θn e2iπ θn·x

ǫ v0(x)

uT

ǫ (x) = ψm

x ǫ , θm e2iπ θm·x

ǫ

vT (x) For this task, we choose a time oscillating potential dǫ(t, x) = ℜ

• ei (λm(θm)−λn(θn))t

ǫ2

e2iπ (θm−θn)·x

ǫ

• d
• t, x, x

ǫ

• ,

where d(t, x, y) is a real potential defined on [0, T] × RN × TN.

Homogenization of Schr¨

• dinger equation

95

• G. Allaire

✞ ✝ ☎ ✆

Assumptions

We assume for p = n, m    (i) λp(θp) is a simple eigenvalue, (ii) θp is a critical point of λp(θ) i.e., ∇θλp(θp) = 0 and a non-resonant assumption (iii) for any p ≥ 1, λp(2θn − θm) = 2λn(θn) − λm(θm).

Homogenization of Schr¨

• dinger equation

96

• G. Allaire
• Theorem. For an initial data u0

ǫ ∈ H1(RN) given by

u0

ǫ(x) = ψn

x ǫ , θn e2iπ θn·x

ǫ v0(x)

with v0 ∈ H1(RN), the solution of the Schr¨

• dinger equation can be written as

uǫ(t, x) = ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ ψn

x ǫ , θn vn(t, x) + ei λm(θm)t

ǫ2

e2iπ θm·x

ǫ

ψm x ǫ , θm vm(t, x) + rǫ(t, x), with lim

ǫ→0

T

• RN |rǫ(t, x)|2 dx = 0,

Homogenization of Schr¨

• dinger equation

97

• G. Allaire

and (vn, vm) ∈ C

• [0, T]; L2(RN)

2 is the unique solution of the homogenized Schr¨

• dinger system

                     i∂vn ∂t − div (A∗

n∇vn) + d∗ nm(t, x) vm = 0

in RN × (0, T) i∂vm ∂t − div (A∗

m∇vm) + d∗ mn(t, x) vn = 0

in RN × (0, T) vn(t = 0, x) = v0(x) in RN vm(t = 0, x) = 0 in RN, with A∗

p = 1 8π2 ∇θ∇θλp(θp), for p = n, m, and ”Fermi golden rule”

d∗

nm(t, x) = d ∗ mn(t, x) = 1

2

• TN d(t, x, y)ψn(y, θn)ψm(y, θm) dy.

Homogenization of Schr¨

• dinger equation

98

• G. Allaire

✞ ✝ ☎ ✆

Comments

➫ The exterior potential dǫ can be light illuminating the semiconductor. ➫ The initial data could be a combination of the to states n and m. ➫ Optical absorption: light excites electrons from the valence band to the conduction band. ➫ Its converse effect is at the root of lasers, light emitting diodes and photo-detectors. ➫ The squared modulus of d∗

nm is called the transition probability per unit time

from state n to m. ➫ Theorem obtained with M. Vanninathan.

Homogenization of Schr¨

• dinger equation

99

• G. Allaire

✞ ✝ ☎ ✆ Sketch of the proof Define two sequences vn

ǫ (t, x) = uǫ(t, x)e−i λn(θn)t

ǫ2

e−2iπ θn·x

ǫ ,

vm

ǫ (t, x) = uǫ(t, x)e−i λm(θm)t

ǫ2

e−2iπ θm·x

ǫ

. They both satisfy the a priori estimate, for p = n, m, vp

ǫ L∞((0,T );L2(RN)) + ǫ∇vp ǫ L2((0,T )×RN) ≤ C

Up to a subsequence they two-scale converge to limits wp(t, x, y) ∈ L2 (0, T) × RN; H1(TN)

• .

Homogenization of Schr¨

• dinger equation

100

• G. Allaire

Recall that      i∂uǫ ∂t − ∆uǫ +

• ǫ−2c

x ǫ

• + dǫ(t, x)
• uǫ = 0

in RN × (0, T) uǫ(t = 0, x) = u0

ǫ(x)

in RN, with the initial data u0

ǫ(x) = ψn

x ǫ , θn e2iπ θn·x

ǫ v0(x)

and a time oscillating potential dǫ(t, x) = ℜ

• ei (λm(θm)−λn(θn))t

ǫ2

e2iπ (θm−θn)·x

ǫ

• d
• t, x, x

ǫ

• ,

where d(t, x, y) is a real potential defined on [0, T] × RN × TN.

