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

Homogenization of Schr¨ odinger equation 17 G. Allaire -II- TWO-SCALE CONVERGENCE METHOD This is a simpler method to prove the convergence theorem. Definition. A sequence of functions u ǫ in L 2 (Ω) is said to two-scale converge to a limit u 0 ( x, y ) belonging to L 2 (Ω × Y ) if, for any Y -periodic smooth function ϕ ( x, y ), it satisfies � x, x � � � � lim u ǫ ( x ) ϕ dx = u 0 ( x, y ) ϕ ( x, y ) dxdy. ǫ ǫ → 0 Ω Ω Y Theorem 1. From each bounded sequence u ǫ in L 2 (Ω) one can extract a subsequence, and there exists a limit u 0 ( x, y ) ∈ L 2 (Ω × Y ) such that this subsequence two-scale converges to u 0 . Theorem 2. Let u ǫ be a bounded sequence in H 1 (Ω). Then, up to a subsequence, u ǫ two-scale converges to a limit u 0 ( x, y ) ≡ u ( x ) ∈ H 1 (Ω), and ∇ u ǫ two-scale converges to ∇ x u ( x ) + ∇ y u 1 ( x, y ) with u 1 ∈ L 2 (Ω; H 1 # ( Y )).

Homogenization of Schr¨ odinger equation 18 G. Allaire 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 L 2 (Ω × Y ), and there exists C > 0 such that x, x � � � | 2 dx ≤ C � ϕ � 2 | ϕ B , ǫ Ω and � x, x � � � � | 2 dx = | ϕ ( x, y ) | 2 dxdy, lim | ϕ ǫ ǫ → 0 Ω Ω Y for any ϕ ( x, y ) ∈ B . Remark. The same works with B = L 2 (Ω; C # ( Y )) and Ω not necessarily bounded.

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Homogenization of Schr¨ odinger equation 20 G. Allaire Proof of Theorem 1. By Schwarz inequality, we have 1 � � � � x, x x, x � � 2 � � � � � � � � u ǫ ( x ) ϕ dx � ≤ C ϕ dx ≤ C � ϕ � B . � � � � ǫ ǫ � � � Ω Ω Thus, the l.h.s. is a continuous linear form on B which can be identified to a duality product � µ ǫ , ϕ � B ′ ,B for some bounded sequence of measures µ ǫ . Since B is separable, one can extract a subsequence and there exists a limit µ 0 such µ ǫ converges to µ 0 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 1 � � � � 2 | ϕ ( x, y ) | 2 dxdy � � |� µ 0 , ϕ � B ′ ,B | ≤ C . � � � � Ω Y Therefore µ 0 is actually a continuous linear form on L 2 (Ω × Y ), by density of B in this space. Thus, there exists u 0 ( x, y ) ∈ L 2 (Ω × Y ) such that � � � µ 0 , ϕ � B ′ ,B = u 0 ( x, y ) ϕ ( x, y ) dxdy. Ω Y

Homogenization of Schr¨ odinger equation 21 G. Allaire Proof of Theorem 2. Since u ǫ and ∇ u ǫ are bounded in L 2 (Ω), up to a subsequence, they two-scale converge to limits u 0 ( x, y ) ∈ L 2 (Ω × Y ) and ξ 0 ( x, y ) ∈ L 2 (Ω × Y ) N . Thus � x, x � � � � ∇ u ǫ ( x ) · � ξ 0 ( x, y ) · � ∀ � Ω; C ∞ # ( Y ) N � � lim ψ dx = ψ ( x, y ) dxdy ψ ∈ D . ǫ ǫ → 0 Ω Ω Y Integrating by parts the left hand side gives x, x x, x x, x � � � � � � � � �� ∇ u ǫ ( x ) · � div y � + ǫ div x � ǫ ψ dx = − u ǫ ( x ) ψ ψ dx. ǫ ǫ ǫ Ω Ω Passing to the limit yields � � u 0 ( x, y ) div y � u 0 ( x, y ) ≡ u ( x ) ∈ L 2 (Ω) . 0 = − ψ ( x, y ) dxdy ⇒ Ω Y Next, we choose � ψ such that div y � ψ ( x, y ) = 0. We obtain � x, x � x, x � � � � ∇ u ǫ ( x ) · � u ǫ ( x ) div x � ψ dx = − ψ dx. ǫ ǫ Ω Ω

Homogenization of Schr¨ odinger equation 22 G. Allaire Passing to the two-scale limit � � � � ξ 0 ( x, y ) · � u ( x ) div x � ψ ( x, y ) dxdy = − ψ ( x, y ) dxdy. Ω Y Ω Y If � ψ does not depend on y , it proves that u ( x ) ∈ H 1 (Ω). Furthermore, � � ( ξ 0 ( x, y ) − ∇ u ( x )) · � ∀ � ψ with div y � ψ ( x, y ) dxdy = 0 ψ = 0 . Ω Y The orthogonal of divergence-free functions are exactly the gradients. Thus, there exists a unique function u 1 ( x, y ) in L 2 (Ω; H 1 # ( Y ) / R ) such that ξ 0 ( x, y ) = ∇ u ( x ) + ∇ y u 1 ( x, y ) .

Homogenization of Schr¨ odinger equation 23 G. Allaire Theorem 4. Let u ǫ ∈ L 2 (Ω) two-scale converge to u 0 ( x, y ) ∈ L 2 (Ω × Y ). 1. Then, u ǫ converges weakly in L 2 (Ω) to u ( x ) = � Y u 0 ( x, y ) dy , and we have ǫ → 0 � u ǫ � 2 L 2 (Ω) ≥ � u 0 � 2 L 2 (Ω × Y ) ≥ � u � 2 lim L 2 (Ω) . 2. Assume further that u 0 ( x, y ) is smooth and that ǫ → 0 � u ǫ � 2 L 2 (Ω) = � u 0 � 2 lim L 2 (Ω × Y ) . Then, we have x, x 2 � � �� � u ǫ ( x ) − u 0 L 2 (Ω) → 0 . � � ǫ � Remark. In the last case we say that u ǫ two-scale converges strongly to u 0 .

Homogenization of Schr¨ odinger equation 24 G. Allaire Proof of Theorem 4. Take a test function depending only on x � � � � lim u ǫ ( x ) ϕ ( x ) dx = u 0 ( x, y ) ϕ ( x ) dxdy = u ( x ) ϕ ( x ) dx. ǫ → 0 Ω Ω Ω Y Thus, u ǫ converges weakly to u in L 2 (Ω). Then, developing the inequality x, x 2 � � �� � � u ǫ ( x ) − ϕ dx ≥ 0 � � ǫ � Ω � � x, x � x, x � � � � | u ǫ ( x ) | 2 dx − 2 | 2 dx ≥ 0 u ǫ ( x ) ϕ dx + | ϕ ǫ ǫ Ω Ω Ω � � � � � | u ǫ ( x ) | 2 dx − 2 | ϕ ( x, y ) | 2 dx dy ≥ 0 lim inf u 0 ( x, y ) ϕ ( x, y ) dx dy + ǫ → 0 Ω Ω Y Ω Y Take ϕ = u 0 to get ǫ → 0 � u ǫ � 2 L 2 (Ω) ≥ � u 0 � 2 lim L 2 (Ω × Y ) .

Homogenization of Schr¨ odinger equation 25 G. Allaire Proof of Theorem 4 (continued). If we assume ǫ → 0 � u ǫ � 2 L 2 (Ω) = � u 0 � 2 lim L 2 (Ω × Y ) , the same computation yields � x, x 2 � � � �� � | u 0 ( x, y ) − ϕ ( x, y ) | 2 dx dy lim inf � u ǫ ( x ) − ϕ dx = � � ǫ � ǫ → 0 Ω Ω Y If u 0 is smooth enough to be a test function ϕ (Carath´ eodory function), it gives the desired result x, x 2 � �� � � u ǫ ( x ) − u 0 L 2 (Ω) → 0 . � � ǫ �

Homogenization of Schr¨ odinger equation 26 G. Allaire Theorem 5. 1. Let u ǫ be a bounded sequence in L 2 (Ω) such that ǫ ∇ u ǫ is also bounded in L 2 (Ω) N . Then, there exists a two-scale limit u 0 ( x, y ) ∈ L 2 (Ω; H 1 # ( Y ) / R ) such that, up to a subsequence, u ǫ two-scale converges to u 0 ( x, y ), and ǫ ∇ u ǫ to ∇ y u 0 ( x, y ). 2. Let u ǫ be a bounded sequence in L 2 (Ω) N such that div u ǫ is also bounded in L 2 (Ω). Then, there exists a two-scale limit u 0 ( x, y ) ∈ L 2 (Ω × Y ) N with div y u 0 = 0 and div x u 0 ∈ L 2 (Ω × Y ) such that, up to a subsequence, u ǫ two-scale converges to u 0 ( x, y ), and div u ǫ to div x u 0 ( x, y ).

