[PDF] - TWO-SCALE ASYMPTOTIC EXPANSION : BLOCH-FLOQUET THEORY IN PERIODIC PDF Document

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TWO-SCALE ASYMPTOTIC EXPANSION : BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES

Our aim is to study an application of Bloch-Floquet Theory in the multiscale analysis of PDE on periodic structures. We will motivate this by an example where we consider a model for an electron moving in a periodic potential created by atoms in a crystal lattice. If we are interested in high frequency wave propagation where the typical wave- length is comparable to the period of the medium, both of which are assumed to be small relative to the length-scale of the physical domain. Then another length scale becomes relevant to the problem and we introduce multiple scales of space and

ing the variations within the microscopic period cell and the slow scale measuring variations within the macroscopic region of interest. Starting from a microscopic description of a problem, we convert to a semiclassical scaling and seek an approx- imate macroscopic description via two scale expansion. Our aim will be to show a stability result which verifies indeed that our approximation has an agreeable degree of accuracy. We consider our physical domain to be a lattice where the structure of our model could resemble the arrangement of atoms in a crystal. We construct our lattice generated by the basis {a1, ..., ad : aj ∈ Rd}, Γ =

njaj, n ∈ Zd , and the unit or period cell of Γ is given by M =

µjaj, µj ∈

2, 1 2

We can think of the unit cell as representing the smallest cell defining the charac- teristic symmetry and where translation reproduces the lattice in that each x ∈ Rd has a unique decomposition x = [x] + γ with [x] ∈ M and γ ∈ Γ. For each lattice Γ we can define its dual or reciprocal lattice Γ⋆ generated by the dual basis {b1, ..., bd : bj ∈ Rd}, where ai · bj = 2πδij. The first Brillouin zone M ⋆

The particular choice of a lattice matters in certain contexts, but for simplicity lets choose Γ = 2πZd, M = [0, 2π)d Γ⋆ = Zd, M ∗ = (−1/2, 1/2]d.

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Let A(x) = (apq(x)) be a symmetric, smooth and uniformly positive definite matrix of 2π-periodic functions, and let our periodic potential W(·) ≥ 0 be a real- valued 2π-periodic, smooth function. We denote the second order elliptic operator by ∇ ·

∂ ∂xp

∂ ∂xq ·

Now let 0 < ε ≪ 1 be the parameter that describes the ratio of the micro- scopic/macroscopic length scales. A rescaling of our PDE into a semi-classical scaling is made by introducing new variables t′ = εt and x′ = εx, rewriting the equation in the new variables and dropping the primes. So lets consider our PDE in a semi-classical scaling regime −iε∂Ψε ∂t − ε2∇ ·

ε )∇Ψε + W x ε

(⋆) Ψε(0, x) = fε(x). We remark that this scaling describes an electron on macroscopic scales where the potential is highly oscillatory with period 2πε. Now lets make a guess for an approximate solution where later we will perform a formal two-scale asymptotic expansion of Ψε(t, x). For computational simplicity later, we now introduce a y-periodic function f(x, y) for y ∈ Rd, and associate to f(x, y) the function f(x, x/ε). So if we consider x and y as independent variables and replace x/ε by y, then by the chain rule the operator ∇ becomes ∇x + 1

For λ ∈ R, lets choose the ansatz Ψε(t, x) = ψ(t, x, y)e−iλt/ε, where we impose the oscillatory term to ensure that there will be no initial layer, a time under which the solution adapts itself to match the initial profile. We will see that this is not only the right choice, but is necessary in general to satisfy initial conditions past O(1), and in looking for expansions valid pointwise in t. So if we divide (⋆) by ε2 and write our operator as Aε := −∇ ·

ε ∂ ∂t + 1 ε2 W

Then plugging in our ansatz AεΨε = 1 ε2 (B − λ)ψ

+1 εB1ψ + B2ψ = 0, where B = −∇y ·

Hence the ansatz requires us to first resolve the eigenvalue problem Bψ = λψ, so we want to examine the spectral resolution of the closure of the following operator in L2(Rd).

3.1. Bloch-Floquet Eigenvalue Problem. The Bloch-Floquet eigenvalue prob- lem, also known as the shifted cell problem, arises from studying the spectrum of the following operator. The spectral decomposition in one dimensional periodic media was first studied by Floquet(1883) and later in a crystal lattice by Bloch(1928).

