Riesz bases of Floquet modes in semi-infinite periodic waveguides and implications

Transcription

1 Riesz bases of Floquet modes in semi-infinite periodic waveguides and implications Thorsten Hohage joint work with Sofiane Soussi Institut für Numerische und Angewandte Mathematik Georg-August-Universität Göttingen Wave propagation in complex media and applications, May 7 11, 2012, Crete

2 outline 1 introduction 2 main theorem a boundary value problem formulation of the main theorem on the radiation condition sketch of the proof 3 implications

3 semi-infinite periodic wave guides v + ω 2 ε p v = 0 in S + := (0, ) (0, L) ε p (x 1 + 1, x 2 ) = ε p (x 1, x 2 )

4 junction of several waveguides S.Fliss, P.Joly, and R.J.Li. Exact boundary conditions for periodic waveguides containing a local perturbation. Comm. Comp. Phys. 1: , 2006.

5 2d periodic half plane The pde on a periodic half plane can be reduced via Floquet transform to a set of pde s in semi-infinite waveguides with quasi-periodic boundary conditions.

6 locally perturbed 2d periodic material S.Fliss and P. Joly. Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media. Appl. Numer. Math., 59: , S.Fliss. Etude mathématique et numérique de la propagation des ondes dans un milieu périodique présentant un défaut. PhD thesis. École Polytechnique, 2009.

7 outline 1 introduction 2 main theorem a boundary value problem formulation of the main theorem on the radiation condition sketch of the proof 3 implications

8 outline 1 introduction 2 main theorem a boundary value problem formulation of the main theorem on the radiation condition sketch of the proof 3 implications

9 differential equation differential equation: v + ω 2 ε p v = 0 in S + where S := R (0, L), S + := [0, ) (0, L), ε p (x 1 + 1, x 2 ) = ε p (x 1, x 2 ), for all (x 1, x 2 ) S, 0 < essinfε p ε p ε a.e. in S

10 boundary value problem v + ω 2 ε p v = 0 in S + γ v = f, v (, 0) = v (, L) = 0, x 2 x 2 v outgoing (BV) with Γ := {0} (0, L), κ > 0, f H 1/2 (Γ) γv := v x 1 Γ + iκv Γ.

11 Floquet modes A Floquet mode is a nontrivial function of the form satisfying v(x 1, x 2 ) = exp(iξx 1 ) m x j 1 u(j) (x 1, x 2 ) j=0 v + ω 2 ε p v = 0 in S v (, 0) = v (, L) = 0 x 2 x 2 u (j) (x 1 + 1, x 2 ) = u (j) (x 1, x 2 ) for all (x 1, x 2 ) S and j = 0,..., m. ξ C is called quasi momentum of the Floquet mode. m {0, 1, 2,... } is called order of the Floquet mode (assuming u (m) 0). Note: ξ is only determined up to additive integer multiples of 2π.

12 energy flux By Green s theorem the following bilinear form is actually independent of x 1 for solutions v, w to the pde: q x1 (v, w) := L 0 ( v (x)w(x) v(x) w ) (x) dx 2 x 1 x 1 Iq(v, v) is (proportional to) the average energy flux in x 1 direction. Let v be a Floquet mode with quasimomentum ξ C. Then Iq(v, v) 0 ξ R Such Floquet modes are called propagating. For a physical propagating Floquet mode in S + we expect Iq(v, v) > 0.

13 incoming and outgoing Floquet modes The dimensions of the space of propagating Floquet modes is a finite even number 2n. There exists a basis {v + 1,, v + n, v 1,, v } of this n space satisfying for j, k {1,..., n} q(v + j, v k ) = 0, q(v + j, v + k ) = iδ j,k, q(v j, v k ) = iδ j,k. (qo) We say a solution v to the pde and the bc is outgoing (or satisfies the radiation condition) if v H 1,+ (S + ) := H 1 (S + ) span{v + 1,, v + n } We define the norm of v = ṽ + n n=1 α nv n + H 1,+ (S + ) by ( v H 1,+ (S + ) := ṽ 2 H 1 (S + ) + ) n 1/2. n=1 α n 2 S.A.Nazarov, B.A.Plamenevsky. Elliptic problems in domains with piecewise smooth boundaries. de Gruyter, Berlin, 1994.