Homogenization of Schr¨ odinger equation 27 G. Allaire Proof of Theorem 5 (first part). We have x, x � � � � � lim u ǫ ( x ) ϕ dx = u 0 ( x, y ) ϕ ( x, y ) dxdy ǫ ǫ → 0 Ω Ω Y � x, x � � � � ǫ ∇ u ǫ ( x ) · � ξ 0 ( x, y ) · � lim ψ dx = ψ ( x, y ) dxdy ǫ ǫ → 0 Ω Ω Y By integration by parts x, x x, x � � � � � � � � ǫ ∇ u ǫ ( x ) · � div y � ψ + ǫ div x � ψ dx = − u ǫ ( x ) ψ dx ǫ ǫ Ω Ω Passing to the two-scale limit � � � � ξ 0 ( x, y ) · � u 0 ( x, y ) div y � ψ ( x, y ) dxdy = − ψ ( x, y ) dxdy Ω Ω Y Y which implies that ξ 0 ( x, y ) = ∇ y u 0 ( x, y ).

Homogenization of Schr¨ odinger equation 28 G. Allaire -III- APPLICATION TO HOMOGENIZATION Conductivity or diffusion equation  x, x � � � � − div A ∇ u ǫ = f in Ω  ǫ u ǫ = 0 on ∂ Ω  with a coefficient tensor A ( x, y ) which is Y -periodic, uniformly coercive and bounded N α | ξ | 2 ≤ � A ij ( x, y ) ξ i ξ j ≤ β | ξ | 2 , ∀ ξ ∈ R N , ∀ y ∈ Y, ∀ x ∈ Ω ( β ≥ α > 0) . i,j =1 A priori estimate. If Ω is bounded, then � u ǫ � H 1 (Ω) ≤ C � f � L 2 (Ω) .

Homogenization of Schr¨ odinger equation 29 G. Allaire ✞ ☎ A priori estimate ✝ ✆ Variational formulation � � x � � ∀ ϕ ∈ H 1 A ∇ u ǫ ( x ) · ∇ ϕ ( x ) dx = f ( x ) ϕ ( x ) dx, 0 (Ω) . ǫ Ω Ω Take ϕ = u ǫ and use coercivity � α �∇ u ǫ � 2 L 2 (Ω) ≤ f ( x ) u ǫ ( x ) dx ≤ � f � L 2 (Ω) � u ǫ � L 2 (Ω) Ω Poincar´ e inequality in Ω � u ǫ � L 2 (Ω) ≤ C (Ω) �∇ u ǫ � L 2 (Ω) Thus �∇ u ǫ � L 2 (Ω) ≤ C (Ω) � f � L 2 (Ω) α

Homogenization of Schr¨ odinger equation 30 G. Allaire ✞ ☎ Two-scale convergence method ✝ ✆ First step. We deduce from the a priori estimates the precise form of the two-scale limit of the sequence u ǫ . By application of Theorem 2, there exist two functions, u ( x ) ∈ H 1 0 (Ω) and u 1 ( x, y ) ∈ L 2 (Ω; H 1 # ( Y ) / R ), such that, up to a subsequence, u ǫ two-scale converges to u ( x ), and ∇ u ǫ two-scale converges to ∇ x u ( x ) + ∇ y u 1 ( x, y ). x, x � � In view of these limits, u ǫ is expected to behave as u ( x ) + ǫu 1 . ǫ

Homogenization of Schr¨ odinger equation 31 G. Allaire ✞ ☎ Two-scale convergence method ✝ ✆ Second step. We multiply the p.d.e. by a test function similar to the limit of x, x , where ϕ ( x ) ∈ D (Ω) and ϕ 1 ( x, y ) ∈ D (Ω; C ∞ � � u ǫ , namely ϕ ( x ) + ǫϕ 1 # ( Y )). ǫ This yields x, x x, x x, x � � � � � � � �� A ∇ u ǫ · ∇ ϕ ( x ) + ∇ y ϕ 1 + ǫ ∇ x ϕ 1 dx ǫ ǫ ǫ Ω � x, x � � �� = f ( x ) ϕ ( x ) + ǫϕ 1 dx. ǫ Ω Regarding A t � x, x � � � x, x �� ∇ ϕ ( x ) + ∇ y ϕ 1 as a test function for the two-scale ǫ ǫ convergence, we pass to the two-scale limit � � � A ( x, y ) ( ∇ u ( x ) + ∇ y u 1 ( x, y )) · ( ∇ ϕ ( x ) + ∇ y ϕ 1 ( x, y )) dxdy = f ( x ) ϕ ( x ) dx. Ω Y Ω

Homogenization of Schr¨ odinger equation 32 G. Allaire ✞ ☎ Two-scale convergence method ✝ ✆ Third step. We read off a variational formulation for ( u, u 1 ). Take ( ϕ, ϕ 1 ) in 0 (Ω) × L 2 � � the Hilbert space H 1 Ω; H 1 # ( Y ) / R endowed with the norm � ( �∇ u ( x ) � 2 L 2 (Ω) + �∇ y u 1 ( x, y ) � 2 L 2 (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 � � A ( x, y ) ( ∇ ϕ ( x ) + ∇ y ϕ 1 ( x, y )) · ( ∇ ϕ ( x ) + ∇ y ϕ 1 ( x, y )) dxdy ≥ Ω Y � � � � � |∇ ϕ ( x )+ ∇ y ϕ 1 ( x, y ) | 2 dxdy = α |∇ ϕ ( x ) | 2 dx + α |∇ y ϕ 1 ( x, y ) | 2 dxdy. α Ω Ω Ω Y Y

Homogenization of Schr¨ odinger equation 33 G. Allaire � � � A ( x, y ) ( ∇ u ( x ) + ∇ y u 1 ( x, y )) · ( ∇ ϕ ( x ) + ∇ y ϕ 1 ( x, y )) dxdy = f ( x ) ϕ ( x ) dx. Ω Y Ω By application of the Lax-Milgram lemma, there exists a unique solution 0 (Ω) × L 2 � � ( u, u 1 ) ∈ H 1 Ω; H 1 # ( Y ) / R . Consequently, the entire sequences u ǫ and ∇ u ǫ converge to u ( x ) and ∇ u ( x ) + ∇ y u 1 ( x, y ). An easy integration by parts shows that the associated p.d.e.’s are the so-called “two-scale homogenized problem”,  − div y ( A ( x, y ) ( ∇ u ( x ) + ∇ y u 1 ( x, y ))) = 0 in Ω × Y     �� �  − div x Y A ( x, y ) ( ∇ u ( x ) + ∇ y u 1 ( x, y )) dy = f ( x ) in Ω  y → u 1 ( x, y ) Y -periodic      u = 0 on ∂ Ω . 

Homogenization of Schr¨ odinger equation 34 G. Allaire ✞ ☎ Two-scale convergence method ✝ ✆ Fourth (and optional) step. Eliminate the y variable and the u 1 unknown N ∂u � u 1 ( x, y ) = ( x ) w i ( x, y ) , ∂x i i =1 where w i ( x, y ) are the unique solutions in H 1 # ( Y ) / R of the cell problems  − div y ( A ( x, y ) ( � e i + ∇ y w i ( x, y ))) = 0 in Y  y → w i ( x, y ) Y -periodic,  at each point x ∈ Ω, and  − div x ( A ∗ ( x ) ∇ u ( x )) = f ( x ) in Ω  u = 0 on ∂ Ω ,  with A ∗ � ij ( x ) = Y A ( x, y ) ( � e i + ∇ y w i ( x, y )) · ( � e j + ∇ y w j ( x, y )) dy.