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Let H1(M) be the Hilbert space of functions periodic with respect to Γ with square integrable first derivatives. For each k ∈ M ⋆, consider the domain D

H1(M) and define B(k) = −

and lets consider the periodic eigenvalue problem B(k)φ(y) = ω2φ(y), y ∈ Rd φ(y + γ) = φ(y), γ ∈ Γ. From [3] it is enough to show that the operator is semi-bounded in the sense of quadratic forms to yield self-adjointness and the compact resolvent necessary for a discrete spectrum. This would follow since we have assumed W ≥ 0, but showing this relies on an application of Friedrich’s extension theorem, the details of which we leave to the reader. For a different treatment of the eigenvalue problem see [6]. As a result, for each k ∈ M ⋆ the eigenvalue problem has a countable sequence

ω2

including multiplicity, and corresponding Γ-periodic eigenfunctions known as Bloch waves φ1(y; k), φ2(y; k), φ3(y; k)... which are smooth in y and form an orthonormal basis of L2

The spectrum may be viewed as a union of intervals or Band spaces σ(B) =

σ(B(k)) =

Em, which differs from the free case, where the spectrum is [0, ∞) as there may be gaps. The interval Em = {ω2

that the functions ω2

we have the following condition ω2

We call Em an isolated Bloch band if for all k ∈ M ⋆ the above condition holds. Lastly it is known that |I| = |{k ∈ M ⋆ : ω2

and it is in this set of measure zero that we encounter what are called band cross-

for simplicity of the model relies on differentiability in k of the eigenvalues and eigenfunctions. 3.2. Bloch Decomposition. For our purposes we will use that the eigenfunctions are complete in L2

in L2(Rd). This decomposition relies on the Bloch-Floquet (or sometimes called Zak) trans- form which is a generalization of the Fourier transform that leaves periodic functions

the Bloch transform given by gb(y; k) =

g(y + γ)e−ik·(y+γ).

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It is not difficult to see that gb(y + γ; k) = gb(y; k) ∀γ ∈ Γ gb(y; k + γ⋆) = e−iγ⋆·ygb(y; k) ∀γ ∈ Γ⋆. From the Bloch transform we see that g(y) =

To verify this we show this for g ∈ S (Rd)

g(y + γ)e−ik·(y+γ)eik·y dk = g(y) +

g(y + γ)

2 , 1 2 ]

e−iknγn dkn = g(y). Moreover the Bloch transform may be seen as an isometry from L2(Rd) to L2(M × M ⋆), since for fixed k ∈ M ⋆ the Bloch Transform is extended to a M- periodic function on Rd. For an interesting and more intimate discussion on the Bloch transform in Lp spaces see [5]. Theorem 3.1. (Bloch Expansion) Let g ∈ L2(Rd). Then its Bloch transform is given by gb(y; k) =

gm

where {φm} are the Bloch eigenfunctions of the shifted operator B(k), and for each k ∈ M ⋆ the m-th Bloch coefficient gm

gm

g(y)e−ik·yφm(y; k) dy, (3.1) and we have the expansion g(y) = lim

gm

(3.2) Moreover, Parseval’s identity holds

|fb(y; k)|2 dy =

|f m

Bloch transform gb(y; k) ∈ L2

with coefficients gm

gb(y; k) =

gm

Now projecting onto φm(y; k) over M and using the definition of Bloch transform we yield the m-th coefficient gm

gb(y; k)φm(y; k) dy =

g(y + γ)e−ik·(y+γ)φm(y; k)dy =

g(y)e−ik·(y)φm(y − γ; k) dy =

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where we used the periodicity of the Bloch waves and a substitution. The last line follows by (⋆) above and the definition of the Bloch Transform. Lastly, Parseval’s Identity follows from both 3.1 and 3.2. We note that as S (Rd) is dense in L2(Rd), the Bloch transform extends to L2(Rd).

