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

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1 TWO-SCALE ASYMPTOTIC EXPANSION : BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES JACK ARBUNICH 1. Setting of 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 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 the typical wavelength 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 time. There are two natural spatial length scales we consider, the fast scale measuring 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 approximate 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 the structure of our model could resemble the arrangement of atoms in a crystal. We construct our lattice generated by the basis {a 1,..., a d : a j R d }, Γ = { γ R d : γ = and the unit or period cell of Γ is given by M = { x R d : x = n j a j, n Z d}, j=1 µ j a j, µ j ( 1, 1 ] }, j=1 We can think of the unit cell as representing the smallest cell defining the characteristic symmetry and translation reproduces the lattice in that each x R d 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 {b 1,..., b d : b j R d }, a i b j = πδ ij. The first Brillouin zone of Γ is the unit cell of the dual lattice. The particular choice of a lattice matters in certain contexts, but for simplicity lets choose Γ = πz d, M = [0, π d Γ = Z d, M = ( 1/, 1/] d. 1

2 J. ARBUNICH. Motivation : Linear Periodic Schrödinger Equation Let A(x = (a pq (x be a symmetric, smooth and uniformly positive definite matrix of π-periodic functions, and let our periodic potential W ( 0 be a realvalued π-periodic, smooth function. We denote the second order elliptic operator by (A(x n ( ( = a pq x. x p x q p,q=1 Now let 0 < ε 1 be the parameter that describes the ratio of the microscopic/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 ε (A( x t ε Ψε + W ( x Ψ ε = 0, t > 0, ( ε Ψ ε (0, x = f ε (x. We remark that this scaling describes an electron on macroscopic scales the potential is highly oscillatory with period πε. Now lets make a guess for an approximate solution later we will perform a formal two-scale asymptotic expansion of Ψ ε (t, x. For computational simplicity later, we now introduce a y-periodic function f(x, y for y R d, 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 ε y. For λ R, lets choose the ansatz Ψ ε (t, x = ψ(t, x, ye iλt/ε, 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 ε and write our operator as A ε := (A(y i ε t + 1 ε W ( y. Then plugging in our ansatz A ε Ψ ε = 1 (B λψ + 1 ε }{{} ε B 1ψ + B ψ = 0, =0 B = y (A(y y + W ( 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 L (R d. 3. The shifted cell problems for a second order elliptic operator 3.1. Bloch-Floquet Eigenvalue Problem. The Bloch-Floquet eigenvalue problem, 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(198.

3 TWO-SCALE ASYMPTOTIC EXPANSION :BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES3 Let H 1 (M be the Hilbert space of functions periodic with respect to Γ with square integrable first derivatives. For each k, consider the domain D ( B(k = H 1 (M and define ( [ ( ] B(k = y + ik A(y y + ik + W (y, and lets consider the periodic eigenvalue problem B(kφ(y = ω φ(y, y R d 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 the eigenvalue problem has a countable sequence of real-valued eigenvalues which accumulate at infinity ω 1(k ω (k ω 3(k..., including multiplicity, and corresponding Γ-periodic eigenfunctions known as Bloch waves φ 1 (y; k, φ (y; k, φ 3 (y; k... which are smooth in y and form an orthonormal basis of L per(m. The spectrum may be viewed as a union of intervals or Band spaces σ(b = σ(b(k = E m, k m N which differs from the free case, the spectrum is [0, as there may be gaps. The interval E m = {ω m(k : k } is called the m-th energy band or Bloch band. We remark that for any m N there exists a closed subset I such that the functions ω m(k are real analytic functions for all k /I, see [], and we have the following condition ω m(k < ω m+1(k < ω m+(k, k /I. We call E m an isolated Bloch band if for all k the above condition holds. Lastly it is known that I = {k : ω m(k = ω m+1(k} = 0, and it is in this set of measure zero that we encounter what are called band crossings. Our assumptions will be driven upon completely avoiding band crossings, for simplicity of the model relies on differentiability in k of the eigenvalues and eigenfunctions. 3.. Bloch Decomposition. For our purposes we will use that the eigenfunctions are complete in L per(m, and the set { e ik y φ m (y; k } forms a generalized basis in L (R d. This decomposition relies on the Bloch-Floquet (or sometimes called Zak transform which is a generalization of the Fourier transform that leaves periodic functions invariant. We write for g L (R d the unique function g b L (M called the Bloch transform given by g b (y; k = γ Γ g(y + γe ik (y+γ.