Homogenization of Schr¨ odinger equation 35 G. Allaire -IV- BLOCH WAVES Theorem 1. For any function u ( y ) ∈ L 2 ( R N ) there exists a unique function u ( y, θ ) ∈ L 2 ( Y × Y ) such that ˆ � u ( y, θ ) e 2 iπθ · y dθ. u ( y ) = ˆ Y u ( y, θ ) is Y -periodic while the function θ → e 2 iπθ · y ˆ The function y → ˆ u ( y, θ ) is Y -periodic. Furthermore, the linear map B , called the Bloch transform and u , is an isometry from L 2 ( R N ) into L 2 ( Y × Y ), i.e. Parseval defined by B u = ˆ formula holds for any u, v ∈ L 2 ( R N ) � � � R N u ( y ) v ( y ) dy = u ( y, θ )ˆ ˆ v ( y, θ ) dy dθ. Y Y

Homogenization of Schr¨ odinger equation 36 G. Allaire Proof. Let u ( y ) be a smooth compactly supported function in R N . Define � u ( y + k ) e − 2 iπθ · ( y + k ) . u ( y, θ ) = ˆ k ∈ Z N 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 ∈ Z N , we have u ( y + k ) e − 2 iπθ · ( y + k ) = e − 2 iπj · y ˆ u ( y, θ + j ) = e − 2 iπj · y � ˆ u ( y, θ ) . k ∈ Z N Thus, θ → e 2 iπθ · y ˆ u ( y, θ ) is Y -periodic. Next, we compute � � � u ( y, θ ) e 2 iπθ · y dθ = T N e − 2 iπθ · k dθ = u ( y ) T N ˆ u ( y + k ) k ∈ Z N since all integrals vanish except for k = 0. This proves the result for smooth compactly supported functions.

Homogenization of Schr¨ odinger equation 37 G. Allaire In particular, it shows that the Bloch transform B is a linear map, well defined on C ∞ c ( R N ) and bounded on L 2 ( R N ). Since C ∞ c ( R N ) is dense in L 2 ( R N ), B can be extended by continuity and the result holds true in L 2 ( R N ).

Homogenization of Schr¨ odinger equation 38 G. Allaire Lemma. Let a ( y ) ∈ L ∞ ( T N ) be a periodic function. For any u ( y ) ∈ L 2 ( R N ), we have B ( au ) = a B ( u ) ≡ a ( y )ˆ u ( y, θ ) . Remark. T N is the flat unit torus, i.e. T N ≡ Y + periodic B.C. Lemma. Let u ( y ) ∈ H 1 ( R N ). The Bloch transform of its gradient is B ( ∇ y u ) = ( ∇ y + 2 iπθ ) B ( u ) ≡ ∇ y ˆ u ( y, θ ) + 2 iπθ ˆ u ( y, θ ) u ( y, θ ) belongs to H 1 ( T N ). and y → B ( u ) ≡ ˆ

Homogenization of Schr¨ odinger equation 39 G. Allaire ✞ ☎ Application ✝ ✆ Find u ∈ H 1 ( R N ) solution of � � in R N , − div y A ( y ) ∇ y u + c ( y ) u = f with f ∈ L 2 ( R N ). We assume that A ( y ) and c ( y ) belong to L ∞ ( T N ) and ∃ ν > 0 such that for a.e. y ∈ T N A ( y ) ξ · ξ ≥ ν | ξ | 2 for any ξ ∈ R N and c ( y ) ≥ c 0 > 0 . The variational formulation is to find u ∈ H 1 ( R N ) such that � � � � ∀ φ ∈ H 1 ( R N ) . A ( y ) ∇ y u · ∇ y φ + c ( y ) uφ dy = R N fφ dy R N

Homogenization of Schr¨ odinger equation 40 G. Allaire Proposition. The p.d.e. in R N is equivalent to the family of p.d.e.’s, indexed by θ ∈ T N , � � in T N , − ( div y + 2 iπθ ) A ( y )( ∇ y + 2 iπθ ) B u + c ( y ) B u = B f which admits a unique solution y → ( B u )( y, θ ) ∈ H 1 ( T N ) for any θ ∈ T N . Proof. We apply the previous Lemmas � � � � ∀ φ ∈ H 1 ( R N ) A ( y ) ∇ y u · ∇ y φ + c ( y ) uφ dy = R N fφ dy R N � � � � � � u · ( ∇ y + 2 iπθ )ˆ u ˆ f ˆ ˆ A ( y )( ∇ y + 2 iπθ )ˆ φ + c ( y )ˆ φ dy dθ = φ dy dθ T N T N Y Y which is a variational formulation in y ∈ T N integrated with respect to θ which is just a parameter.

Homogenization of Schr¨ odinger equation 41 G. Allaire For a given θ ∈ Y , consider the Green operator G θ  L 2 ( T N ) L 2 ( T N ) →  g ( y ) → G θ g ( y ) = v ( y )  where v ∈ H 1 ( T N ) is the unique solution of � � in T N . − ( div y + 2 iπθ ) A ( y )( ∇ y + 2 iπθ ) v + c ( y ) v = g One can check that G θ is a self-adjoint compact complex-valued linear operator acting on L 2 ( T N ). As such it admits a countable sequence of real increasing eigenvalues ( λ n ) n ≥ 1 (repeated with their multiplicity) and normalized eigenfunctions ( ψ n ) n ≥ 1 with � ψ n � L 2 ( T N ) = 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 .

Homogenization of Schr¨ odinger equation 42 G. Allaire In other words, the eigenvalues and eigenfunctions satisfy the so-called Bloch (or shifted) spectral cell equation � � in T N . − ( div y + 2 iπθ ) A ( y )( ∇ y + 2 iπθ ) ψ n + c ( y ) ψ n = λ n ( θ ) ψ n θ ∈ Y is the Bloch frequency (or quasi momentum). A Bloch wave is φ n ( y ) = ψ n ( θ, y ) e 2 iπθ · y which satisfies in R N . − div y ( A ( y ) ∇ y φ n ) + c ( y ) φ n = λ ( θ ) φ n

Homogenization of Schr¨ odinger equation 43 G. Allaire Theorem 2. For any u ( y ) ∈ L 2 ( R N ), ∃ ˆ u n ( θ ) ∈ L 2 ( T N ), n ≥ 1, such that � � u n ( θ ) ψ n ( y, θ ) e 2 iπθ · y dθ. u ( y ) = T N ˆ n ≥ 1 Furthermore, the linear map B , called the Bloch transform and defined by u n ) n ≥ 1 , is an isometry from L 2 ( R N ) into ℓ 2 L 2 ( T N ) � � B u = (ˆ , i.e. Parseval formula holds for any u, v ∈ L 2 ( R N ) � � � R N u ( y ) v ( y ) dy = u n ( θ )ˆ ˆ v n ( θ ) dθ. Y n ≥ 1 Proof. We decompose each ˆ u ( y, θ ) on the corresponding eigenbasis � � u ( y, θ ) = ˆ u n ( θ ) ψ n ( y, θ ) ˆ with ˆ u n ( θ ) = T N ˆ u ( y, θ ) ψ n ( y, θ ) dy. n ≥ 1 Commuting the sum with respect to n and the integral with respect to θ is a standard Fubini type result.

Homogenization of Schr¨ odinger equation 44 G. Allaire ✞ ☎ Application ✝ ✆ � � in R N , − div y A ( y ) ∇ y u + c ( y ) u = f with f ∈ L 2 ( R N ). We obtain an explicit algebraic formula for the solution ˆ f n ( θ ) u n ( θ ) = ˆ ∀ n ≥ 1 , ∀ θ ∈ Y, λ n ( θ ) which is a generalization of a similar formula, using Fourier transform, for a constant coefficient p.d.e..

Homogenization of Schr¨ odinger equation 45 G. Allaire ✞ ☎ Smoothness of the Bloch eigenvalues ✝ ✆ Lemma. θ → λ n ( θ ) is Lipschitz. Remark: for multiple eigenvalues θ → λ n ( θ ) is usually not smoother. Lemma (Kato, Rellich). If an eigenvalue λ n ( θ ) is simple at the value θ = θ n , then it remains simple in a small neighborhood of θ n and the n -th eigencouple is analytic in this neighborhood of θ n .