Homogenization of Schr¨

• dinger equation

101

• G. Allaire

First step. We multiply the p.d.e. by the complex conjugate of ǫ2φ(t, x, x ǫ )ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ

This yields the variational formulation of the cell problem −( divy + 2iπθn)

• (∇y + 2iπθn)wn
• + c(y)wn = λn(θn)wn

in TN. By simplicity of λn(θn), there exists vn(t, x) ∈ L2 (0, T) × RN such that wn(t, x, y) = vn(t, x)ψn(y, θn). The same argument works replacing n by m ! wm(t, x, y) = vm(t, x)ψm(y, θm).

Homogenization of Schr¨

• dinger equation

102

• G. Allaire

Second step. We multiply the p.d.e. by the complex conjugate of Ψǫ = ei λn(θn)t

ǫ2

e2iπ θn·x

ǫ

• ψn(x

ǫ , θn)φ(t, x) + ǫ

N

• k=1

∂φ ∂xk (t, x)ζk

n(x

ǫ )

• We perform the same algebra as before except for the new term

T

• RN d
• t, x, x

ǫ 1 2

• ei (λm(θm)−2λn(θn))t

ǫ2

e2iπ (θm−2θn)·x

ǫ

+ e−i λm(θm)t

ǫ2

e−2iπ θm·x

ǫ

• uǫ
• ψn(x

ǫ , θn)φ(t, x) + O(ǫ)

• dt dx

= T

• RN d
• t, x, x

ǫ 1 2

• v2n−m

ǫ

+ vm

ǫ

ψn(x ǫ , θn)φ(t, x) + O(ǫ)

• dt dx

where we introduced a new sequence v2n−m

ǫ

defined by v2n−m

ǫ

(t, x) = uǫ(t, x)e−i (2λn(θn)−λm(θm))t

ǫ2

e−2iπ (2θn−θm)·x

ǫ

.

Homogenization of Schr¨

• dinger equation

103

• G. Allaire

Applying the same arguments as in the first step, this sequence v2n−m

ǫ

two-scale converges to w2n−m(t, x, y) which satisfies −( divy + 2iπ(2θn − θm))

• (∇y + 2iπ(2θn − θm))w2n−m
• + c(y)w2n−m

= (2λn(θn) − λm(θm))w2n−m in TN. Because of the non-resonance assumption 2λn(θn) − λm(θm) = λp(2θn − θm) ∀p the above spectral problem has no solution other than 0 w2n−m(t, x, y) ≡ 0.

Homogenization of Schr¨

• dinger equation

104

• G. Allaire

Finally the two-scale limit is −i

• RN v0φ(t = 0) dx − i

T

• RN vn

∂φ ∂t dt dx − T

• RN A∗

nvn · ∇∇φdt dx

+ T

• RN d∗

nm(t, x)vmφ dt dx = 0.

with d∗

nm(t, x) = 1

2

• TN d(t, x, y)ψn(y, θn)ψm(y, θm) dy.

A symmetric argument works for vm (changing n in m in the test function Ψǫ).

Homogenization of Schr¨

• dinger equation

105

• G. Allaire
• IX- CONCLUSION AND GENERALIZATIONS

✗ Asymptotic of wave packets: uǫ(t, x) ≈ ei λ1(θn)t

ǫ2

e2iπ θn·x

ǫ ψn

x ǫ

• v
• t, x + V

ǫ t

• ✗ Justification of the notion of effective mass.

✗ This is not G- or H-convergence. ✗ WKB ansatz for periodic coefficients. Other generalizations: ☞ Parabolic equation (A., Capdeboscq, Piatnistki, Siess, Vanninathan, ARMA 174, 2004). ☞ Wave equation. (work in progress) ☞ Nonlinear Schr¨

• dinger equation (Ch. Sparber, 2006).