4 4 J. ARBUNICH It is not difficult to see that g b (y + γ; k = g b (y; k γ Γ g b (y; k + γ = e iγ y g b (y; k γ Γ. From the Bloch transform we see that g(y = g b (y; ke ik y dk. ( To verify this we show this for g S (R d g b (y; ke ik y dk = g(y + γe ik (y+γ e ik y dk γ Γ = g(y + g(y + γ γ Γ γ 0 d ik e nγ n n=1 ( 1, 1 ] dk n = g(y. Moreover the Bloch transform may be seen as an isometry from L (R d to L (M, since for fixed k the Bloch Transform is extended to a M- periodic function on R d. For an interesting and more intimate discussion on the Bloch transform in L p spaces see [5]. Theorem 3.1. (Bloch Expansion Let g L (R d. Then its Bloch transform is given by g b (y; k = gb m (kφ m (y; k, m=1 {φ m } are the Bloch eigenfunctions of the shifted operator B(k, and for each k the m-th Bloch coefficient gb m (k are given by gb m (k := lim g(ye ik y φ m (y; k dy, (3.1 N and we have the expansion g(y = lim N y N N gb m (ke ik y φ m (y; k dk. (3. Moreover, Parseval s identity holds f b (y; k dy = M fb m (k. Proof. It is enough to prove (3.1 and (3. for g S (R d. Since for k, the Bloch transform g b (y; k L per(m, then we can be expand g b (y; k in Bloch waves with coefficients gb m (k so that g b (y; k = gb m (kφ m (y; k. m=1 Now projecting onto φ m (y; k over M and using the definition of Bloch transform we yield the m-th coefficient gb m (k = g b (y; kφ m (y; k dy = g(y + γe ik (y+γ φ m (y; kdy M = γ Γ M+γ M γ Γ m=1 g(ye ik (y φ m (y γ; k dy = g(ye ik y φ m (y; k dy R d

5 TWO-SCALE ASYMPTOTIC EXPANSION :BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES5 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.. We note that as S (R d is dense in L (R d, the Bloch transform extends to L (R d. We remark the reason for the terminology shifted cell problem is that the two operators B and B(k are related by and so Be ik y = e ik y B(k, Be ik y φ m (y; k = e ik y B(kφ m (y; k = e ik y ω m(kφ m (y; k. So for g S (R d the above remark gives the spectral resolution of B by Bf(y = fb m (ke ik y ωm(kφ m (y; k dk. 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 Schrödinger equation we are dealing with t u(t, x+ibu(t, x = 0, so if we decompose u L (R d we yield from the resolution ( t u m b (t, + iωmu m b (t, e ik x φ m (x; k dk = 0, which implies t u m b (t, k = iω m(ku m b (t, k by using 3.1 of the above theorem. Hence we have after returning to semi-classical scaling and localizing about some k 0 u ε (t, x = e i(k x ω m (kt/ε u m b (0, φ m (x/ε; k dk = = e i(k0 x ω m (k0t/ε e i[(k k0 x (ω e i(k0 x ω m (k0t/ε ϕ m (t, x, x/ε, the last line is done more explicitly in chapter 4.3 of [1]. m (k ω m (k0t]/ε u m b (0, φ m (x/ε; k dk 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 solids. We now make a remark if we are dealing with an isolated band or are away from an intersection then both ωn (k and φ N(y; k are analytic for k. Hence differentiating the eigenvalue problem once yields [ ] B(k ω N k φ N + [ k B(k k ωn ] φn = 0, k B(k = ia(y ( y + ik ( i ( y + ik ( A(y kp B(k = i q=1 [( ] ( [ ] a pq (y + ik q + + ik q a pq (y yq yq Projecting onto φ N we obtain the group velocity at k 0 c N (k 0 = k ωn(k 0 = k B(k 0 φ N (, k 0, φ N (, k 0 L (M