Homogenization of Schr¨ odinger equation 46 G. Allaire Assumption: λ n ( θ n ) is a simple eigenvalue. � � Operator A n ( θ ) ψ = − ( div y + 2 iπθ ) A ( y )( ∇ y + 2 iπθ ) ψ + c ( y ) ψ − λ n ( θ ) ψ, Then, in a neighborhood of θ n , one can differentiate A n ( θ ) ∂ψ n = 2 iπe k A ( y )( ∇ y +2 iπθ ) ψ n +( div y +2 iπθ ) ( A ( y )2 iπe k ψ n )+ ∂λ n ( θ ) ψ n , ∂θ k ∂θ k and the second derivative is A n ( θ ) ∂ 2 ψ n � � = 2 iπe k A ( y )( ∇ y + 2 iπθ ) ∂ψ n ∂ψ n + ( div y + 2 iπθ ) A ( y )2 iπe k ∂θ k ∂θ l ∂θ l ∂θ l � � +2 iπe l A ( y )( ∇ y + 2 iπθ ) ∂ψ n ∂ψ n + ( div y + 2 iπθ ) A ( y )2 iπe l ∂θ k ∂θ k + ∂λ n ( θ ) ∂ψ n + ∂λ n ( θ ) ∂ψ n ∂θ k ∂θ l ∂θ l ∂θ k − 4 π 2 e k A ( y ) e l ψ n − 4 π 2 e l A ( y ) e k ψ n + ∂ 2 λ n ( θ ) ψ n ∂θ l ∂θ k

Homogenization of Schr¨ odinger equation 47 G. Allaire -V- SCHR¨ ODINGER EQUATION Schr¨ odinger equation in a periodic medium: i∂u ǫ � x � x x, x  � � � � � � �� in R N × R + ǫ − 2 c ∂t − div A ∇ u ǫ + + d u ǫ = 0   ǫ ǫ ǫ  u ǫ ( t = 0 , x ) = u 0 in R N , ǫ ( x )  Complex-valued unknown: u ǫ ( t, x ) : R + × R N → C (0 , 1) N -periodic, real, measurable and bounded. ☞ y → A ( y ) , c ( y ) , d ( x, y ) ☞ A is a N × N symmetric, uniformly coercive, tensor. ☞ c and d are scalar functions with no sign. eodory function, u 0 ǫ ∈ H 1 ( R N ). ☞ d ( x, y ) is a Carath´

Homogenization of Schr¨ odinger equation 48 G. Allaire 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∂u ǫ � x  � in R N × R + ∂t − ( div + iǫ A ǫ ) ( ∇ + iǫ A ǫ ) u ǫ + ǫ − 2 c u ǫ = 0   ǫ u ǫ ( t = 0 , x ) = u 0  in R N , ǫ ( x )  t, x, x � � where A ǫ = A is the electromagnetic vector potential such that the ǫ electric field E and magnetic field B are E = − ∂ A ǫ and B = curl A ǫ . ∂t ✗ Periodic homogenization: ǫ → 0 is the period. ✗ Singular perturbation: potential of order ǫ − 2 .

Homogenization of Schr¨ odinger equation 49 G. Allaire ✞ ☎ SCALING ✝ ✆ ✗ This is not the scaling of semi-classical analysis: i ∂u ǫ � x � x x, x � � � � � � �� ǫ − 2 c ∂t − div A ∇ u ǫ + + d u ǫ = 0 ǫ ǫ ǫ ǫ ✗ This is a parabolic scaling (for longer times). ✗ Scaling of a long time asymptotic: change of variables y = x/ǫ and τ = t/ǫ 2 ⇒ cells of size 1, no more oscillations i∂u ǫ � � c ( y ) + ǫ 2 d ( ǫy, y ) ∂τ − div y ( A ( y ) ∇ y u ǫ ) + u ǫ = 0 ⇒ small and slowly varying exterior potential. Goal: find homogenized models, justify the notion of electron effective mass.

Homogenization of Schr¨ odinger equation 50 G. Allaire ✞ ☎ A priori estimates ✝ ✆ For any final time T > 0, there exists a constant C > 0 that does not depend on ǫ such that � u 0 � u ǫ � L ∞ ((0 ,T ); L 2 ( R N )) = ǫ � L 2 ( R N ) , � u 0 ǫ � L 2 ( R N ) + ǫ �∇ u 0 � � ǫ �∇ u ǫ � L ∞ ((0 ,T ); L 2 ( R N ) N ) ≤ C ǫ � L 2 ( R N ) N . Proof. Multiply the equation by u ǫ and by ∂u ǫ ∂t , integrate by parts and take the real part. No strong convergence a priori.

Homogenization of Schr¨ odinger equation 51 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 in the unit torus T N . − div y ( A ( y ) ∇ y ψ n ) + c ( y ) ψ n = λ n ψ n First eigenvalue λ 1 is simple and first eigenfunction ψ 1 ( y ) > 0 is positive by Krein-Rutman theorem (maximum principle). Interpretation of ψ 1 : periodic ground state. ǫ 2 u ǫ ( t, x ) Change of unknowns: v ǫ ( t, x ) = e − i λ 1 t ψ 1 ( x ǫ ) Algebraic trick: ψ 2 � � ψ 1 div ( A ∇ ( ψ 1 v )) = ψ 1 v div ( A ∇ ψ 1 ) + div 1 A ∇ v

Homogenization of Schr¨ odinger equation 52 G. Allaire A simpler equation for v ǫ : � ∂v ǫ i | ψ 1 | 2 � x � x x, x  � � � � � in R N × R + ( | ψ 1 | 2 A ) + ( | ψ 1 | 2 d ) ∂t − div ∇ v ǫ v ǫ = 0   ǫ ǫ ǫ u 0 ǫ ( x ) in R N . v ǫ ( t = 0 , x ) =   ψ 1 ( x ǫ ) Theorem. (Standard periodic homogenization) � x If u 0 ǫ ( x ) = v 0 ( x ) ψ 1 � , then v ǫ , converges weakly in L 2 � (0 , T ); H 1 ( R N ) � to the ǫ solution v of the homogenized problem  i∂v in R N × (0 , T ) ∂t − div ( A ∗ ∇ v ) + d ∗ ( x ) v = 0   v ( t = 0 , x ) = v 0 ( x ) in R N ,   T N | ψ 1 | 2 ( y ) d ( x, y ) dy and A ∗ is the “usual” homogenized tensor where d ∗ ( x ) = � for | ψ 1 ( y ) | 2 A ( y ).

Homogenization of Schr¨ odinger equation 53 G. Allaire ✞ ☎ INTERPRETATION ✝ ✆ Convergence result: � x u ǫ ( t, x ) ≈ e i λ 1 t � ǫ 2 ψ 1 v ( t, x ) ǫ ☞ It is crucial that ψ 1 ( y ) > 0 (maximum principle). ☞ This trick is called factorization principle. ☞ Equivalent results for parabolic or hyperbolic equations, transport equation (A.-Bal, A.-Capdeboscq ...). � x u 0 ǫ ( x ) converges weakly to v 0 ( x ) in L 2 ( R N ). ☞ Still true if ψ 1 � ǫ What happens if the weak limit is v 0 ≡ 0 ?

Homogenization of Schr¨ odinger equation 54 G. Allaire WKB METHOD A different kind of ansatz. Geometric optics or WKB method (Wentzel, Kramers, Brillouin). i ∂u ǫ x, x + 1 x, x  � � � � � � in R N × (0 , T ) , ∂t − div A ∇ u ǫ ǫ 2 c u ǫ = 0  ǫ ǫ ǫ u ǫ ( t = 0 , x ) = u 0 in R N ǫ ( x )  ☞ 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...

Homogenization of Schr¨ odinger equation 55 G. Allaire ☞ Introduce the operator � � A ( x, θ ) ψ ≡ − ( div y + 2 iπθ ) A ( x, y )( ∇ y + 2 iπθ ) ψ + c ( x, y ) ψ ☞ Eigenvalues λ n ( x, θ ) and eigenfunctions ψ n ( x, y, θ ). ☞ Choose an energy level n such that λ n ( x, θ ) is simple ∀ θ ∈ T N , ∀ x ∈ R N . ☞ Replace θ par ∇ S 0 and consider an initial data with oscillating phase x, x ǫ ( x ) = e 2 iπ S 0( x ) � � u 0 u 0 ( x ) ψ n ǫ , ∇ S 0 ( x ) ǫ

Homogenization of Schr¨ odinger equation 56 G. Allaire ✞ ☎ High frequency asymptotic expansion or WKB ansatz ✝ ✆ t, x, x t, x, x u ǫ ( t, x ) = e 2 iπ S ( t,x ) � � � � � � v + ǫv 1 + ... ǫ ǫ ǫ The first derivatives are � � ǫ∂u ǫ 2 iπ ( v + ǫv 1 ) ∂S ∂t + ǫ∂v ∂t = e 2 iπ S ( t,x ) ∂t + O ( ǫ 2 ) ǫ � � ǫ ∇ u ǫ = e 2 iπ S ( t,x ) 2 iπ ( v + ǫv 1 ) ∇ S + ∇ y v + ǫ ( ∇ x v + ∇ y v 1 ) + O ( ǫ 2 ) ǫ We compute also the second derivatives, plug the ansatz in the p.d.e. and deduce a cascade of equations.

