Spectral properties of a 2D scalar wave equation with 1D-periodic coefficients: application to SH elastic waves
SSpectral properties of a 2D scalar wave equation with1D-periodic coefficients: application to SH elastic waves
A.A. Kutsenko a , A.L. Shuvalov a , A.N. Norris b , O. Poncelet aa Institut de M´ecanique et d’Ing´enierie de Bordeaux,Universit´e de Bordeaux, UMR CNRS 5295, Talence 33405, France. b Department of Mechanical and Aerospace Engineering,Rutgers University, Piscataway, NJ 08854-8058, USAOctober 27, 2018
Abstract
The paper provides a rigorous analysis of the dispersion spectrum of SH (shearhorizontal) elastic waves in periodically stratified solids. The problem consists of anordinary differential wave equation with periodic coefficients, which involves two freeparameters ω (the frequency) and k (the wavenumber in the direction orthogonal tothe axis of periodicity). Solutions of this equation satisfy a quasi-periodic boundarycondition which yields the Floquet parameter K . The resulting dispersion surface ω ( K, k ) may be characterized through its cuts at constant values of
K, k and ω thatdefine the passband (real K ) and stopband areas, the Floquet branches and the isofre-quency curves, respectively. The paper combines complementary approaches basedon eigenvalue problems and on the monodromy matrix M . The pivotal object is theLyapunov function ∆ (cid:0) ω , k (cid:1) ≡ trace M = cos K which is generalized as a functionof two variables. Its analytical properties, asymptotics and bounds are examined andan explicit form of its derivatives obtained. Attention is given to the special case ofa zero-width stopband. These ingredients are used to analyze the cuts of the surface ω ( K, k ) . The derivatives of the functions ω ( k ) at fixed K and ω ( K ) at fixed k and ofthe function K ( k ) at fixed ω are described in detail. The curves ω ( k ) at fixed K areshown to be monotonic for real K, while they may be looped for complex K (i.e. inthe stopband areas). The convexity of the closed (first) real isofrequency curve K ( k )is proved thus ruling out low-frequency caustics of group velocity. The results are rele-vant to the broad area of applicability of ordinary differential equation for scalar wavesin 1D phononic (solid or fluid) and photonic crystals. a r X i v : . [ m a t h - ph ] M a r Introduction
The wave equation with periodic coefficients is ubiquitous in physics and engineering. Itsapplications in acoustics of solids have gained a new momentum since the introduction ofartificial periodic materials such as phononic crystals. A common mathematical frameworkis the Floquet-Bloch theory of partial differential equations with periodic coefficients [16].It does not however yield many explicit results for the general case of 2D or 3D periodicityand vector waves. The notable exception allowing an explicit analysis is the case of 1Dperiodicity and scalar waves which is governed by Hill’s equation [17]. The spectral proper-ties of Hill’s equation are very well understood for the situation where the wave propagatesalong some fixed direction (parallel to the periodicity axis or not). This case implies asingle spectral parameter. The objective of the present paper is to take on a broader per-spective of arbitrary (2D) propagation of scalar waves in 1D periodic media. This setupimplicates dependence on two spectral parameters and thus leads to more elaborate wavespectral properties. The specific problem to be addressed is described next.Consider SH (shear horizontal) wave motion of the form u z ( x, y, t ) = U ( y ) exp [ i ( kx − ωt )]which travels in the symmetry plane XY of a stratified monoclinic elastic solid with pe-riodic density ρ ( y ) = ρ ( y + T ) and stiffness c ijkl ( y ) = c ijkl ( y + T ). The elastodynamicequation yields a second-order ordinary differential equation for the amplitude U ( y ) ,∂ j ( c ijkl ∂ l u k ) = ρ ¨ u i ⇒ (cid:0) c U (cid:48) + ikc U (cid:1) (cid:48) + ik (cid:0) c U (cid:48) + ikc U (cid:1) = − ρω U, (1)where ∂ ≡ ∂/∂x, ∂ ≡ ∂/∂y, (cid:48) ≡ d/dy and Voigt’s indices 4 = yz, xz are used [3].It is convenient to pass from U to u = U e iϕ with ϕ ( y ) = ik (cid:82) y ( c /c ) d y which reduces(1) to the Sturm-Liouville form (cid:0) µ ( y ) u (cid:48) ( y ) (cid:1) (cid:48) − k µ ( y ) u ( y ) = − ω ρ ( y ) u ( y ) , (2)where µ = c and µ = c − c /c denote the shear moduli. Equation (2) is the objectof our study. The coefficients µ , ( y ) and ρ ( y ) are T -periodic strictly positive piecewisecontinuous functions of y ∈ R , and k, ω are two real parameters (unless otherwise specified).The functions u ( y ) and µ ( y ) u (cid:48) ( y ) are assumed absolutely continuous. They satisfy thequasi-periodic boundary conditions u ( T ) = e iKT u (0) , µ ( T ) u (cid:48) ( T ) = e iKT µ (0) u (cid:48) (0) (3)with the Floquet parameter K ∈ C , which by periodicity of e iKT may be defined on thestrip Re KT ∈ [ − π, π ] called the Brillouin zone. Note that Eq. (2) admits equivalentrepresentations obtained by changing the function and/or variable. For instance, replacingthe variable y ⇒ (cid:101) y = (cid:82) y µ − ( ς ) d ς recasts (2) in the form of a weighted Schr¨odingerequation u (cid:48)(cid:48) ( (cid:101) y ) + ω Z u ( (cid:101) y ) = 0 , with ω Z = (cid:0) ω − µ k /ρ (cid:1) Z , Z = ρµ . (4)2ote that this transformation does not require reinforcing the above-imposed conditionof piecewise continuity of µ ( y ). The coefficients Z and Z ( Z = Z at k = 0) have thephysical meaning of, respectively, impedance and normal impedance that we will find usefulfor interpretations.There exists a comprehensive spectral theory describing the eigenvalues ω n ( n ∈ N ) of(2), (3) as functions of K at fixed k, e.g. [6, 15, 17, 22, 18, 1]. From this perspective, thespectrum for real K ∈ R is represented by the Floquet branches ω n ( K ) on the ( ω, K )-plane.Each branch spans a finite range on the ω -axis, called a passband, with a correspondingbounded solution u n ( y ). Separating them are the ranges of ω , called stopbands, where ω ∈ R and KT ∈ π Z + i ( R \ . Properties of the functional dependence of ω n ( K ) atfixed k can be described by various analytical means. One of the key ingredients of thistheory is the so-called Lyapunov real-valued function ∆( ω ) defined as the half trace of themonodromy matrix (the propagator