14 well-posedness Proposition The boundary value problem (BV) is well-posed, i.e. for every right hand side f H 1/2 ((0, L)) there exists a unique solution v H 1,+ (S + ), and the solution depends continuously on the data. In much more generality shown in S.A.Nazarov, B.A.Plamenevsky. Elliptic problems in domains with piecewise smooth boundaries. de Gruyter, Berlin, An independent proof drops out of our analysis. T.Hohage, S.Soussi. Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides. arxiv:

15 outline 1 introduction 2 main theorem a boundary value problem formulation of the main theorem on the radiation condition sketch of the proof 3 implications

16 monodromy operator (left) translation operator: (T v)(x) := v(x 1 + 1, x 2 ) Assume that T (H 1,+ (S + )) H 1,+ (S + ). (This is always the case if all v + j are of order 0.) Let V H 1,+ (S + ) be the space of weak solutions to (BV). By well-posedness γ V : V H 1/2 (Γ) has a bounded inverse (γ V ) 1 : H 1/2 (Γ) V. Hence, we can define the monodromy operator R := γ T (γ V ) 1 : H 1/2 (Γ) H 1/2 (Γ).

17 main theorem Theorem 1 There exist Floquet modes v + n, n N, which form a Riesz basis of V. W.r.t. this basis T : V V is represented by an infinite Jordan matrix. All Jordan blocks are finite, and at most a finite number has size > 1. 2 {γ v + n : n N} is a Riesz basis of H 1/2 (Γ). R : H 1/2 (Γ) H 1/2 (Γ) is represented by the same Jordan matrix. T.Hohage, S.Soussi. Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides. arxiv:

18 other lateral boundary conditions The main theorem has also been shown for other lateral boundary conditions: Dirichlet: v(, 0) = v(, L) = 0 mixed: β-quasiperiodic: v(, 0) = 0 and v x 2 (, L) = 0 v(, L) = e iβ v(, 0) and v iβ v (, L) = e (, 0) x 2 x 2 For β = 0 and β = π we cannot exclude the existence of infinitely many Jordan blocks of size > 1 (unless we impose additional symmetry assumptions).

19 other boundary condition on Γ The main theorem holds true for arbitrary trace operators v γv := θ D v(0, ) + θ N x 1 (0, ) with θ D, θ N C, θ D + θ N > 0 if solutions to the corresponding boundary value problem (BV) are unique. For general γ solutions to (BV) may not be unique for all ω! Proposition Assume that γ is either the Dirichlet or the Neumann trace and that ε p satisfies the symmetry condition ε p (1 x 1, x 2 ) = ε p (x 1, x 2 ). Then the only solution v H 1,+ γ (S + ) to (BV) with f = 0 is v = 0.

20 outline 1 introduction 2 main theorem a boundary value problem formulation of the main theorem on the radiation condition sketch of the proof 3 implications

21 Floquet transform With Ω := R/Z (0, L) the Floquet transform is defined by F : L 2 (S) L 2 (( π, π) Ω) Fv(ξ, x) := 1 v(x 1 + l, x 2 )e iξ(x 1+l) 2π l Z F is isometric, and its inverse is given by v(x) = 1 2π π π Fv(ξ, x)e iξx 1 dξ, x S. The operator ξ, defined by the property ξ (Fv(ξ, )) = (F v)(ξ, ), is given by ξ = e iξx 1 e iξx 1 = ( x1 + iξ) x 2, ξ [ π, π].

22 band structure of the spectrum pde can be written as eigenvalue equation Av = ω 2 v for the operator A := 1 ε p : H 2 N(S) L 2 (S) where H 2 N (S) := {v H2 (S) : x2 v(, 0) = x2 v(, L) = 0}. Due to isometry of F, the spectrum of A is the union of the spectra of the operators A ξ := 1 ε p ξ : H 2 N(Ω) L 2 (Ω), for ξ R. The operators A ξ are positive and self-adjoint in the weighted space L 2 (Ω, ε p ) and have a compact resolvent. The order of the eigenpairs (λ m (ξ), w m,ξ ) of A ξ can be arranged such that ξ (λ m (ξ), w m,ξ ) is analytic for all m.

23 group velocity If λ m (ξ ) = ω 2, then v(x) := w m,ξ (x)e iξ x 1 is a Floquet mode with quasi momentum ξ. The group velocity of such a Floquet mode is defined by dω dξ = d λ m (ξ) ξ=ξ dξ = λ m(ξ ) ξ=ξ 2 λ m (ξ ).