Homogenization of Schr¨ odinger equation 57 G. Allaire ✞ ☎ Cascade of equations ✝ ✆ Order ǫ − 2 : A ( x, ∇ S ) v = 2 π ∂S T N . ∂t v in This is a spectral problem, y being the space variable and ( t, x ) being fixed parameters. Thus, we deduce 2 π ∂S ∂t = λ n ( x, ∇ S ) and by simplicity of the eigenvalue v ( t, x, y ) = u ( t, x ) ψ n ( x, y, ∇ S ( t, x )) . We have obtained an eikonal equation (Hamilton-Jacobi) to compute the phase with the initial data S (0 , x ) = S 0 ( x ).

Homogenization of Schr¨ odinger equation 58 G. Allaire Order ǫ − 1 : T N , A ( x, ∇ S ) v 1 = λ n ( x, ∇ S ) v 1 + f in with f = − i∂v � � � � ∂t + div y + 2 iπ ∇ S ( A ∇ x v ) + div x A ( ∇ y + 2 iπ ∇ S ) v . To solve for v 1 the Fredholm alternative requires that � T N f ( t, x, y ) ψ n ( x, y, ∇ S ) dy = 0 . Recall that v ( t, x, y ) = u ( t, x ) ψ n ( x, y, ∇ S ) and Fredholm alternative for ∇ θ ψ n � −∇ x u ·∇ θ λ n = 2 iπ T N [( div y + 2 iπθ ) ( A ( y ) ∇ x uψ n ) + A ( y )( ∇ y + 2 iπθ ) ψ n · ∇ x u ] dy We deduce an homogenized transport equation ∂u ∂t − ∇ θ λ n ( ∇ S ) · ∇ x u + b ∗ u = 0 2 π

Homogenization of Schr¨ odinger equation 59 G. Allaire ✄ � Reminder on Bloch waves ✂ ✁ ✖ Bloch eigenvalue λ n ( x, θ ) and eigenvector ψ n ( x, y, θ ) ✖ Assumption: λ n ( θ ) is a simple eigenvalue. ✖ Operator � � A n ( x, θ ) ψ = − ( div y + 2 iπθ ) A ( x, y )( ∇ y + 2 iπθ ) ψ + c ( x, y ) ψ − λ n ( x, θ ) ψ, ✖ Bloch spectral problem A n ( x, θ ) ψ n = 0 ✖ We differentiate with respect to θ A n ( x, θ ) ∂ψ n = 2 iπe k A ( ∇ y + 2 iπθ ) ψ n + ( div y + 2 iπθ ) ( A 2 iπe k ψ n ) + ∂λ n ψ n , ∂θ k ∂θ k ✖ Fredholm alternative: the r.h.s. is orthogonal to ψ n .

Homogenization of Schr¨ odinger equation 60 G. Allaire ✘ Group velocity V = ∇ θ λ n ( ∇ S ) 2 π ✘ If we write v ( t, x, y ) = u ( t, x, θ ) ψ n ( x, y, θ ) with θ = ∇ S , then we deduce an homogenized Liouville equation in the phase space ∂ | u | 2 − V · ∇ x | u | 2 + ∇ x λ n · ∇ θ | u | 2 = 0 ∂t

Homogenization of Schr¨ odinger equation 61 G. Allaire ✄ � Conclusion ✂ ✁ x, x � � u ǫ ( t, x ) ≈ e 2 iπ S ( t,x ) ψ n ǫ , ∇ S ( t, x ) u ( t, x, ∇ S ( t, x )) ǫ The semi-classical limit is given by the dynamic of the following Hamiltonian system in the phase space ( x, θ ) ∈ R N × T N  1 x = ˙ 2 π ∇ θ λ n ( x, θ )  ˙ θ = −∇ x λ n ( 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.

Homogenization of Schr¨ odinger equation 62 G. Allaire ✞ ☎ Special case ✝ ✆ Monochromatic initial data. Periodic coefficients. Assume S 0 ( 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 + 2 πλ n ( θ ) t Furthermore V is constant, b ∗ = 0 and ∇ x λ n ( θ ) = 0, thus u ( t, x ) = u 0 ( x + V t ) In such a simpler case we can find a better ”long time” ansatz of the solution.

Homogenization of Schr¨ odinger equation 63 G. Allaire -VI- HOMOGENIZATION We consider a monochromatic wave packet as initial data � x e 2 iπ θn · x ǫ , θ n � v 0 ∈ H 1 ( R N ) . u 0 ǫ v 0 ( x ) ǫ ( x ) = ψ n with Assumption: the eigenvalue λ n ( θ n ) 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).

Homogenization of Schr¨ odinger equation 64 G. Allaire For θ ∈ Y the Bloch (or shifted) spectral cell equation is � � in T N − ( div y + 2 iπθ ) A ( y )( ∇ y + 2 iπθ ) ψ n + c ( y ) ψ n = λ n ( θ ) ψ n Schr¨ odinger equation in a periodic medium: i∂u ǫ � x � x x, x  � � � � � � �� in R N × R + ǫ − 2 c ∂t − div A ∇ u ǫ + + d u ǫ = 0   ǫ ǫ ǫ u ǫ ( t = 0 , x ) = u 0  in R N , ǫ ( x ) 

Homogenization of Schr¨ odinger equation 65 G. Allaire Theorem 1. Assume further that the group velocity vanishes ∇ θ λ n ( θ n ) = 0 . � x u ǫ ( t, x ) = e i λn ( θn ) t e 2 iπ θn · x ǫ , θ n � ǫ ψ n Then v ( t, x ) + r ǫ ( t, x ) with ǫ 2 � R N | r ǫ ( t, x ) | 2 dx = 0 , ǫ → 0 sup lim t ∈ [0 ,T ] (0 , T ); L 2 ( R N ) � � and v ∈ C is the unique solution of the homogenized Schr¨ odinger equation  i∂v in R N × (0 , T ) ∂t − div ( A ∗ n ∇ v ) + d ∗ n ( x ) v = 0   v ( t = 0 , x ) = v 0 ( x ) in R N ,   1 � T N d ( x, y ) | ψ n ( y ) | 2 dy . with A ∗ 8 π 2 ∇ θ ∇ θ λ n ( θ n ) and d ∗ n = n ( x ) =

Homogenization of Schr¨ odinger equation 66 G. Allaire ✞ ☎ 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.

Homogenization of Schr¨ odinger equation 67 G. Allaire | x |→ + ∞ d ( x, y ) = d ∞ ( y ) uniformly in T N . Define the Theorem 2. Assume lim group velocity or drift V = 1 2 π ∇ θ λ n ( θ n ) ∈ R N . � � � x t, x + V u ǫ ( t, x ) = e i λn ( θn ) t e 2 iπ θn · x ǫ , θ n � ǫ ψ n Then v ǫ t + r ǫ ( t, x ) with ǫ 2 � T � R N | r ǫ ( t, x ) | 2 dxdt = 0 , where v ∈ C (0 , T ); L 2 ( R N ) � � lim is the unique ǫ → 0 0 solution of the homogenized Schr¨ odinger equation  i∂v in R N × (0 , T ) ∂t − div ( A ∗ n ∇ v ) + d ∗ n v = 0   v ( t = 0 , x ) = v 0 ( x ) in R N ,   1 � T N d ∞ ( y ) | ψ n ( y ) | 2 dy . with A ∗ 8 π 2 ∇ θ ∇ θ λ n ( θ n ) and d ∗ n = n =