6 6 J. ARBUNICH Now if we differentiate the eigenvalue problem again we yield the effective mass tensor M = (M pq : M pq (k 0 = 1 ( ω N = a pq φ N, φ N 1 [ kp x p x q B(k 0 ω N (k 0 ] φ N, φ N k p k q 1 [ kqb(k 0 ω N (k 0 ] φ N, φ N k q k p When this tensor exists we have ( ω N x p x q 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. 4. Asymptotic Approximation 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 z ε (t, x = e i(k0 x ω N (k0t/ε[ ψ 0 (t, x, y + εψ 1 (t, x, y ψ 0, ψ 1 are to be determined below. Then if Ψ ε (t, x is the exact solution solving (, we have that Ψ ε (t, z ε (t, Cε, L (R d sup ] y=x/ε for some 0 < T < and ε sufficiently small, C is a constant depending on the initial data f(x, but not on ε Formal Expansion. Lets assume that for some k 0 we can find an isolated Bloch band so that ωn (k 0 is a distinct eigenvalue. Lets specify that our initial data lives in this Bloch band E N and is a spatially modulated plane wave with rapidly varying phase. Consider the initial value problem A ε Ψ ε (t, x = (A( x ε Ψε i Ψ ε ε t Ψ ε (0, x = f(xe ik0 x/ε φ N (x/ε; k + O(ε, + 1 ε W Γ (x Ψ ε = 0, t > 0, ( ε f C0 (R d. We now choose a similar ansatz as before, but now expand ψ in powers of ε with λ = ωn (k 0 given by Ψ ε (t, x e i(k0 x ω N (k0t/ε ψ(t, x, y, ψ(t, x, y = ε j ψ j (t, x, y. After plugging in the ansatz we have A ε Ψ ε = 1 ε B 0ψ ( ( B0 ψ 1 + B 1 ψ 0 + B0 ψ + B 1 ψ 1 + B ψ 0 ε + ε ( B 0 ψ 3 + B 1 ψ + B ψ 1 + O(ε = 0 j=0,

7 TWO-SCALE ASYMPTOTIC EXPANSION :BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES7 which induces a hierarchy of equations: B 0 ψ 0 = 0 (4.1 B 0 ψ 1 + B 1 ψ 0 = 0 (4. B 0 ψ + B 1 ψ 1 + B ψ 0 = 0 (4.3 B 0 ψ 3 + B 1 ψ + B ψ 1 = 0 (4.4 B 0 = ( [ ( ] y + ik 0 A(y y + ik 0 + WΓ (y ωn(k 0 = B(k 0 ωn (k 0 ( B 1 = i t + kb(k 0 x B = x (A(y x. Hence we see from the first equation that by separation of variables ψ 0 (t, x, y = u(t, xφ N (y; k 0 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 ψ 1. And since B 0 has a one-dimensional kernel then by Friedholm s alternative B 1 ψ 0 ker ( B 0, and so it follows A 0 u = ib 1 ψ 0 (t, x,, φ N (, k 0 L (M = t u(t, x + k B(k 0 φ N (, k 0, φ N ( ; k 0 x u(t, x = t u(t, x + c N (k 0 x u(t, x = 0. Hence the slowly modulated amplitude u satisfies a homogenized transport equation, which implies that u(t, x = f(x c N t, c N = c N (k 0 R d is the group velocity vector. Thus we have to leading order ( Ψ ε (t, x e i k 0 x ω 0 (k0t /ε f ( x c N t φ m (x/ε; k 0 + O(ε. Now that ψ 0 satisfies the solvability condition then we may decompose ψ 1 = ψ 1 + ψ 1, ψ 1 = v(t, xφ N (y; k 0 ker(b 0 = i ( B0 1 ( cn k B(k 0 φ N u ker (, B 0 ψ 1 we note by Fredholm s alternative that B 1 0 = (B ω N (k 0 1 is bounded. Suppose we wish to write ψ 1 = χ(y; k 0 u(t, x, χ : M R d is a periodic complex analytic function in k and smooth in y in each component. Then by plugging ψ 1 and our now determined ψ 0 into 4. we see that χ must satisfy a familiar looking cell problem [ B(k0 ωn ] [ χ = i cn k B(k 0 ] φ N.

