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 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 time. There are two natural spatial length scales we consider, the fast scale measur- 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 { a 1 , ..., a d : a j ∈ R d } , d � n j a j , n ∈ Z d � � γ ∈ R d : γ = Γ = , j =1 and the unit or period cell of Γ is given by d � − 1 2 , 1 �� � x ∈ R d : x = � M = µ j a j , µ j ∈ , 2 j =1 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 ∈ 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 } , where a i · b j = 2 πδ ij . The first Brillouin zone M ⋆ 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 = 2 π Z d , M = [0 , 2 π ) d Γ ⋆ = Z d , M ∗ = ( − 1 / 2 , 1 / 2] d . 1
2 J. ARBUNICH 2. Motivation : Linear Periodic Schr¨ odinger Equation Let A ( x ) = ( a pq ( x )) be a symmetric, smooth and uniformly positive definite matrix of 2 π -periodic functions, and let our periodic potential W ( · ) ≥ 0 be a real- valued 2 π -periodic, smooth function. We denote the second order elliptic operator by n � ∂ � � ∂ � � � � ∇ · A ( x ) ∇ · = a pq x · . ∂x p ∂x q p,q =1 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∂ Ψ ε A ( x � x � ε ) ∇ Ψ ε � � Ψ ε = 0 , t > 0 , ∂t − ε 2 ∇ · + W ( ⋆ ) ε Ψ ε (0 , x ) = f ε ( x ) . We remark that this scaling describes an electron on macroscopic scales where the potential is highly oscillatory with period 2 πε . Now lets make a guess for an approximate solution where 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, 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 ε 2 and write our operator as − i ∂t + 1 ∂ A ε := −∇ · � � � � A ( y ) ∇ · ε 2 W y . ε Then plugging in our ansatz A ε Ψ ε = 1 +1 ε 2 ( B − λ ) ψ ε B 1 ψ + B 2 ψ = 0 , � �� � =0 where � � � � B = −∇ y · A ( y ) ∇ y · + W y . Hence the ansatz requires us to first resolve the eigenvalue problem B ψ = λψ, so we want to examine the spectral resolution of the closure of the following operator in L 2 ( R d ). 3. The shifted cell problems for a second order elliptic operator 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).
TWO-SCALE ASYMPTOTIC EXPANSION :BLOCH-FLOQUET THEORY IN PERIODIC STRUCTURES 3 Let H 1 ( 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 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 ( k ) φ ( y ) = ω 2 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 ∈ M ⋆ the eigenvalue problem has a countable sequence of real-valued eigenvalues which accumulate at infinity ω 2 1 ( k ) ≤ ω 2 2 ( k ) ≤ ω 2 3 ( k ) ≤ ..., including multiplicity, and corresponding Γ-periodic eigenfunctions known as Bloch waves φ 1 ( y ; k ) , φ 2 ( y ; k ) , φ 3 ( y ; k ) ... which are smooth in y and form an orthonormal basis of L 2 per ( M ) . The spectrum may be viewed as a union of intervals or Band spaces � � σ ( B ) = σ ( B ( k )) = E m , k ∈ M ⋆ m ∈ N which differs from the free case, where the spectrum is [0 , ∞ ) as there may be gaps. The interval E m = { ω 2 m ( k ) : k ∈ M ⋆ } is called the m -th energy band or Bloch band. We remark that for any m ∈ N there exists a closed subset I ⊂ M ⋆ such that the functions ω 2 m ( k ) are real analytic functions for all k ∈ M ⋆ /I , see [2], and we have the following condition ω 2 m ( k ) < ω 2 m +1 ( k ) < ω 2 m +2 ( k ) , k ∈ M ⋆ /I. We call E m an isolated Bloch band if for all k ∈ M ⋆ the above condition holds. Lastly it is known that | I | = |{ k ∈ M ⋆ : ω 2 m ( k ) = ω 2 m +1 ( k ) }| = 0 , and it is in this set of measure zero that we encounter what are called band cross- ings. 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. 3.2. Bloch Decomposition. For our purposes we will use that the eigenfunctions � � are complete in L 2 e ik · y φ m ( y ; k ) per ( M ), and the set forms a ”generalized” basis in L 2 ( R d ). 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 invariant. We write for g ∈ L 2 ( R d ) the unique function g b ∈ L 2 ( M × M ⋆ ) called the Bloch transform given by � g ( y + γ ) e − ik · ( y + γ ) . g b ( y ; k ) = γ ∈ Γ
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