Homogenization of Schr¨ odinger equation 68 G. Allaire ✞ ☎ Generalization for a multiple eigenvalue ✝ ✆ What happens if λ n ( θ n ) is not simple ? In general, if λ n ( θ n ) is of multiplicity p > 1, we expect an homogenized system of p equations. Assume that λ n ( θ n ) = λ n +1 ( θ n ) is of multiplicity 2 and that, up to a convenient relabelling, λ n ( θ ) and λ n +1 ( θ ) are smooth branches of eigenvalues (and eigenfunctions). (Very strong assumption !) We can prove that the homogenized system is  i∂v 1 in R N × R + ∂t − div ( A ∗ n ∇ v 1 ) + d ∗ 11 ( x ) v 1 + d ∗ 12 ( x ) v 2 = 0       i∂v 2 in R N × R + A ∗ + d ∗ 21 ( x ) v 1 + d ∗ � � ∂t − div n +1 ∇ v 2 22 ( x ) v 2 = 0     ( v 1 , v 2 )( t = 0 , x ) = ( v 0 1 , v 0 in R N . 2 )( x )  

Homogenization of Schr¨ odinger equation 69 G. Allaire ✞ ☎ Formal proof of Theorem 1 ✝ ✆ We make an ansatz for the solution + ∞ t, x, x u ǫ ( t, x ) = e i λn ( θn ) t e 2 iπ θn · x � � � ǫ i u i ǫ 2 ǫ ǫ i =0 where the oscillating phase is deduced from WKB. We recall the operator � � A n ( θ ) ψ ≡ − ( div y + 2 iπθ ) A ( y )( ∇ y + 2 iπθ ) ψ + c ( y ) ψ − λ n ( θ ) ψ We plug the ansatz in the equation and get a cascade of equations.

Homogenization of Schr¨ odinger equation 70 G. Allaire ✞ ☎ Cascade of equations ✝ ✆ Order ǫ − 2 : A n ( θ n ) u 0 ( t, x, y ) = 0 from which we deduce, by simplicity of λ n ( θ n ), that u 0 ( t, x, y ) ≡ v ( t, x ) ψ n ( y, θ n ) . Order ǫ − 1 : A n ( θ n ) u 1 ( t, x, y ) = div x ( A ( y )( ∇ y + 2 iπθ n ) ψ n v ) + ( div y + 2 iπθ n ) ( A ( y ) ∇ x vψ n ) from which we deduce (up to the addition of a multiple of ψ n ) N 1 ∂v ( t, x ) ∂ψ n � ( y, θ n ) u 1 ( t, x, y ) ≡ 2 iπ ∂x k ∂θ k k =1

Homogenization of Schr¨ odinger equation 71 G. Allaire Order ǫ 0 : A n ( θ n ) u 2 ( t, x, y ) = f from which we deduce that N ∂ 2 v ( t, x ) ∂ 2 ψ n u 2 ( t, x, y ) ≡ − 1 � 4 π 2 ∂x k ∂x l ∂θ k ∂θ l k,l =1 and, more important, by the Fredholm alternative, � T N f ( t, x, y ) ψ n ( t, x, y ) dy = 0 . This last condition yields the homogenized equation.

Homogenization of Schr¨ odinger 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¨ odinger equation 73 G. Allaire ✞ ☎ STEP 1. SPECTRAL CELL PROBLEM ✝ ✆ We apply two-scale convergence to the bounded sequence v ǫ defined by v ǫ ( t, x ) = u ǫ ( t, x ) e − i λn ( θn ) t e − 2 iπ θn · x ǫ , ǫ 2 which admits a two-scale limit v ∗ ( t, x, y ) ∈ L 2 � (0 , T ) × R N ; H 1 ( T N ) � . We multiply the Schr¨ odinger equation by the complex conjugate of ϕ ǫ ≡ ǫ 2 φ ( t, x, x ǫ ) e i λn ( θn ) t e 2 iπ θn · x ǫ 2 ǫ

Homogenization of Schr¨ odinger equation 74 G. Allaire Variational formulation � T � i∂u ǫ � � ∂t ϕ ǫ + A ǫ ∇ u ǫ · ∇ ϕ ǫ + ( ǫ − 2 c ǫ + d ǫ ) u ǫ ϕ ǫ dt dx = 0 . R N 0 Replace u ǫ by v ǫ and ϕ ǫ by ǫ 2 φ ǫ � T � ǫ + ( c ǫ − λ n ( θ n )) v ǫ φ � ǫ � A ǫ ( ǫ ∇ + 2 iπθ n ) v ǫ · ( ǫ ∇ − 2 iπθ n ) φ dt dx = O ( ǫ 2 ) . R N 0 Passing to the two-scale limit yields the variational formulation of � A ( y )( ∇ y + 2 iπθ n ) v ∗ � + c ( y ) v ∗ = λ n ( θ n ) v ∗ − ( div y + 2 iπθ n ) in T N . Since λ n ( θ n ) is simple, there exists a scalar function v ( t, x ) ∈ L 2 ((0 , T ) × Ω) such that v ∗ ( t, x, y ) = v ( t, x ) ψ n ( y, θ n ) .

Homogenization of Schr¨ odinger equation 75 G. Allaire ✞ ☎ STEP 2. HOMOGENIZED PROBLEM ✝ ✆ We multiply the Schr¨ odinger equation by the complex conjugate of � � N ψ n ( x ∂φ ( t, x ) ζ k ( x Ψ ǫ = e i λn ( θn ) t e 2 iπ θn · x � ǫ , θ n ) φ ( t, x ) + ǫ ǫ ) ǫ 2 ǫ ∂x k k =1 where ζ k ( y ) is the solution of A ( θ n ) ζ k = e k A ( y )( ∇ y + 2 iπθ n ) ψ n + ( div y + 2 iπθ n ) ( A ( y ) e k ψ n ) in T N with the operator A ( θ ) defined on L 2 ( T N ) by � � A ( θ n ) ψ = − ( div y + 2 iπθ n ) A ( y )( ∇ y + 2 iπθ n ) ψ + c ( y ) ψ − λ n ( θ n ) ψ The existence of ζ k is guaranteed because ∂ψ n ∇ θ λ n ( θ n ) = 0 = 2 iπζ k since ∂θ k

Homogenization of Schr¨ odinger 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 ψ n . ☞ The ǫ − 1 terms cancel out because of the equation for ζ k . ☞ We pass to the two-scale limit in the ǫ 0 terms. ☞ We obtain a very weak form of the homogenized equation. 1 ☞ The homogenized tensor A ∗ 8 π 2 ∇ θ ∇ θ λ n ( θ n ) because of the n is equal to ∂ 2 ψ n compatibility condition (Fredholm alternative) for ∂θ k ∂θ l .

Homogenization of Schr¨ odinger equation 77 G. Allaire R N A ǫ ( ∇ + 2 iπ θ n ǫ )( φv ǫ ) · ( ∇ − 2 iπ θ n � � ǫ R N A ǫ ∇ u ǫ · ∇ Ψ ǫ dx = ǫ ) ψ n R N A ǫ ( ∇ + 2 iπ θ n v ǫ ) · ( ∇ − 2 iπ θ n ǫ )( ∂φ � ǫ + ǫ ǫ ) ζ k ∂x k v ǫ · ( ∇ − 2 iπ θ n � ∂φ ǫ R N A ǫ e k − ǫ ) ψ n ∂x k R N A ǫ ( ∇ + 2 iπ θ n ǫ )( ∂φ � ǫ + v ǫ ) · e k ψ n ∂x k � R N A ǫ v ǫ ∇ ∂φ � R N A ǫ v ǫ ∇ ∂φ ǫ ǫ · ( ǫ ∇ − 2 iπθ n ) ζ − · e k ψ n − k ∂x k ∂x k k ( ǫ ∇ + 2 iπθ n ) v ǫ · ∇ ∂φ � ǫ R N A ǫ ζ + ∂x k

Homogenization of Schr¨ odinger equation 78 G. Allaire � x � Variational formulation for ψ n ǫ R N A ǫ ( ∇ + 2 iπ θ n n · ( ∇ − 2 iπ θ n � ǫ )Φ + 1 � R N ( c ǫ − λ n ( θ n )) ψ ǫ ǫ ) ψ ǫ n Φ = 0 ǫ 2 Take Φ = φv ǫ ⇒ cancellation of the ǫ − 2 terms