over a period). By this definition, ∆( ω ) = cos KT determines the passbands and stopbands as the ranges (cid:12)(cid:12) ∆( ω ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ∆( ω ) (cid:12)(cid:12) > k is considered as an independent variable on top of ω and K . Keeping ω as an eigenvalueof Eqs. (2)-(3) now implies its dependence on two parameters: ω n = ω n ( K, k ). For K real, ω n ( K, k ) is a multisheet surface whose sheets projected on the ( ω, k )-plane span thepassband areas bounded by the cutoff lines ( | ∆ | = 1) and separated by the stopband areas.Cutting this surface by the planes k = const and ω = const produces the Floquet branchesand the isofrequency (a.k.a. slowness) curves, respectively. Clearly, such perspective isconsiderably richer than the one restricted to the Floquet curves at fixed k. It is alsoimportant to note that the present study differs from the two-parameter Sturm-Liouvilleproblem with Dirichlet, Neumann and Robin boundary conditions, which has been studiedelsewhere, see e.g. [4, 24].The structure and main results of the paper are as follows. Section 2 introduces com-plementary approaches based on differential operators A K ( k ) , B K ( ω ) defined by (2), (3)and on the matricant M ( y, y ) of the equivalent differential system. The operators A K ( k ) , B K ( ω ) are self-adjoint and have a complete orthogonal system of joint eigenfunctions, asshown in Appendix A1 by explicit construction of their resolvent operators. The eigenval-ues ω n and k n of A K ( k ) and B K ( ω ) are then linked to the monodromy matrix M ( T, ± iK via the generalized (depending on two parameters) Lyapunov func-tion ∆( ω , k ) ≡ trace M ( T,
0) = cos KT . Section 3 describes this function in some detail . It is shown in § ω , k ) inside the passbands | ∆ | < ω and k , and that ∆( ω ) for fixed k and ∆( k ) at fixed ω each satisfiesLaguerre’s theorem (by virtue of the estimates of ∆( ω , k ) given in Appendix A2). Thesetwo fundamental facts explain the regular structure of the passband/stopband spectrumon the ( ω, k )-plane. The WKB approach [10] is used in § § | ∆ | = 1 with the ( ω, k )-plane. It is shown thatZWS may or may not exist for an arbitrary periodic profile of ρ ( y ) and µ , ( y ), are likelyto exist for any profile that is even about the period midpoint, and always exist for a peri-odically bilayered structure. In the model cases, ZWS may also form infinite lines on the( ω, k )-plane. Closed-form expressions for the partial derivatives of ∆( ω , k ) are obtainedin § M of the matricant M taken at different points y within the periodand weighted by ρ ( y ) and/or µ ( y ) . An alternative representation is derived for the first-order derivatives of ∆( ω , k ) within the passbands by using the eigenfunctions of A K ( k )and B K ( ω ). The two equivalent formulas obtained for the first derivatives of ∆( ω , k )provide an explicit meaning to their sign-definiteness and offer useful complementary in-sight. In particular, it reveals some interesting attributes of the function M ( y + 1 , y ) , whose zeros ( ω, k ) are y -dependent solutions of the Dirichlet problem on [ y, y + T ] , see § ω , k ) (= cos KT ) and the expressionsfor its derivatives established in Section 3 are then used in Section 4 to analyze principalcuts of the dispersion surface ω n ( K, k ) . In § ω ( k ) for fixed K is studied.It is shown that if K is real then the curves ω n ( k ) are monotonic (this may not be so forcomplex K ) and they tend to the same linear asymptote k min y ∈ [0 ,T ] [ µ ( y ) /ρ ( y )] whichis independent of n. In § ω ( K ) at fixed k is discussed. For real K ,the first non-zero derivative of Floquet branches ω n ( K ) is provided (it is a first derivativeinside the passbands and a second one at the cutoffs); for the stopbands, the condition on ω realizing maximum of | Im K ( ω ) | is formulated. The real isofrequency curves K ( k ) at fixed ω are considered in §§ ω less than the first cutoff ω (cid:0) πT − , (cid:1) . It is proved that, whatever thedistortion of its shape due to unidirectional periodicity may be, this isofrequency curve isalways convex and hence low-frequency caustics of the group velocity ∇ ω are impossible.Finally, useful bounds on the first eigenvalue ω ( K, k ) for KT ∈ [ − π, π ] and any k areprovided in Appendix A3.Without loss of generality, in the following we take T = 1; more precisely, this impliesthe redefinitions y ⇒ y/T ≡ y, ω ⇒ ωT ≡ ω, k ⇒ kT ≡ k and K ⇒ KT ≡ K so that thevariables y and ω, k, K are hereafter non-dimensional. We also assume throughout that T = 1 is a minimal possible period. Equation (2) with the conditions (3) can be considered in either of the equivalent forms A K u = ω u, B K u = k u, u ∈ D K (5)4ith the operators A K ≡ A K ( k ) and B K ≡ B K ( ω ) A K u = − ρ (cid:0) µ u (cid:48) (cid:1) (cid:48) + k µ ρ u, B K u = 1 µ (cid:0) µ u (cid:48) (cid:1) (cid:48) + ω ρµ u. (6)Their common domain is D K = (cid:8) u ∈ D : η (1) = e iK η (0) (cid:9) ,D = { u ∈ AC [0 , , µ u (cid:48) ∈ AC [0 , } , η ( y ) = (cid:18) u ( y ) iµ ( y ) u (cid:48) ( y ) (cid:19) , (7)where K ∈ C and AC [0 ,
1] is the space of all absolutely continuous functions from [0 ,
1] to C (note that using ” i ” in the definition of η and hence in (10) is a conventional option whichis useful for a compact form of (13) and similar identities). Let ( · , · ) ρ, µ and (cid:107)·(cid:107) ρ, µ bea standard inner product and norm in the Hilbert space H ρ, µ = L ρ, µ (0 ,