24 equivalence results Proposition For a Floquet mode of the form v(x) = w m,ξ (x)e iξ x 1, λ m (ξ ) = ω 2 with non vanishing group velocity, i.e. λ m(ξ ) 0 the following statements are equivalent: 1 v has positive energy flux, i.e. Iq(v, v) > 0. 2 v has positive group velocity, i.e. λ m(ξ ) > 0. 3 v satisfies the limit absorption principle. 4 v satisfies the limit amplitude principle. Moreover, if ṽ(x) = w n,ξ (x)e iξ x 1 is another Floquet mode with λ n (ξ ) = ω 2 and m n, then q(v, ṽ) = 0. T.Hohage, S.Soussi. Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides. arxiv: M.Radosz. The principles of limit absorption and limit amplitude for periodic operators. Ph.D. thesis, Karlsruhe Institute of Technology, S.Fliss. Etude mathématique et numérique de la propagation des ondes dans un milieu périodique présentant un défaut. PhD thesis. École Polytechnique, 2009.

25 If two or more bands cross at ω 2 (i.e. λ m (ξ ) = λ n (ξ ) = ω 2 for some ξ [ π, π)) and if λ m(ξ )λ n(ξ ) < 0, then there are systems of Floquet modes satisfying (qo), but not the limiting absorption principle. This disproves a conjecture in discussion M.Ehrhardt, J.Sun, C.Zheng. Evaluation of scattering operators for semiinfinite periodic arrays. Commun. Math. Sci. 7: , open problem: Characterization of physical modes if λ m(ξ ) = 0.

26 outline 1 introduction 2 main theorem a boundary value problem formulation of the main theorem on the radiation condition sketch of the proof 3 implications

27 characteristic values and Floquet modes Let X, Y be Banach spaces and B : C L(X, Y), ξ B ξ a holomorphic mapping. Then ξ 0 C is called a characteristic value if B ξ0 is not injective. The following statements are equivalent: ξ 0 is a characteristic value of ξ ξ + ω 2 ɛ p. There exists a Floquet mode with quasimomentum ξ 0. There exists a notion of the multiplicity of characteristic values and a correspondence to Floquet modes of higher order. For given ξ 0 a basis of the space of all Floquet modes with quasi-momentum ξ 0 can be chosen such that the translation operator T has Jordan form w.r.t. this basis. The set of characteristic values is symmetric w.r.t. the real axis.

28 the unperturbed case ω = 0 The characteristic values of ( ξ ) are precisely the numbers ξ m,n := 2πm + i πn L and ξ m,n with m Z and n {0, 1, 2,... }. A Floquet mode corresponding to ξ m,n for any m is given v n + (x) := exp ( πn L x ) ( 1 cos πn L x 2), n = 0, 1, The functions {v n + / v n + H 1 : n = 0, 1,... } form an orthonormal basis of the solution space V H 1 (S + ) span{1}. - The translation operator T is diagonal w.r.t. this basis. Therefore, the main theorem holds true for ω = 0.

29 a generalized Rouché theorem For a simple, closed, rectifiable contour Γ let N ((B ξ ); Γ) denote the number of characteristic values of (B ξ ) enclosed by Γ, counted with multiplicity. Theorem (generalized Rouché Theorem) Let Γ be a simple, closed, rectifiable contour. Let (A ξ ) be a holomorphic family of Fredholm operators of index 0 in a neighborhood of the interior of Γ. Let (S ξ ) be a holomorphic family of operators satisfying A 1 ξ S ξ < 1 for all ξ Γ. Then ξ (A ξ + S ξ ) 1 is holomorphic in a neighborhood of Γ and N ((A ξ + S ξ ); Γ) = N ((A ξ ); Γ). I.C.Gohberg and E.I.Sigal. An operator generalization of the logarithmic residue theorem and Rouché s theorem. Mat. Sb. (N.S.), 84: , 1971.

30 estimating locations of characteristic values by Rouché s Theorem left: contours for estimating locations of large char. val. right: contour for estimating number of small char. val.

31 sketch of the proof Choose a basis {v + 1, v + 2,... } of the span of right propagating and decaying Floquet modes where v + 1,..., v + are the right propagating modes and the n decaying modes are order with increasing imaginary part of quasimomentum. The set {v n + operator : n N} is a Riesz basis of V if and only if the T : l 2 (N) H 1,+ γ (S + ), (a n ) is a norm isomorphism from l 2 (N) to V. a n v n + n=1 Our strategy is to compare T to the operator T 0 for the case ω = 0. We have seen that T 0 is isometric.