8 8 J. ARBUNICH Recalling what was done to find the wave velocity we see that B 0 χ = i [ ω N B(k 0 ] k φ N = B 0 1 i kφ N, and so its easy to see that if ( χ, φ N = 0 that is χ ( ker ( B0 d then Thus we may write χ(y; k 0 = 1 i kφ N (y; k 0 + Cφ N (y; k 0, C = 1 ( k φ N (, k 0 φ N ( ; k 0 φ N (y; k 0. i ψ 1 (t, x, y = u(t, x χ(y; k 0 + v(t, xφ N (y; k 0, it is our aim to determine the slowly varying amplitude v = v(t, x, we impose that v(0, x = 0, so that ψ 1 (0, x, y = f(x χ(y; k 0. Now we consider equation 4.3 in the hierarchy and force the solvability condition so that after a bit of rearranging φn B 1 ψ 1 B ψ 0, φ N A 1 v = + ] = A 0 v x [ A( φ N, φ N x u = i ( t v + c N x v + p,q=1 M pq + B 1 (χ u, φ N u = 0 x p xq Thus we yield the following inhomogeneous transport equation t v(t, x + c N x v(t, x = i M f pq x p xq (x c N t = if (x c N t p,q=1 v(0, x = 0 which by Duhamel s principle has the following form v(t, x = itf (x c N t. Therefore, we have our solution to second order ψ 0 (t, x, y = f ( x c N t φ N (y; k 0 ψ 1 (t, x, y = x f(x c N t χ(y; k 0 + itf (x c N tφ N (y; k Accuracy of Approximation. Now we are in a position to justify our expansion. We first require the following lemmas: Lemma 4.. Consider the following I.V.P. S ε Ψ ε = i Ψε + 1 t ε Hε per = 0, t > 0, ( Ψ ε (0, x = f(xe ik0 x/ε φ N (x/ε; k + O(ε Hper ε = ε [A( x ε ] ( + W x Γ ε, and f C 0 (R d. ( Then! Ψ ε (t, x C [0, ; L (R d. Proof. This follows by noting that the on the domain C 0 (R d the operator H ε per is essentially self-adjoint, and so by Stone s theorem we have s strongly continuous unitary group U ε (t = e i H εt/ε, H ε is the unique self-adjoint extension of H ε. As C 0 (R d is dense in L (R d we can extend the unitary group to L (R d.

9 TWO-SCALE ASYMPTOTIC EXPANSION :BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES9 Hence we see by unitarity that for all t R, U ε (tψ ε (0 = Ψ ε (0. Before we proceed we will need some estimates that will be used in the main proof to follow: Lemma 4.3. Let T > 0, and recall the functions ψ 0, ψ 1 of the leading order and second order terms in our approximation. Then sup B 1 ψ 1 (t C 1 (I sup B ψ 0 (t C (II B ψ 1 (t C 3 (III sup C 1, C, C 3 are constants that depend on f. Proof. This follows by a somewhat straight forward computation we note that χ, φ N <. Firstly, we compute and t ψ 1 (t (I B1 ψ 1 t ψ 1 + ( k B(k 0 x ψ 1, χ ( p t f(x c N t x + φn F + t ( c N F p p K 1 c N χ f H + φ N ( K f H + t c N F K 3 (1 + T f H 3. ( k B(k 0 x ψ 1 (t p,q=1 kq B(k 0 χ f p + t F ( k B(k 0 φ N x q x p K 4 f H + T K 5 k B(k 0 φ N f H 3 K 6 (1 + T f H 3. Hence it is now clear that B1 ψ 1 K 7 (1 + T f H 3 = C 1. Secondly, we compute (II B ψ 0 (t φn Lastly, we see that (III B ψ 1 (t p,q,s=1 p,q=1 apq f φ N a pq ( fh x p = C. xq apq (y ( χs 3 f(x c N t + t φn F (x c N t x p x q x s d x p x q K 8 (1 + T f H 4 We are now in a position to prove the main stability result. We remark that is in general a prototype of what one can hope for in general.