1) of functionswith quadratically summable measure ρ ( y )d y and µ ( y )d y, respectively; so that( u, v ) ρ = (cid:90) ρ ( y ) u ( y ) v ∗ ( y )d y, (cid:107) u (cid:107) ρ = ( u, u ) ρ , ( u, v ) µ = (cid:90) µ ( y ) u ( y ) v ∗ ( y )d y, (cid:107) u (cid:107) µ = ( u, u ) µ , (8)where ∗ means complex conjugation.The operator (2) on L ( R ) with eigenvalues ω (or k ) can be represented as a directintegral decomposition ⊕ K ∈ [0 , π ] A K (or ⊕ K ∈ [0 , π ] B K ) [22]. Therefore the spectrum of theoperator (2) is a union of all eigenvalues of A K (or B K ) for K ∈ [0 , π ] and hence for all K ∈ R since A K = A K +2 π , B K = B K +2 π . The operators A K and B K are symmetric if K ∈ R , i.e. ( A K u, v ) ρ = ( u, A K v ) ρ , ( B K u, v ) µ = ( u, B K v ) µ for u, v ∈ D K , and they bothhave compact and self-adjoint resolvents that satisfy the Hilbert-Schmidt theorem (seeAppendix A1). Therefore A K and B K are self-adjoint with purely discrete spectra σ ( A K )and σ ( B K ) containing an infinite number of real eigenvalues ω n ( K, k ) and k n ( K, ω ) ( n ∈ N ), and corresponding eigenfunctions u n ( ≡ u n, A and u n, B ) forming a complete orthogonalsystem in the spaces H ρ and H µ , respectively. The operator A K is positive for any k ∈ R (i.e. for any k ≥ A K u, u ) ρ ≥ > k (cid:54) = 0) , (9)so its spectrum σ ( A K ) consists of non-negative eigenvalues ω n ( K, k ) (strictly positive at k (cid:54) = 0), which are hereafter numbered in increasing order ω ≤ ω ≤ . . . By contrast, B K is not sign-definite and hence its spectrum σ ( B K ) includes both positive and negativeeigenvalues k n ( K, ω ). Note that real eigenvalues of A K and B K are also admitted atIm K (cid:54) = 0 (see Definition 4(c) below).Equation (2) can be recast as η (cid:48) ( y ) = Q ( y ) η ( y ) with Q ( y ) = i (cid:18) − µ − µ k − ρω (cid:19) (10)5or η ( y ) introduced in (7) . Given an initial condition η ( y ), Eq. (10) has a unique solution η ( y ) = M ( y, y ) η ( y ) (11)defined through the propagator matrix, or matricant, M ( y, y ) ≡ (cid:18) M ( y, y ) M ( y, y ) M ( y, y ) M ( y, y ) (cid:19) = (cid:98)(cid:90) yy [ I + Q ( ς ) d ς ]= I + (cid:90) yy Q ( ς ) d ς + (cid:90) yy Q ( ς ) d ς (cid:90) ς y Q ( ς ) d ς + . . . , (12)where (cid:98)(cid:82) is the multiplicative integral evaluated by the Peano series [21] and I is the 2 × M ( y, y ) = 1 due to tr Q = 0 , where tr means the trace. By(10) Q = − TQ + T for ω , k ∈ R and so M − ( y, y ) = TM + ( y, y ) T ⇒ Im M , ( y, y ) = 0 , Re M , ( y, y ) = 0 , (13)where + denotes Hermitian transpose and T is the 2 × Q ( y ) is also even about the midpoint of the interval [ y , y ] then M ( y, y ) = TM T ( y, y ) T ⇒ M ( y, y ) = M ( y, y ) , (14)where T denotes transpose. The properties (13) and (14) are actually valid for matrices Q and M of arbitrary n × n size (see [24] for details), while (13 ) and (14) are attributesof the 2 × is evident from the definition(7) of η with a real scalar u ).Assume a periodic Q ( y ) so that Q ( y ) = Q ( y + 1) and hence M ( y, y ) = M ( y + 1 , y + 1).The propagator M ( y + 1 , y ) over a period [ y , y + 1] is called the monodromy matrix.For any y ≡ y, denote its elements as M ( y + 1 , y ) = (cid:18) m ( y ) im ( y ) im ( y ) m ( y ) (cid:19) , m , ( y ) = M , ( y + 1 , y ) ,im , ( y ) = M , ( y + 1 , y ) , (15)where Im m j ( y ) = 0 , j = 1 .. , for ω , k ∈ R by (13) . The assumed periodicity with useof the chain rule implies the identity M ( y + 1 , y ) = M ( y + 1 , M (1 , M (0 , y ) = M ( y, M (1 , M − ( y, . (16) Remark 1
The trace and eigenvalues of M ( y + 1 , y ) are independent of y by virtue of(16). Hereafter, unless otherwise specified, we set y = 0 and define the monodromy matrix as M (1 ,
0) with respect to the period [0 ,
1] (as in (7), (8)).6earing in mind det M = 1 , denote the eigenvalues of M (1 ,
0) by q and q − . Introducethe generalized Lyapunov function∆( ω , k ) ≡
12 tr M (1 ,
0) = 12 (cid:0) q + q − (cid:1) , (17)which is analytic in ω , k by (10) , (12) and real for ω , k ∈ R by (13) . As notedabove, the function ∆( ω , k ) is independent of the interval on which the unit period isdefined. It is also invariant for any similarity equivalent formulation of the system matrix (cid:101) Q ( y ) = C − Q ( y ) C because tr (cid:102) M = tr (cid:0) C − MC (cid:1) = tr M , leaving ∆( ω , k ) unchanged. Proposition 2
For any complex numbers k, ω, K , the following statements are equiva-lent: (i) ω is an eigenvalue of the operator A K ( k ); (ii) k is an eigenvalue of the operator B K ( ω ); (iii) k, ω and K are connected by the equality ∆( ω , k ) − cos K = 0 . (18) Proof.
The link (i) ⇒ (ii) follows from Eq. (5). Consider (i),(ii) ⇒ (iii). According to (i) or (ii), ω or k is an eigenvalue of, respectively, A K ( k ) or B K ( ω ). Then there exists u ( y ) ∈ D K that satisfies (5) hence (2), and consequently the vector η ( y ) , generated by u ( y ) according to (7) , is a solution of Eq. (10). So, by (11), η (1) = M (1 , η (0) . On theother hand, as indicated in (7) , u ( y ) ∈ D K implies that η (1) = e iK η (0). Hence e iK is aneigenvalue q of M (1 , (iii) . Nowconsider (iii) ⇒ (i),(ii) . From (18) and the definition (17), the eigenvalue q of M (1 ,
0) is q = e iK , and corresponding eigenvector w exists such that M (1 , w = e iK w . Let u ( y ) bethe first component of the solution η ( y ) = M ( y, w of Eq. (10) with the initial condition η (0) = w . From the above, u ( y ) belongs to D K and satisfies Eq. (5), which implies (i),(ii) . (cid:4) Corollary 3
Each eigenfunction u of A K and B K is equal to the first component of thevector η ( y ) = M ( y, w , where w is the eigenvector of M (1 , corresponding to the eigen-value q = e iK . Definition 4
Passband areas, cutoffs and stopband areas are defined for ω , k ∈ R (andhence real ∆( ω , k ) ) as follows: ( ω, k ) : | ∆ | ≤ ⇔ K ∈ R ) passbands , ∆ = ± ⇔ K ∈ π Z ) cutoffs , | ∆ | > ⇔ K ∈ π Z + i ( R \ stopbands . Before discussing general properties of the Lyapunov function ∆( ω , k ) , it is expedientto mention its explicit properties at ω = 0 and/or k = 0. Obviously ∂ ∆ /∂ω = 0 at ω = 07nd ∂ ∆ /∂k = 0 at k = 0 . By (10) , (12) and (17),∆( ω , k ) = 1 + (cid:10) µ − (cid:11) (cid:0) (cid:104) µ (cid:105) k − (cid:104) ρ (cid:105) ω (cid:1) + O (cid:0) ( ω + k ) (cid:1) with (cid:104)·(cid:105) ≡ (cid:90) ( · ) d y ; ∂ ∆ /∂ ( ω ) = − (cid:104) ρ (cid:105) (cid:10) µ − (cid:11) , ∂ ∆ /∂ (cid:0) k (cid:1) = (cid:10) µ − (cid:11) (cid:104) µ (cid:105) at ω = 0 , k = 0 , (19)where the identity (cid:82) d ς (cid:82) ς [ f ( ς ) f ( ς ) + f ( ς ) f ( ς )] d ς = (cid:104) f (cid:105) (cid:104) f (cid:105) was used in (19) .Note that ∆(0 , k ) > k > (cid:2) ∂ ∆ /∂ ( ω ) (cid:3) ω =0 < k ≥ , whereas the boundsof ∆( ω ,
0) and the sign of (cid:2) ∂ ∆ /∂ (cid:0) k (cid:1)(cid:3) k =0 are not fixed for ω > . Also note the explicitnon-semisimple form of the matrix M ( y,
0) = (cid:18) − i (cid:82) y µ − ( ς ) d ς (cid:19) at ω = 0 , k = 0 . (20) ∆( ω , k ) We proceed with some observations on the analytical properties of the function ∆( ω , k )that underlie the alternating structure of the passbands and stopbands. Lemma 5 If ω / ∈ R or k / ∈ R then ∆ / ∈ [ − , . Proof.