32 sketch of the proof, cont d Estimate the perturbation of eigenvectors of ( ξ + ω 2 ε p ) for characteristic values ξ with large imaginary part to show that T is well-defined and T T 0 is compact. Show that γ T 0 : l 2 (N) H 1/2 (Γ) is a norm isomorphism. Show that T is injective. Using Riesz theory and well-posedness of the boundary value problem we find that both γ T : l 2 (N) H 1/2 (Γ) and T : l 2 (N) V are norm isomorphisms. By construction the operators T and R are represented by Jordan matrices.

33 outline 1 introduction 2 main theorem a boundary value problem formulation of the main theorem on the radiation condition sketch of the proof 3 implications

34 Riesz basis for finite sections of periodic wave guides Let {vn : n N} denote a basis of Floquet modes in S := (, 0] (0, L) which propagate left or decay as x 1. Corollary Let a > 0 and consider the space V H 1 ((0, a) (0, L)) of all functions w satisfying w + ω 2 ε p w = 0 in (0, a) (0, L) w (, 0) = w (, L) = 0. x 2 x 2 Then for an H 1 -normalization of v n ± the set { v + n : n N } { vn : n N } is a Riesz basis of V.

35 dual basis Consider for simplicity a Dirichlet boundary condition on Γ. Assumption: (BV) well posed and all characteristic values have multiplicity 1. Main Theorem: {v + n (0, ) : n N} Riesz basis of H 1/2 (Γ). Question: What is the dual basis of {v + n (0, ) : n N}? Proposition Given proper scaling the dual basis to { v + n (0, ) : n N } is { v n Γ DtN +( } v ) n Γ : n N H 1/2 (Γ). x 1 If ε p (x 1, x 2 ) = ε p (1 x 1, x 2 ) this reduces to { } 2 v n x 1 Γ : n N. Here DtN + : H 1/2 (Γ) H 1/2 (Γ), DtN + (v Γ ) := v x 1 Γ Dirichlet-to-Neumann operator for S +.

36 dual basis Consider for simplicity a Dirichlet boundary condition on Γ. Assumption: (BV) well posed and all characteristic values have multiplicity 1. Main Theorem: {v + n (0, ) : n N} Riesz basis of H 1/2 (Γ). Question: What is the dual basis of {v + n (0, ) : n N}? Proposition Given proper scaling the dual basis to { v + n (0, ) : n N } is { v n Γ DtN +( } v ) n Γ : n N H 1/2 (Γ). x 1 If ε p (x 1, x 2 ) = ε p (1 x 1, x 2 ) this reduces to { } 2 v n x 1 Γ : n N. Here DtN + : H 1/2 (Γ) H 1/2 (Γ), DtN + (v Γ ) := v x 1 Γ Dirichlet-to-Neumann operator for S +.

37 characterization of adjoint of monodromy operator Proof based on the following characterization of the L 2 adjoint R of the monodromy operator R: Lemma For φ H 1/2 (Γ) we have R φ = w x 1 Γ where w H 1,+ (S + ) satisfies w + ω 2 ε p w = 0 in S + \ {1} (0, L), w = 0 on Γ, [ ] w (1, ) := w (1+, ) w (1, ) = 0, x 1 x 1 x 1 w (, 0) = w (, L) = 0, x 2 x 2 w outgoing

38 explicit formula for DtN operator Proposition Let φ n := v n x 1 Γ DtN +( vn ) Γ (or φn := 2 v n x 1 Γ if ε p (x 1, x 2 ) = ε p (1 x 1, x 2 )) and assume that φ n is scaled s.t. v n + Γ, φ n L 2 (Γ) = 1. Then DtN + f := v n + f, φ n L 2 (Γ) Γ, f H 1/2 (Γ). x 1 n=1 A truncated sum may be used for implementation. Better: Replace φ n and v + n x 1 Γ for large n by trigonometric polynomials. Error estimates with explicit constants provided by our analysis.

39 conclusions Semi infinite periodic waveguides are building blocks of more complicated periodic structures. We proved that there exist Riesz bases of Floquet modes of the space of solutions to a time harmonic wave equation both for semi-infinite wave guides and finite sections of wave guides. Traces also form bases of trace spaces. We derived explicit formulas for dual basis in a trace space and an explicit formula for the DtN operator. Thank you for your attention!