Homogenization of Schr¨ odinger equation 79 G. Allaire � x � Variational formulation for ζ k ǫ R N A ǫ ( ∇ + 2 iπ θ n k · ( ∇ − 2 iπ θ n � ǫ )Φ + 1 � R N ( c ǫ − λ n ( θ n )) ζ ǫ ǫ ) ζ ǫ k Φ = ǫ 2 R N A ǫ ( ∇ + 2 iπ θ n n · ( ∇ − 2 iπ θ n � � ǫ − 1 n · e k Φ − ǫ − 1 ǫ ) ψ ǫ R N A ǫ e k ψ ǫ ǫ )Φ . ∂x k v ǫ ⇒ cancellation of the ǫ − 1 terms ∂φ Take Φ =

Homogenization of Schr¨ odinger equation 80 G. Allaire ✞ ☎ STEP 3. STRONG CONVERGENCE ✝ ✆ This is a consequence of the energy conservation � v ǫ ( t ) � L 2 ( R N ) = � u ǫ ( t ) � L 2 ( R N ) = � u 0 ǫ � L 2 ( R N ) → � ψ n v 0 � L 2 ( R N × T N ) = � v 0 � L 2 ( R N ) and of the notion of strong two-scale convergence. Recall that � x e 2 iπ θn · x ǫ , θ n � u 0 ǫ v 0 ( x ) ǫ ( x ) = ψ n

Homogenization of Schr¨ odinger equation 81 G. Allaire ✞ ☎ Proof of Theorem 2 ✝ ✆ New ingredient: two-scale convergence with drift. Proposition (Marusic-Paloka, Piatnitski). Let V ∈ R N be a given drift velocity. Let u ǫ be a bounded sequence in L 2 ((0 , T ) × R N ). Up to a subsequence, there exist a limit u 0 ( t, x, y ) ∈ L 2 ((0 , T ) × R N × T N ) such that u ǫ two-scale converges with drift weakly to u 0 in the sense that � T � � � t, x + V ǫ t, x lim R N u ǫ ( t, x ) φ dt dx = ǫ ǫ → 0 0 � T � � T N u 0 ( t, x, y ) φ ( t, x, y ) dt dx dy 0 R N for all functions φ ( t, x, y ) ∈ L 2 � (0 , T ) × R N ; C ( T N ) � .

Homogenization of Schr¨ odinger equation 82 G. Allaire Lemma (Marusic-Paloka, Piatnitski). Let φ ( t, x, y ) ∈ L 2 � (0 , T ) × R N ; C ( T N ) � . Then � T � T 2 � �� � t, x + V ǫ t, x � � � � � T N | φ ( t, x, y ) | 2 dt dx dy. lim � φ dt dx = � � ǫ ǫ → 0 � R N R N 0 0 Proof. Change of variables x ′ = x + V ǫ t � T � T 2 2 t, x ′ , x ′ � �� � �� � � � t, x + V ǫ t, x � ǫ − V dt dx ′ � � � � � φ dt dx = � φ ǫ 2 t � � � � ǫ � � 0 R N 0 R N We mesh R N with cubes of size ǫ , R N = ∪ i ∈ Z Y ǫ i with Y ǫ i = x ǫ i + (0 , ǫ ) N 2 2 t, x ′ , x ′ i , x ′ � �� � �� � ǫ − V � ǫ − V � � � � � � t, x ǫ � � φ ǫ 2 t dx = � φ ǫ 2 t dx + O ( ǫ ) � � � � � Y ǫ � R N i i � � � i , y ) | 2 dy + O ( ǫ ) = | φ ( x, y ) | 2 dx dy + O ( ǫ ) � ǫ N T N | φ ( x ǫ = Ω Y i ∈ Z

Homogenization of Schr¨ odinger equation 83 G. Allaire -VII- LOCALIZATION We come back to the semi-classical scaling and to locally periodic coefficients i ∂u ǫ x, x x, x � � � � � � + ǫ − 2 c ∂t − div A ∇ u ǫ u ǫ = 0 ǫ ǫ ǫ We choose well-prepared initial data, Bloch wave packets.

Homogenization of Schr¨ odinger equation 84 G. Allaire ✞ ☎ ASSUMPTIONS ✝ ✆ We choose a point ( x n , θ n ) ∈ R N × T N in the phase space such that λ n ( x n , θ n ) is a simple eigenvalue and ∇ θ λ n ( x n , θ n ) = ∇ x λ n ( x n , θ n ) = 0 We consider well-prepared initial data ǫ v 0 � x − x n x n , x e 2 iπ θn · x � ǫ , θ n � � u 0 ǫ ( x ) = ψ n √ ǫ with v 0 ∈ H 1 ( R N ) (degenerate case for WKB !) Notations. New scale z = x − x n √ ǫ

Homogenization of Schr¨ odinger equation 85 G. Allaire ✞ ☎ Main result ✝ ✆ Theorem (A.-Palombaro). t, x − x n t, x − x n x n , x u ǫ ( t, x ) = e i λn ( xn,θn ) t e 2 iπ θn · x � ǫ , θ n � � � � � ǫ ψ n √ ǫ √ ǫ v + r ǫ ǫ � R N | r ǫ ( t, z ) | 2 dz = 0 , uniformly in time, and v ( t, z ) is the unique with lim ǫ → 0 solution of the homogenized Schr¨ odinger equation  i∂v in R N × R + ∂t − div z ( A ∗ ∇ z v ) + div z ( vB ∗ z ) + c ∗ v + vD ∗ z · z = 0   v (0 , z ) = v 0 ( z ) in R N   where c ∗ is a constant coefficient and A ∗ , B ∗ , D ∗ are constant matrices defined by 1 2 iπ ∇ θ ∇ x λ n ( x n , θ n ) , D ∗ = 1 1 A ∗ = 8 π 2 ∇ θ ∇ θ λ n ( x n , θ n ) , B ∗ = 2 ∇ x ∇ x λ n ( x n , θ n ) .

Homogenization of Schr¨ odinger equation 86 G. Allaire ✞ ☎ Homogenized problem ✝ ✆  i∂v in R N × R + ∂t − div z ( A ∗ ∇ z v ) + div z ( vB ∗ z ) + c ∗ v + vD ∗ z · z = 0   v (0 , z ) = v 0 ( z ) in R N   Lemma. The operator A ∗ : L 2 ( R N ) → L 2 ( R N ) defined by A ∗ ϕ = − div ( A ∗ ∇ ϕ ) + div( ϕB ∗ z ) + c ∗ ϕ + ϕD ∗ z · z is self-adjoint. Corollary. The homogenized Schr¨ odinger equation is well-posed in C ( R + ; L 2 ( R N )) and satisfies the energy conservation ∀ t ∈ R + . || v ( t, · ) || L 2 ( R N ) = || v 0 || L 2 ( R N )

Homogenization of Schr¨ odinger equation 87 G. Allaire Proof of the corollary. By self-adjointness � R N A ∗ vv dz = �A ∗ v, v � = � v, A ∗ v � = �A ∗ v, v � ∈ R . � ∂v � R N A ∗ vv dz = 0 i ∂t v dz + R N Take the imaginary part to get 1 d � R N | v | 2 dz = 0 2 dt

Homogenization of Schr¨ odinger equation 88 G. Allaire Proof of the lemma. By integration by parts and since B ∗ ∈ i R , B ∗ = − B ∗ � div( vB ∗ z ) w − 1 � div( wB ∗ z ) v − 1 � � � � 2tr B ∗ vw 2tr B ∗ wv dz = dz R N R N Thus � � � � A ∗ ∇ v · ∇ w + D ∗ z · z vw + div( vB ∗ z ) w − 1 R N A ∗ vw dz = 2tr B ∗ vw dz R N � � 1 2tr B ∗ + c ∗ � + vw dz R N which is symmetric (obvious for the blue terms) since 1 2 i tr B ∗ + Im c ∗ = 0 ∂ 2 ψ n as a consequence of the Fredholm alternative for ∂z k ∂θ l .