If ∆ ∈ [ − ,
1] then according to Proposition 2 the identity (18) holds for K ∈ R and hence ω or k is an eigenvalue of A K ( k ) or B K ( ω ), respectively. It was shown (see(11) and below) that the eigenvalues of A K ( k ) are positive and the eigenvalues of B K ( ω )are real. (cid:4) Proposition 6
The derivatives ∂ ∆ /∂ ( ω ) and ∂ ∆ /∂ (cid:0) k (cid:1) do not vanish within an openpassband interval ∆( ω , k ) ∈ ( − , . Proof.
By Lemma 5, if ∆ ∈ ( − ,
1) then ω , k ∈ R . Suppose that ∂ ∆ /∂ ( ω ) = 0 forsome real value ω . Then, because ∆( ω ) ( ≡ ∆( ω , k ) at fixed k ) is an analytic function,there exists complex (cid:101) ω in the vicinity of ω for which ∆( (cid:101) ω ) ∈ ( − ,
1) . This contradictsLemma 5, and hence ∂ ∆ /∂ (cid:0) ω (cid:1) (cid:54) = 0 . The same reasoning proves that ∂ ∆ /∂ (cid:0) k (cid:1) (cid:54) = 0.Consequently, Eq. (18) at fixed ω > k ) has only real and simple roots k n (or ω n ) if cos K ∈ ( − , . (cid:4) Proposition 6 plays a pivotal role in explaining the origin of the Floquet stopbands bythe following simple reasoning. Consider ρ ( y ) , µ , ( y ) resulting from an arbitrary peri-odic perturbation of some reference constant values ρ and µ , , so that ∆( ω , k ) is aperturbation of ∆ ( ω , k ) = cos K with K = ρ µ ω − µ µ k . Since the first derivativesof ∆( ω , k ) do not vanish within ( − , , the perturbed extreme values ∆ = ± ± − , , thereby leading to complex values K ∈ π Z + i ( R \ roposition 7 For ω , k ∈ R , the derivatives of any order n ∈ N of the functions ∆( ω ) and ∆( k ) ( ≡ ∆( ω , k ) at fixed k and fixed ω , respectively) have only real and simple zeros,each lying between consecutive zeros of the ( n − th derivative of the same function. Inparticular, the first derivatives of ∆( ω ) and ∆( k ) have a single and simple zero betweenconsecutive zeros of ∆( ω , k ) and do not vanish elsewhere.Proof. It is shown in Appendix A2 that the functions ∆( ω ) and ∆( k ) are entire functionsof order of growth . Their zeros are the eigenvalues of the operators A π/ ( k ) and B π/ ( ω ),and are therefore real and simple. Hence both functions satisfy the conditions of Laguerre’stheorem (e.g. [27]), implying that the derivatives of ∆( ω ) and of ∆( k ) are also entirefunctions with order of growth and they have the desired properties. (cid:4) Propositions 6 and 7 define the basic form of the function ∆( ω , k ) at fixed ω or k . Itis exemplified in Fig. 1 for a piecewise continuous profile of material coefficients chosen as µ ( y ) = µ ( y ) = 14 (1 + 3 y ) (2 + y ) , ρ ( y ) = 2 + y for y ∈ [0 ,
1] (21)(taking µ , in GPa and ρ in g/cm implies ωT ≡ ω in MHz · mm in this and subsequentfigures). Note that ∆( ω ) has an infinite number of zeros that are strictly positive andmove rightwards as k increases, whereas ∆( k ) has an infinite number of negative zeros at ω = 0 which move one by one on the positive semi-axis k > ω increases. -101 0 20 40 60 ∆ ω k = 0 -6-5-4-3-2-101230 10 20 30 40 50 ∆ ω k = 0 k = 6 (a) -6-5-4-3-2-10123-100 -50 0 50 100 150 200 k ω = 0 ω = 7.3 ω = 17 (b) ∆ Figure 1:
Generalized Lyapunov function ∆( ω , k ) for the profile (21): (a)∆( ω ) (= ∆( − ω )) at different fixed values of k (a fragment of ∆( ω ) at k = 0 for ω ≷ k ) at different fixed values ω . Since zeros of the first derivatives of ∆( ω , k ) cannot be points of inflection or zero-curvature by Proposition 7, we can now refine the numbering of branches ω n ( K, k ) =9 ω n ( K, k ) ( ≥
0) in the passbands as follows:0 < ω ( K, k ) < ω ( K, k ) < . . . if K ∈ R , K / ∈ π Z ;0 ≤ ω (0 , k ) < ω (0 , k ) ≤ ω (0 , k ) < ω (0 , k ) ≤ . . . if K ∈ π Z ;0 < ω ( π, k ) ≤ ω ( π, k ) < ω ( π, k ) ≤ ω ( π, k ) < . . . if K ∈ π + 2 π Z . (22)With reference to (19) and Proposition 6, the sign of first derivatives of ∆( ω , k ) along ω n ( K, k ) in the n th open passband | ∆ | < )) issgn (cid:2) ∂ ∆ /∂ ( ω ) (cid:3) = − sgn (cid:2) ∂ ∆ /∂ (cid:0) k (cid:1)(cid:3) = ( − n . (23)The possibility of equality of two cutoffs (see (22) , ), i.e. of a double root of the equation∆( ω ) = ±
1, implies a zero-width stopband addressed in detail in § ω ,n and ω ,n of (2) satisfying the conditions u (0) = 0 , u (1) = 0 and u (cid:48) (0) = 0 ,u (cid:48) (1) = 0, respectively. It is known that ω D ,n and ω N ,n are simple zeros of the functions M (1 ,
0) and M (1 ,
0) of ω, which occur once per each stopband complemented by cut-offs (except the first stopband devoid of ω D ,n ). The branches ω D , ( k ) < ω D , ( k ) ... and ω N , ( k ) < ω N , ( k ) ... are thus related to the passband eigenvalues ω n ( K, k ) of (22) as ω D , j ( k ) , ω N , j +1 ( k ) ∈ [ ω j (0 , k ) , ω j +1 (0 , k )]; ω D , j − ( k ) , ω N , j ( k ) ∈ [ ω j − ( π, k ) , ω j ( π, k )] , (24)where j ∈ N and ω N , ( k ) ∈ [0 , ω (0 , k )]. Recall that the stopbands and cutoffs are invariantwith respect to the choice of the period interval [ y , y + 1] ≡ [0 ,
1] (see Remark 1); however,the branches ω D ,n ( k ) and ω N ,n ( k ) within this area certainly depend on the choice of thepoint y ≡
0. In other words, some fixed values ω, k realize the Dirichlet or Neumannconditions at the edges of [ y , y + 1] iff y is a zero of the function M ( y + 1 , y ) ≡ im ( y )or M ( y + 1 , y ) ≡ im ( y ), respectively (see § Q ( y ) is an even function about the midpoint of the period [ y , y + 1] for some y ,then the Dirichlet and Neumann branches ω D ,n ( k ) and ω N ,n ( k ) satisfying m ( y ) = 0 and m ( y ) = 0 coincide with the cutoff curves. We note the useful identity m ( y ) m ( y ) > | ∆ | < M = 1,and hence for any | ∆ | < m ( y ) and m ( y ) are strictly non-zero insidethe passbands by (24). ∆ Some insight into the high-frequency spectrum in the case of continuous and piecewisecontinuous periodicity can be gained from the WKB asymptotics [10] of the Lyapunovfunction ∆( ω , k ) at fixed k . To this end recall the impedance Z = Z (cid:112) − µ k /ρω with Z = √ ρµ introduced in (4). For