Beik·y = eik·yB(k), and so Beik·yφm(y; k) = eik·yB(k)φm(y; k) = eik·yω2

So for g ∈ S (Rd) the above remark gives the spectral resolution of B by Bf(y) =

f m

We can utilize the Bloch decomposition to filter out the nature of the expansion we may hope to make in the next section. We recall for the unscaled Schr¨

equation we are dealing with ∂tu(t, x)+iBu(t, x) = 0, so if we decompose u ∈ L2(Rd) we yield from the resolution

which implies ∂tum

Hence we have after returning to semi-classical scaling and localizing about some k0 ∈ M ⋆ uε(t, x) =

ei(k·x−ω2

m(k)t)/εum

=

ei(k0·x−ω2

m(k0)t)/ε

m(k)−ω2 m(k0))t]/εum

=

ei(k0·x−ω2

m(k0)t)/εϕm(t, x, x/ε),

where the last line is done more explicitly in chapter 4.3 of [1]. 3.3. Group velocity and Effective Mass Tensor. We want to introduce two basic quantities that one can use to study complex phenomena with transport in

from an intersection then both ω2

differentiating the eigenvalue problem once yields

where ∇kB(k) = −iA(y)

apq(y) ∂ ∂yq + ikq

∂ ∂yq + ikq

cN(k0) = ∇kω2

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Now if we differentiate the eigenvalue problem again we yield the effective mass tensor M = (Mpq): Mpq(k0) = 1 2 ∂ω2

∂xp∂xq

2

∂kp ∂φN ∂kq , φN

2

∂kq ∂φN ∂kp , φN

∂ω2

∂xp∂xq −1 = (apq)−1, and say it is the effective mass tensor of the N-th band, which reflects the band curvature or physically how the electron moves under interaction of a periodic potential.

We will examine a simplified model that can be generalized rather easily as our aim will be to emphasize the basic justification in this type of analysis. For the more general setting of the general geometric optics expansion see chapter 4.5 in [1]. Lets motivate the formal two-scale expansion by stating the main theorem we wish to justify and prove: Theorem 4.1. Lets define our approximate solution zε(t, x) = ei(k0·x−ω2

N(k0)t)/ε

ψ0(t, x, y) + εψ1(t, x, y)

where ψ0, ψ1 are to be determined below. Then if Ψε(t, x) is the exact solution solving (⋆), we have that sup

for some 0 < T < ∞ and ε sufficiently small, where C is a constant depending on the initial data f(x), but not on ε. 4.1. Formal Expansion. Lets assume that for some k0 ∈ M ⋆ we can find an isolated Bloch band so that ω2

initial data lives in this Bloch band EN and is a spatially modulated plane wave with rapidly varying phase. Consider the initial value problem AεΨε(t, x) = −∇ ·

ε )∇Ψε − i ε ∂Ψε ∂t + 1 ε2 WΓ x ε

(⋆) Ψε(0, x) = f(x)eik0·x/εφN(x/ε; k) + O(ε), where f ∈ C∞

We now choose a similar ansatz as before, but now expand ψ in powers of ε with λ = ω2

Ψε(t, x) ∼ ei(k0·x−ω2

N(k0)t)/εψ(t, x, y), ψ(t, x, y) =

εjψj(t, x, y). After plugging in the ansatz we have AεΨε = 1 ε2 B0ψ0 + 1 ε

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which induces a hierarchy of equations: B0ψ0 = 0 (4.1) B0ψ1 + B1ψ0 = 0 (4.2) B0ψ2 + B1ψ1 + B2ψ0 = 0 (4.3) B0ψ3 + B1ψ2 + B2ψ1 = 0 (4.4) where B0 = −

B1 = −i ∂ ∂t + ∇kB(k0) · ∇x

Hence we see from the first equation that by separation of variables ψ0(t, x, y) = u(t, x)φN(y; k0) u(0, x) = f(x). To determine the slowly varying amplitude u = u(t, x) we note that our second equation is an inhomogeneous equation for ψ1. And since B0 has a one-dimensional kernel then by Friedholm’s alternative B1ψ0 ∈ ker

⊥, and so it follows A0u =

= ∂tu(t, x) +

= ∂tu(t, x) + cN(k0) · ∇xu(t, x) = 0. Hence the slowly modulated amplitude u satisfies a homogenized transport equa- tion, which implies that u(t, x) = f(x − cNt), where cN = cN(k0) ∈ Rd is the group velocity vector. Thus we have to leading order Ψε(t, x) ∼ ei

0(k0)t

Now that ψ0 satisfies the solvability condition then we may decompose ψ1 = ˜ ψ1 + ψ⊥

where ˜ ψ1 = v(t, x)φN(y; k0) ∈ ker(B0) ψ⊥

⊥, where we note by Fredholm’s alternative that B−1 = (B − ω2

Suppose we wish to write ψ⊥

where χ : M → Rd is a periodic complex analytic function in k and smooth in y in each component. Then by plugging ψ1 and our now determined ψ0 into 4.2 we see that χ must satisfy a familiar looking cell problem