10 10 J. ARBUNICH Theorem 4.4. Lets define z ε (t, x = e i(k0 x ω N (k0t/ε[ ψ 0 (t, x, y + εψ 1 (t, x, y ψ 0, ψ 1 satisfy the above solvability conditions. Then if Ψ ε (t, x is the exact solution solving (, we have that Ψ ε (t, z ε (t, L (R d Cε, sup ] y=x/ε for some 0 < T < and ε sufficiently small, C is a constant depending on the initial data f(x, but not on ε. Proof. Let define the difference of the exact and approximate solution ϕ(t, x := ( Ψ ε z ε (t, x. By how we constructed z ε through the solvability conditions we note that [ 1 S ε z ε = εa ε z ε = ε B }{{ 0ψ } 0 + ( ( ] ( B 0 ψ 1 + B 1 ψ }{{} 0 +ε B1 ψ 1 +B ψ 0 +ε B ψ 1 e i k 0 x ω N /ε (k0t =0 =0 and so ( S ε ϕ(t, x = ε A }{{ ε Ψ } ε A ε z ε = r1(t, ε x, y =0 ϕ 0 = ϕ(0, x = r(x, ε y,, r ε 1(t, x, y = e i(k0 x ω N (k0t/ε[ ε ( B ψ 0 + B 1 ψ 1 + ε B ψ 1 ] r ε (x, y = εe ik0 x/ε ψ 1 (0 = εe ik0 x/ε( x f(x χ(y; k 0 L (R d Hence we have reduced our approximation to the following PDE with inhomogeneous residual terms { i ϕ t + 1 ε Hε perϕ = r ε 1(t, x, y, t > 0, ϕ(0, x = r ε (x, y Now by Duhamel s Principle using the linear propagator in Lemma 4.1 we have ϕ(t, x = U ε (tϕ 0 i t 0 U ε (s tr ε 1(s, x, y ds, which implies by unitarity of U ε (t and using the above lemma to bound the remainder terms Ψ ε (t, z ε (t, = ϕ(t U ε (tr ε + = ε ψ1 (0 + t 0 t 0 ε ψ 1 (0 + εt ε ( C + ε C Cε U ε (tr1(s ε ds r ε 1 (s ds ( ( sup B1 ψ 1 + B ψ 0 (t + ε (B ψ 1 (t C is a constant that depends on f. accuracy after taking supremum over t. Hence we have proved the desired

11 TWO-SCALE ASYMPTOTIC EXPANSION :BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES 11 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 ψ. 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 ε in 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 r ε 1 and the estimate above will boil down to applying Gronwall s Lemma to gain the final stability result. References [1] Bensoussan, A. Lions, J.L. Papanicolaou, G., Asymptotic Analysis of Periodic Structures. AMS Chelsea Publishing Series,(1978, pp [] C.H. Wilcox, Theory of Bloch Waves, J.Anal. Math., 33(1978, pp [3] V. Kondratiev, M. Shubin, Discreteness of Spectrum for the Magnetic Schrödinger Operators, Commun. Partial Diff. Eqns 7, , (00. [4] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators, Academic Press, New York, (1978. [5] B. Bernhard, The Bloch Transform on L p -Spaces,Dissertation,IANA,(013. [6] P. Kuchment. Floquet Theory for Partial Differential Equations,Birkhauser Verlag, Basel (1993 (J. Arbunich Department of Mathematics, Statistics, and Computer Science, M/C 49, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA address: jarbun@uic.edu