any fixed k, let ω > k max y ∈ [0 , ( µ /ρ ) so that Z ( y ) is real (the so-called supersonic regime). Suppose for brevity that the overall periodic10rofile of Z ( y ) has at most one point of discontinuity per period. If so, the zero-order WKBapproximation ∆ (0)WKB of ∆ takes an especially simple form∆ (0)WKB = 12 (cid:0) [ Z ] / + [ Z ] − / (cid:1) cos (cid:0) ω (cid:90) µ − Z d y (cid:1) , (25)where ± iωµ − Z are the eigenvalues of the matrix Q defined in (10) and [ Z ] = Z (cid:0) y − d (cid:1) /Z (cid:0) y + d (cid:1) with Z (cid:0) y ± d (cid:1) ≡ lim ε → Z ( y d ± ε ) is the relative jump of Z at the possible point y d of its pe-riodic discontinuity. Assume first that Z ( y ) is strictly continuous for any y (not restrictedto [0 , Z ] = 1 . Then Eq. (25) yields (cid:12)(cid:12)(cid:12) ∆ (0)WKB (cid:12)(cid:12)(cid:12) ≤ | ∆ | > , whose widths (the frequency gaps between cutoffs, see(19) , ) may well be nonzero at finite ω. Thus if Z ( y ) is continuous then Eq. (25) merely im-plies that the stopband widths tend to zero at any fixed k as ω tends to infinity. The latterconclusion is also valid even if µ /ρ has periodic jumps but ρµ is continuous throughout,so that [ Z ] (cid:54) = 1 indicates existence of nonzero stopbands at finite ω but [ Z ] → [ Z ] = 1 at ω → ∞ . On the other hand, if ρµ does have a jump and so [ Z ] (cid:54) = 1 , then Eq. (25) showsthat the stopband widths remain nonzero as ω → ∞ . Having stated this, we hasten to addthat a physically sensible profile model should be related to the frequency ω in that a finite ω implies that a probing wave ”sees” appropriately abrupt variations of material propertiesas jumps, which are of course smoothed out by the ’infinite zoom’ of the limit ω → ∞ . Theabove WKB conclusions on the high-frequency trends of cutoffs agree with a less generalframework of, specifically, small periodic perturbations that provides expressions for thestopband widths through the Fourier series coefficients, see [3, 6].As an example, consider again Fig. 1, which is plotted for a piecewise continuousprofile (21) that gives [ Z ] = 12 (cid:112) (1 − k /ω ) / (4 − k /ω ) (note that a ’single periodicdiscontinuity y d ’ is located at the edges of the period T = 1 by (21); however, similarlyto Remark 1, ∆ (0)WKB does not depend on the choice of the period [0 ,
1] relative to y d ). Itis easy to check that the exact curves ∆ shown in Fig. 1a are well fitted by the WKBapproximation (25) (not displayed to avoid overloading the plot) once ω is greater enoughthan k max (cid:112) µ /ρ = 2 k . It is also seen from Fig. 1a that increasing ω makes the curves∆ for different fixed k tend to that related to k = 0 , as predicted by Eq. (25).In the case of two or more discontinuity points per period, applying the WKB asymp-totics separately along each range of continuity modifies (25) to the form with two or morephase terms corresponding to the reflection-transmission at each discontinuity. For moreexamples of using the WKB approach to the periodic profile, see [23].11 .3 Zero-width stopband The following definition of a zero-width stopband (ZWS) is motivated by the possibleoccurrence of the second and third cases in (22). Definition 8 If ω = ω n (0 , k ) = ω n +1 (0 , k ) or ω = ω n − ( π, k ) = ω n ( π, k ) for some ω, k ∈ R and n ∈ N , then this cutoff point ( ω, k ) is called a ZWS. It is essential that the cutoff curves are analytic (as any ω n ( K, k ) with fixed K ∈ R is,see § ω, k )-plane and hence a saddle point | ∆ | = 1 on the Lyapunov-function surface ∆( ω , k ) . For the same reason, if, exceptionally(see § ω ( k ) of local extremum | ∆ | = 1 of ∆( ω , k ) , then suchline cannot have an edge point.A comprehensive account of the properties of ZWS is based on the next proposition. Proposition 9
The following statements are equivalent: (i) ( ω, k ) is a ZWS; (ii) ∆( ω , k ) = ± and ∂ ∆( ω , k ) /∂ ( ω ) = 0 ; (iii) ∆( ω , k ) = ± and ∂ ∆( ω , k ) /∂ ( k ) = 0 ; (iv) M (1 ,
0) = ± I .Proof. The link (i) ⇔ (ii) follows from Definition 8 and Proposition 7. The link (i) ⇒ (iv) canbe inferred e.g. via (24), which tells us that assuming (i) entails M (1 ,
0) = M (1 ,
0) = 0and hence M (1 , M (1 ,
0) = det M = 1 , where M , M are real by (13) . Since (i) also means tr M (1 ,
0) = ± , it follows that M (1 ,
0) = ± I as stated. Next let us show (iv) ⇒ (ii) . Assume M (1 ,
0) = ± I for some (cid:101) ω, (cid:101) k ∈ R . Note that ∆( (cid:101) ω , (cid:101) k ) = ± q = e iK = ± M (1 ,
0) = ± I has geometrical multiplicity2, hence (cid:101) ω is an eigenvalue of A K ( (cid:101) k ) of multiplicity 2 by Corollary 3. Now considersome K (cid:48) ∈ R arbitrary close to K that yields cos K (cid:48) = ∆( ω , (cid:101) k ) ∈ ( − , . Since (cid:101) ω isa double eigenvalue of A K ( (cid:101) k ), the self-adjoint operator A K (cid:48) ( (cid:101) k ) has two distinct simpleeigenvalues ω (cid:16) K (cid:48) , (cid:101) k (cid:17) close to (cid:101) ω , and, by Propositions 2 and 6, these are distinct simplezeros of ∆( ω , (cid:101) k ) − cos K (cid:48) . Therefore ∆( (cid:101) ω , (cid:101) k ) = ± ω , (cid:101) k ) , i.e. ∂ ∆ /∂ ( ω ) = 0 at (cid:101) ω , (cid:101) k , which is equivalent to (ii). Note that reversing the abovereasoning proves (ii) ⇒ (iv) without appeal to (24), and that invoking B K ( ω ) in place of A K ( k ) provides a similar proof of (iii) ⇔ (iv) (see also Proposition 16 below). (cid:4) Note that the point ω = 0 , k = 0 which yields ∆ = 1 is not a ZWS since it does not satisfyany of the above statements, which is evident from (19)-(20).Proposition 9 implies that the multiplicity of ω , k as the roots of equation ∆( ω , k ) − cos K at K ∈ R is the same as their multiplicity as the eigenvalues of A K ( k ) , B K ( ω ) (this It is understood that a ZWS is actually not a ’stopband’ (in the sense of Definition 4). Note that asimilar notion of ’zero-width passband’ is inconceivable due to Proposition 7.