Homogenization of Schr¨ odinger equation 89 G. Allaire ✞ ☎ Compactness and localization ✝ ✆    ∇ x ∇ x λ n ∇ θ ∇ x λ n  ( x n , θ n ) . ∇∇ λ n = ∇ θ ∇ x λ n ∇ θ ∇ θ λ n Lemma. If the matrix ∇∇ λ n is positive definite, then there exists an orthonormal basis { ϕ n } n ≥ 1 in L 2 ( R N ) of eigenfunctions of the homogenized problem. Moreover for each n there exists a real constant γ n > 0 such that e γ n | z | ϕ n , e γ n | z | ∇ ϕ n ∈ L 2 ( R N ) . This is localization ! (cf. Anderson in a stochastic framework)

Homogenization of Schr¨ odinger equation 90 G. Allaire Proof. For simplicity assume that Re( c ∗ ) = 0. � � A ∗ ∇ v · ∇ v + D ∗ z · z | v | 2 − iB ∗ Im( vz · ∇ v ) �A ∗ v, v � = R N A ∗ vv dz = � � dz R N Recall that 1 1 D ∗ = 1 A ∗ = B ∗ = 8 π 2 ∇ θ ∇ θ λ n , 2 iπ ∇ θ ∇ x λ n , 2 ∇ x ∇ x λ n . � t � Define Φ( z ) = 2 iπv ( z ) z , ∇ v ( z ) . Then � 1 � � �A ∗ v, v � = ||∇ v || 2 L 2 ( R N ) + || z v || 2 8 π 2 ∇∇ λ n Φ · Φ dz ≥ C L 2 ( R N ) R N The space { v ∈ H 1 ( R N ) s.t. ( zv ) ∈ L 2 ( R N ) } is compactly embedd in L 2 ( R N ). 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¨ odinger equation 91 G. Allaire ✞ ☎ Proof of the homogenization theorem ✝ ✆ Variational proof with an oscillating test function. 1. Change of variables z = x − x n √ 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¨ odinger equation 92 G. Allaire Change of variables: z = x − x n √ ǫ , Change of unknowns: w ǫ ( t, z ) := e 2 iπ θn · z √ ǫ v ǫ ( t, z ) = e − i λn ( θn ) t u ǫ ( t, x ) . ǫ New equation: � √ ǫz, z/ √ ǫ ǫ [ c ( √ ǫz, z/ √ ǫ ) − λ n ( θ n )] w ǫ = 0 i∂w ǫ ∇ w ǫ ] + 1  � ∂t − div[ A   ǫ ( √ ǫz ) w ǫ (0 , z ) = u 0   The oscillations are at scale √ ǫ : this is the parabolic scaling !

Homogenization of Schr¨ odinger equation 93 G. Allaire Taylor expansion around zero for the macroscopic variable � √ ǫz, z + √ ǫz ·∇ x c z · z + O ( √ ǫ | z | ) 3 . � � 0 , z � � 0 , z � +1 � 0 , z � c √ ǫ = c √ ǫ √ ǫ 2 ǫ ∇ x ∇ x c √ ǫ We obtain c ( √ ǫz, z/ √ ǫ ) − λ n ( θ n ) 1 � � = ǫ c (0 , z/ √ ǫ ) − λ n ( θ n ) √ ǫz · ∇ x c (0 , z/ √ ǫ ) + 1 z · z + O ( √ ǫ ) 1 + 1 � 0 , z � � � 2 ∇ x ∇ x c √ ǫ ǫ as usual + new singular term to cancel + bounded harmonic potential New test function: � 1 N n φ ( t, z ) + √ ǫ ∂ψ ǫ ∂ψ ǫ ∂φ Ψ ǫ ( t, z ) = e 2 iπθ n · z � �� � ψ ǫ n n √ ǫ ( t, z ) + z k φ ( t, z ) 2 iπ ∂θ k ∂z k ∂x k k =1

Homogenization of Schr¨ odinger equation 94 G. Allaire -VIII- TIME OSCILLATING POTENTIAL i∂u ǫ � x  � � � in R N × (0 , T ) ǫ − 2 c ∂t − ∆ u ǫ + + d ǫ ( t, x ) u ǫ = 0   ǫ u ǫ ( t = 0 , x ) = u 0  in R N , ǫ ( x )  Goal: transfer some energy from the initial data to some target final data � x � x e 2 iπ θn · x e 2 iπ θm · x ǫ , θ n � ǫ , θ m � v T ( x ) u 0 ǫ v 0 ( x ) u T ǫ ( x ) = ψ n ǫ ( x ) = ψ m ǫ For this task, we choose a time oscillating potential t, x, x e i ( λm ( θm ) − λn ( θn )) t e 2 iπ ( θm − θn ) · x � � � � d ǫ ( t, x ) = ℜ d , ǫ 2 ǫ ǫ where d ( t, x, y ) is a real potential defined on [0 , T ] × R N × T N .

Homogenization of Schr¨ odinger equation 95 G. Allaire ✞ ☎ Assumptions ✝ ✆ We assume  λ p ( θ p ) is a simple eigenvalue, ( i )  for p = n, m θ p is a critical point of λ p ( θ ) i.e., ∇ θ λ p ( θ p ) = 0 ( ii )  and a non-resonant assumption λ p (2 θ n − θ m ) � = 2 λ n ( θ n ) − λ m ( θ m ) . ( iii ) for any p ≥ 1 ,

Homogenization of Schr¨ odinger equation 96 G. Allaire Theorem. For an initial data u 0 ǫ ∈ H 1 ( R N ) given by � x e 2 iπ θn · x ǫ , θ n � with v 0 ∈ H 1 ( R N ) , u 0 ǫ v 0 ( x ) ǫ ( x ) = ψ n the solution of the Schr¨ odinger equation can be written as � x u ǫ ( t, x ) = e i λn ( θn ) t e 2 iπ θn · x ǫ , θ n � ǫ ψ n v n ( t, x ) ǫ 2 � x + e i λm ( θm ) t e 2 iπ θm · x ǫ , θ m � ψ m v m ( t, x ) + r ǫ ( t, x ) , ǫ 2 ǫ with � T � R N | r ǫ ( t, x ) | 2 dx = 0 , lim ǫ → 0 0

Homogenization of Schr¨ odinger equation 97 G. Allaire � 2 is the unique solution of the homogenized [0 , T ]; L 2 ( R N ) � and ( v n , v m ) ∈ C Schr¨ odinger system i∂v n  in R N × (0 , T ) ∂t − div ( A ∗ n ∇ v n ) + d ∗ nm ( t, x ) v m = 0       i∂v m   in R N × (0 , T ) ∂t − div ( A ∗ m ∇ v m ) + d ∗  mn ( t, x ) v n = 0   v n ( t = 0 , x ) = v 0 ( x ) in R N        in R N ,  v m ( t = 0 , x ) = 0  1 with A ∗ 8 π 2 ∇ θ ∇ θ λ p ( θ p ), for p = n, m , and ”Fermi golden rule” p = mn ( t, x ) = 1 � ∗ d ∗ T N d ( t, x, y ) ψ n ( y, θ n ) ψ m ( y, θ m ) dy. nm ( t, x ) = d 2

Homogenization of Schr¨ odinger equation 98 G. Allaire ✞ ☎ Comments ✝ ✆ ➫ The exterior potential d ǫ 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¨ odinger equation 99 G. Allaire ✞ ☎ Sketch of the proof ✝ ✆ Define two sequences ǫ ( t, x ) = u ǫ ( t, x ) e − i λn ( θn ) t e − 2 iπ θn · x ǫ , v n ǫ 2 ǫ ( t, x ) = u ǫ ( t, x ) e − i λm ( θm ) t e − 2 iπ θm · x v m . ǫ 2 ǫ They both satisfy the a priori estimate, for p = n, m , � v p ǫ � L ∞ ((0 ,T ); L 2 ( R N )) + ǫ �∇ v p ǫ � L 2 ((0 ,T ) × R N ) ≤ C Up to a subsequence they two-scale converge to limits w p ( t, x, y ) ∈ L 2 � (0 , T ) × R N ; H 1 ( T N ) � .

Homogenization of Schr¨ odinger equation 100 G. Allaire Recall that i∂u ǫ � x  � � � in R N × (0 , T ) ǫ − 2 c ∂t − ∆ u ǫ + + d ǫ ( t, x ) u ǫ = 0   ǫ u ǫ ( t = 0 , x ) = u 0  in R N , ǫ ( x )  with the initial data � x e 2 iπ θn · x ǫ , θ n � u 0 ǫ v 0 ( x ) ǫ ( x ) = ψ n and a time oscillating potential t, x, x e i ( λm ( θm ) − λn ( θn )) t e 2 iπ ( θm − θn ) · x � � � � d ǫ ( t, x ) = ℜ d , ǫ 2 ǫ ǫ where d ( t, x, y ) is a real potential defined on [0 , T ] × R N × T N .