χ = i

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Recalling what was done to find the wave velocity we see that B0χ = i

1 i ∇kφN, and so its easy to see that if

⊥d then

χ(y; k0) = 1

i ∇kφN(y; k0) + CφN(y; k0), where

i

Thus we may write ψ1(t, x, y) = ∇u(t, x) · χ(y; k0) + v(t, x)φN(y; k0), where it is our aim to determine the slowly varying amplitude v = v(t, x), where we impose that v(0, x) = 0, so that ψ1(0, x, y) = ∇f(x) · χ(y; k0). Now we consider equation 4.3 in the hierarchy and force the solvability condition so that after a bit of rearranging A1v =

Mpq ∂2u ∂xp∂xq = 0 Thus we yield the following inhomogeneous transport equation    ∂tv(t, x) + cN · ∇xv(t, x) = i

Mpq

v(0, x) = 0 which by Duhamel’s principle has the following form v(t, x) = itF(x − cNt). Therefore, we have our solution to second order where ψ0(t, x, y) = f

ψ1(t, x, y) = ∇xf(x − cNt) · χ(y; k0) + itF(x − cNt)φN(y; k0). 4.2. Accuracy of Approximation. Now we are in a position to justify our ex-

Lemma 4.2. Consider the following I.V.P.    SεΨε = −i∂Ψε ∂t + 1 εHε

(⋆) Ψε(0, x) = f(x)eik0·x/εφN(x/ε; k) + O(ε) where Hε

x

Then !∃Ψε(t, x) ∈ C

is essentially self-adjoint, and so by Stone’s theorem we have s strongly continuous unitary group U ε(t) = e−i ¯

where ¯ Hε is the unique self-adjoint extension of Hε. As C∞

we can extend the unitary group to L2(Rd).

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Hence we see by unitarity that for all t ∈ R,

proof to follow: Lemma 4.3. Let T > 0, and recall the functions ψ0, ψ1 of the leading order and second order terms in our approximation. Then sup

sup

sup

where C1, C2, C3 are constants that depend on f.

that χ∞, φN∞ < ∞. Firstly, we compute (I)

where ∂tψ1(t)2 ≤

∂xp

≤ K1|cN|

and (∇kB(k0) · ∇x)ψ1(t)2 ≤

∂xq∂xp

≤ K4

Hence it is now clear that

Secondly, we compute (II)

∂2f ∂xp∂xq

Lastly, we see that (III)

∂xp∂xq∂xs

d

∂xp∂xq

in general a prototype of what one can hope for in general.

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Theorem 4.4. Lets define zε(t, x) = ei(k0·x−ω2

N(k0)t)/ε

ψ0(t, x, y) + εψ1(t, x, y)

where ψ0, ψ1 satisfy the above solvability conditions. Then if Ψε(t, x) is the exact solution solving (⋆), we have that sup

for some 0 < T < ∞ and ε sufficiently small, where C is a constant depending on the initial data f(x), but not on ε.

ϕ(t, x) :=

(t, x). By how we constructed zε through the solvability conditions we note that Sεzε = εAεzε = 1 ε B0ψ0

+

N(k0)t

and so Sεϕ(t, x) = ε

−Aεzε = rε

ϕ0 = ϕ(0, x) = rε

where rε

N(k0)t)/ε

ε

∇xf(x) · χ(y; k0)

Hence we have reduced our approximation to the following PDE with inhomo- geneous residual terms

∂t + 1 εHε

ϕ(0, x) = rε

Now by Duhamel’s Principle using the linear propagator in Lemma 4.1 we have ϕ(t, x) = U ε(t)ϕ0 − i t U ε(s − t)rε

which implies by unitarity of U ε(t) and using the above lemma to bound the re- mainder terms

t

= ε

t

≤ ε

sup

˜ C + ε ˜ ˜ C

where C is a constant that depends on f. Hence we have proved the desired accuracy after taking supremum over t.

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Remark 4.5. We can actually prove this estimate up to time T/ε but this would require us to go one step further in the expansion and determine ψ2. Of course the price to pay would be a more rigorous computation in the estimation. We see that we cannot go up to time T/ε in our estimate for we would pick up a factor of 1

the last line of the above proof and would only satisfy the estimate to O(1), which would be insufficient. Remark 4.6. As a closing remark, in general if we had a nonlinear term in our PDE it would show up in the residual term rε

down to applying Gronwall’s Lemma to gain the final stability result. References