K / ∈ R , where a double root ω or k of Eq. (18) isnot a double eigenvalue of, respectively, A K ( k ) or B K ( ω ) which are no longer self-adjointfor K / ∈ R . It is also pointed out that the eigenvalue q = e iK of M (1 ,
0) has an algebraicmultiplicity 2 at any cutoff, while its geometrical multiplicity is 2 only at cutoffs that areZWS.
Corollary 10
The matrix M (1 , is non-semisimple for any cutoff ( ω, k ) unless it is aZWS. We note that the non-semisimple nature of the monodromy matrix at the cutoffs hasimportant ramifications for the interpretation of its matrix logarithm, which has beenproposed as the basis for dynamic effective medium models, see [25, 26].
To begin with, it is recalled that the period T = 1 is everywhere understood as a minimal possible period, so that trivial ZWS which turn up when T is a multiple of the minimalperiod are disregarded.Given an arbitrary periodic Q ( y ) , the condition M (1 ,
0) = ± I stipulating existenceof ZWS imposes three real constraints on two parameters ω, k and hence is unlikely tohold. However, if the profile Q ( y ) is symmetric (even) about the midpoint of the period[0 , M (1 ,
0) implies only two constraintsand thus such profile can be expected to yield a set of ZWS points (intersections of cutoffcurves | ∆ | = 1) on the ( ω, k )-plane. More precisely, since the cutoffs are independent ofhow the period interval is fixed (see Remark 1), ZWS are expected to exist if a given profile Q ( y ) admits such a choice of the period interval [ y , y + 1] ≡ [0 ,
1] within which Q ( y ) issymmetric.Note that by definition any ZWS is also an intersection of Dirichlet and Neumannbranches (24) while the inverse is generally not true. Moreover, in contrast to ZWS, theDirichlet and Neumann branches and hence their intersections (cid:8) ω, k (cid:9) D=N depend on thechoice of the period interval. For instance, let Q ( y ) be symmetric with respect to a fixedperiod [0 , . Then the Dirichlet and Neumann branches coincide with the cutoff curvesand hence any intersection (cid:8) ω, k (cid:9)
D=N is a ZWS (see e.g. Fig. 1 of [24]) . However, if for agiven Q ( y ) = Q ( y + 1) the period is shifted so that Q ( y ) is not even about its midpoint,then a new set (cid:8) ω, k (cid:9) D=N includes but generally does not coincide with the (unchanged)set of ZWS.As a simple explicit example, consider a periodically bilayered structure where Q ( y )takes two alternating constant values within two layers j = 1 , , M (1 ,
0) = (cid:32) cos ψ cos ψ − Z Z sin ψ sin ψ Z cos ψ sin ψ + i Z sin ψ cos ψ i Z cos ψ sin ψ + i Z sin ψ cos ψ cos ψ cos ψ − Z Z sin ψ sin ψ (cid:33) , (26)where Z j is the layer impedance defined in (4) and ψ j = ωZ j d j /µ j with d j for thelayer thickness. The set of Dirichlet/Neumann intersections (cid:8) ω, k (cid:9) D=N is defined bysimultaneous vanishing of both off-diagonal components of (26), which implies the fol-lowing three options: (i) { sin ψ = 0 , sin ψ = 0 } , (ii) { cos ψ = 0 , cos ψ = 0 } and (iii) { Z = Z , sin ( ψ + ψ ) = 0 } , where (iii) may or may not hold for real ω, k [2] . It is seenthat (i) and (iii) yield M (1 ,
0) = ± I . Thus (i) and maybe (iii) define ZWS, while (ii) doesnot.Recall that an infinite periodically bilayered structure can always be considered overa three-layered period where the same stepwise profile Q ( y ) is symmetric. Hence the factthat any bilayered profile always admits ZWS (see e.g. Fig. 2b in § Q ( y ) that can be definedas symmetric over some interval [ y , y + 1] . • Uniform normal impedance: Z ≡ ρ ( y ) µ ( y ) = const at any y ∈ [0 , . Let k = 0 . The coefficient in (4) at k = 0 is Z ( (cid:101) y ) = Z ( (cid:101) y ) , which is constant at Z ( y ) = const by virtue of µ > . Alternatively, note from (10) that Q ( y ) with k = 0 and Z = const has constant eigenvectors. Either of these observations readily shows that, for k = 0, a dependence of ω on K > K ∈ [0 , π ]) is a straight line and thusall stopbands are ZWS, that is, there is no stopbands at all. The only difference with thecase of constant ρ and µ is the slope of ω ( K,
0) which is specified as follows: ω ( K,
0) = KZ / (cid:104) ρ (cid:105) = K/Z (cid:10) µ − (cid:11) , (27) • Uniform speed: c ≡ µ ( y ) /ρ ( y ) = const at any y ∈ [0 ,
1] ( µ ( y ) is arbitrary).The Lyapunov function is then ∆( ω , k ) = ∆( ω − c k , , and consequently ω n ( K, k ) = (cid:112) ω n ( K,
0) + c k . (28)Hence if ω n ( πm,
0) with m = 0 or 1 is a zero-width stopband, that is, if ω n ( πm,
0) = ω n +1 ( πm, ω n ( πm, k ) = ω n +1 ( πm, k ) ∀ k, i.e. the entire line (cid:0) ω n ( πm, k ) , k (cid:1) for any k ∈ R is a locus of ZWS. Note from (28) and (20) that the first cutoff (which isnot a ZWS) is ω (0 , k ) = ck = ω N , ( k ), where ω N , ( k ) is the first Neumann solution for y ∈ [0 , Uniform normal impedance and speed: Z = const and c = const at any y ∈ [0 , . Now Eqs. (27) and (28) together imply that all stopbands are ZWS for any k ∈ R . Notethat the inverse statement is true under an additional condition of absolute continuity of Z , by the Borg theorem [5]. ∆ Theorem 11
The derivatives of ∆( ω , k ) at any ω , k ∈ C (hence in both the passbandsand the stopbands at ω , k ∈ R ) are given by the formula ∂ n + m ∆( ω , k ) ∂ ( ω ) n ∂ ( k ) m = 12 ( − i ) n i m n ! m ! (cid:90) d ς (cid:90) ς d ς . . . (cid:90) ς n + m − d ς n + m × F ( ς , . . . , ς n + m ) M ( ς n + m + 1 , ς ) M ( ς , ς ) . . . M ( ς n + m − , ς n + m ) , (29) where M ( y i , y j ) is a right off-diagonal component of the matricant M ( y i , y j ) , and (cid:122) ( ς , . . . , ς n + m ) ≡ (cid:88) σ ∈ Ω f σ ( ς ) . . . f σ n + m ( ς n + m ) , f ( ς ) ≡ ρ ( ς ) , f ( ς ) ≡ µ ( ς ) ;Ω ≡ (cid:110) ( σ , . . . , σ n + m ) : σ i = 0 , (cid:88) σ i = m (cid:111) , (30) i.e. Ω is a set of C nn + m = ( n + m )! /n ! m ! permutations of a set ( σ , . . . , σ n + m ) , in whicheach σ i is either or and their sum is m .Proof. The expression (29) follows from the following property of matricants of relatedsystems [21]: let Q ( y ) M ( y, y ) = dd y M ( y, y ) and (cid:101) Q ( y ) (cid:102) M ( y, y ) = dd y (cid:102) M ( y, y ) where (cid:101) Q ( y ) = Q ( y ) + Q ( y ); then (cid:102) M ( y, y ) = M ( y, y ) (cid:98)(cid:90) yy [ I + M ( y , ς ) Q ( ς ) M ( ς, y ) d ς ]= M ( y, y ) + (cid:90) yy M ( y, ς ) Q ( ς ) M ( ς , y ) d ς + . . . (31)+ (cid:90) yy d ς . . . (cid:90) ς j − y d ς j M ( y, ς ) Q ( ς ) M ( ς , ς ) Q ( ς ) . . . M ( ς j , y ) + . . . . Next note that Q (cid:0) y ; ω , k (cid:1) ≡ Q (cid:2) ω , k (cid:3) defined by (10) is linear in both ω and k . Denote small perturbations of ω and k by ε ω and ε k . From (10) , Q (cid:2) ω + ε ω , k + ε k (cid:3) = Q (cid:2) ω , k (cid:3) + i ( µ ε k − ρε ω ) Γ , Γ = (cid:18) (cid:19) . (32)15quation (31) with Q ≡ i ( µ ε k − ρε ω ) Γ is therefore a Taylor series of (cid:102) M ≡ M (cid:2) ω + ε ω , k + ε k (cid:3) about the point ε ω = 0 , ε k = 0, and hence the derivatives of the monodromy matrix M (1 , ω and k are ∂ n + m M (1 , ∂ ( ω ) n ∂ ( k ) m = ( − i ) n i m n ! m ! (cid:90) d ς . . . (cid:90) ς n + m − d ς n + m × F ( ς , . . . , ς n + m ) M (1 , ς ) ΓM ( ς , ς ) Γ . . . M ( ς n + m ,
0) (33)with F defined in (30). Note that F = ρ ( ς ) . . . ρ ( ς n ) at m = 0 and F = µ ( ς ) . . . µ ( ς m )at n = 0 . Equation (33) and the definition ∆( ω , k ) = tr M (1 ,
0) together imply ∂ n + m ∆( ω , k ) ∂ ( ω ) n ∂ ( k ) m = 12 ∂ n + m tr M (1 , ∂ ( ω ) n ∂ ( k ) m = ( − i ) n i m n ! m !2 (cid:90) d ς . . . (cid:90) ς n + m − d ς n + m × F ( ς , . . . , ς n + m ) tr (cid:2) M ( ς n + m + 1 , ς ) ΓM ( ς , ς ) Γ . . . M ( ς n + m − , ς n + m ) Γ (cid:3) , (34)where we have used the identity tr [ M (1 , ς ) . . . M ( ς n + m , M ( ς n + m , M (1 , ς ) . . . ]and the fact that M ( ς n + m ,
0) = M ( ς n + m + 1 ,
1) due to periodicity. By definition of Γ , MΓ = (cid:18) M M (cid:19) ⇒ tr (cid:104) M ( i ) Γ . . . M ( k ) Γ (cid:105) = M ( i )2 . . . M ( k )2 , (35)which reduces (34) to the desired form (29). (cid:4) Corollary 12
The first-order derivatives of ∆( ω , k ) follow from (29) as ∂ ∆ ∂ ( ω ) = 12 (cid:90) ρ ( y ) m ( y )d y, ∂ ∆ ∂ ( k ) = − (cid:90) µ ( y ) m ( y )d y, (36) where im ( y ) = M ( y + 1 , y ) , see (15). Interestingly, the expression (29) for any derivative of ∆( ω , k ) involves, apart from ρ ( y ) and/or µ ( y ), only a single, right off-diagonal, element M ( ς i , ς j ) of the matricant.Recall that Re M = 0 by (13) , which conforms that (29) is real as it must be. Next wewill obtain a different representation for the first derivatives of ∆( ω , k ) that is expressedvia an eigenfunction u ( y ) of (5). In contrast to (29), this representation is restricted tothe passbands (cid:12)(cid:12) ∆( ω , k ) (cid:12)(cid:12) ≤ ω , k ∈ R . We note that the components ofeigenvectors of M (1 , M (1 , Theorem 13
The first derivatives of ∆( ω , k ) within the open passband intervals ∆ ∈ ( − , (and hence ω , k ∈ R ) satisfy the formulas ∂ ∆ ∂ ( ω ) = sin K w + Tw (cid:90) ρ ( y ) | u ( y ) | d y, ∂ ∆ ∂ ( k ) = − sin K w + Tw (cid:90) µ ( y ) | u ( y ) | d y, (37)16 here w is an eigenvector of M (1 , corresponding to the eigenvalue q = e iK , and u ( y ) isthe first component of the vector η ( y ) = M ( y, w = ( u, iµ u (cid:48) ) T . At the cutoffs ∆ = ± , Eq. (37) yields zero derivatives in the exceptional case of a ZWS, and is otherwise modifiedto ∂ ∆ ∂ ( ω ) = 12 i w + d Tw g (cid:90) ρ ( y ) | u ( y ) | d y, ∂ ∆ ∂ ( k ) = − i w + d Tw g (cid:90) µ ( y ) | u ( y ) | d y, (38) where w d and w g are the proper and generalized eigenvectors of M (1 , that realize itsJordan form (see (44)), and u ( y ) is equal to the first component of the vector η ( y ) = M ( y, w d . Proof of (37).
The monodromy matrix M (1 ,
0) at | ∆ | (cid:54) = 1 has distinct eigenvalues q (cid:54) = q − and hence linear independent eigenvectors w , w . Specify their numbering as M (1 , w = q w , M (1 , w = q − w with q = e iK (cid:54) = q − = e − iK . (39)According to (31) and (32), ∂ M (1 , ∂ ( ω ) = (cid:90) M (1 , y ) ∂ Q ( y ) ∂ ( ω ) M ( y, y = − i M (1 , (cid:90) P ( y )d y, where P ( y ) ≡ ρ ( y ) M − ( y, ΓM ( y, (cid:0) ⇒ tr P ( y ) = ρ ( y )tr Γ = 0 (cid:1) . (40)Hence, the derivative of ∆ = tr M (1 ,
0) at | ∆ | (cid:54) = 1 is ∂ ∆ ∂ ( ω ) = 12 i (cid:20) q (cid:90) P ( y )d y + 1 q (cid:90) P ( y )d y (cid:21) = sin KT (cid:90) P ( y )d y, (41)where P is the upper diagonal element of P ( y ) in the base of vectors w and w . Forthe passband case ∆ ∈ ( − ,
1) being considered, the identity M − = TM + T (see (13) )implies that w +1 Tw = 0; w +1 Tw , w +2 Tw (cid:54) = 0 (cid:2) ( w +1 Tw )( w +2 Tw ) < (cid:3) . (42)Using (42), the equality w +1 TM − = ( Mw ) + T (following from (13) ) and the definitionof Γ given in (32), we find that P ( y ) (cid:12)(cid:12)(cid:12) ∆ ∈ ( − , = w +1 TP ( y ) w w +1 Tw = ρ ( y ) η +1 ( y ) TΓ η ( y ) w +1 Tw = ρ ( y ) | u ( y ) | w +1 Tw , (43)where η ( y ) = M ( y, w = ( u, iµ u (cid:48) ) T . Based on the numbering in (39) it follows that η (1) = e iK w and so u is an eigenfunction of (5) (see Corollary 3). Substituting (43)into (41) and setting w defined in (39) as w ≡ w leads to (37) . The proof of (37)
17s the same. Note that the sign alternation (23) of both derivatives at successive cutoffsis described in (37) by the factor ( w + Tw ) − sin K as follows: using K ∈ [0 , π ] impliessin K ≥ w + Tw (due to switching between right- and leftwardmodes at successive cutoffs); while using unrestricted K > w + Tw < K. (cid:4) Proof of (38) . Consider a cutoff ∆ = ± M (1 , . Denote M (1 , w d = q d w d , M (1 , w g = q d w g + w d at ∆ ≡ q d = ± , (44)which defines (not uniquely) the pair w d and w g as a basis in which M (1 ,
0) at ∆ = ± ∂ ∆ ∂ ( ω ) = 12 tr ∂ M (1 , ∂ ( ω ) = 12 i (cid:90) P ( y )d y, (45)where P is the left off-diagonal of P ( y ) at ∆ = ± w d and w g . Theidentity M − = TM + T for a non-semisimple M (1 ,
0) implies that w + d Tw d = 0; w + d Tw g (cid:54) = 0 (cid:2) Re w + d Tw g = 0 for det M = 1 (cid:3) . (46)By (46) and the definition (40) of P ( y ), P ( y ) (cid:12)(cid:12)(cid:12) ∆= ± = w + d TP ( y ) w d w + d Tw g = ρ ( y ) η + d ( y ) TΓ η d ( y ) w + d Tw g = ρ ( y ) | u ( y ) | w + d Tw g , (47)where η d ( y ) = M ( y, w d = ( u, iµ u (cid:48) ) T . Inserting (47) in (45) provides (38) . The proofof (38) is the same. (cid:4) Note that (38) can also be obtained directly from (37) by taking its limit as | ∆ | < | ∆ | = ± . To do so, proceed from (39) with q, q − tending to q d . It is al-ways possible to choose w , w so that they have w d as a common limit and then( w − w ) / (cid:0) q − q − (cid:1) tends to w g , where w d and w g satisfy (44). By using this limit-ing definition of w g and the property w +1 Tw = 0 (see (42) ), the limit of the pre-integralfactor in (37) with w ≡ w corresponding to q = e iK is found to besin K w +1 Tw = q − q − i w +1 T ( w − w ) → ∆ →± i w + d Tw g . (48)The factor w + d Tw g may also be expressed in terms of the elements M i (1 , ≡ M i ofthe matrix M (1 ,
0) which satisfies (13). Using (44) yields two alternative forms of thisexpression as follows: w + d Tw g = | w d | M ∗ | M − q d | + | M | = | w d | M ∗ | M − q d | + | M | . (49)18f M , M (cid:54) = q d then M , M (cid:54) = 0 , and so both formulas in (49) are equivalent, whichfollows from tr M (1 ,
0) = 2 q d , det [ M (1 , − q d I ] = 0 and (13) . If M = q d hence M = q d (or vice versa), then either M = 0 or M = 0, as occurs for instance if Q ( y ) is evenabout the midpoint of the period [0 , § M , M is ruled out for a non-semisimple M (1 , Corollary 14
The right-hand sides of (36) are equal to those of (37) in the passbands ∆ ∈ ( − , , and to those of (38) at the cutoffs ∆ = ± (unless the cutoff is a ZWS). m ( y ) An important role of the function m ( y ) defined in (15) is revealed by the fact that,according to (36), the first derivative of ∆( ω , k ) in ω or k is an integral of m ( y ) witha positive weight factor ρ ( y ) or µ ( y ) . Recall also that zeros of m ( y ) are the Dirichletsolutions for the interval [ y, y + 1], see § Theorem 15
The continuous function m ( y ) = m ( y +1) satisfies the following properties:(i) if ∆( ω , k ) ∈ ( − , then m ( y ) has no zeros for y ∈ [0 , ; (ii) if ∆( ω , k ) = ± then m ( y ) ≥ for any y ∈ [0 , or m ( y ) ≤ for any y ∈ [0 , ; (iii) if ∆( ω , k ) / ∈ ( − , and ω , k ∈ R , then m ( y ) has only finite number of zeros in [0 , .Proof. Consider (i).
Suppose that ∆ ∈ ( − ,
1) and there exists (cid:101) y such that m ( (cid:101) y ) = 0 . Then M ( (cid:101) y + 1 , (cid:101) y ) has eigenvalues m ( (cid:101) y ) and m ( (cid:101) y )(= m − ( (cid:101) y ) by det M = 1). Therefore,with reference to Remark 1, ∆ = (cid:2) m ( (cid:101) y ) + m − ( (cid:101) y ) (cid:3) , where m according to (15) is real(since ω , k ∈ R by Lemma (5)). Hence | ∆ | ≥ , which contradicts the initial assumption . The statement (ii) follows from (i) and the analyticity of ∆( ω , k ) . Consider (iii).
Firstnote an identity M (cid:48) ( y + 1 , y ) = Q ( y ) M ( y + 1 , y ) − M ( y + 1 , y ) Q ( y ) ⇒ m (cid:48) ( y ) = m ( y ) − m ( y ) µ ( y ) , (50)where (cid:48) ≡ d / d y (if y is a point discontinuity of a piecewise continuous Q ( y ) , then d / d y is aright or left derivative). Since µ ( y ) >
0, it follows that m (cid:48) ( y ) = 0 iff m ( y ) = m ( y ) . Nowlet us suppose the inverse of (iii) , i.e., that ∆ / ∈ ( − ,
1) admits the existence of an infinite set { y n } ∞ for which m ( y n ) = 0. Without loss of generality we may assume that lim n →∞ y n = y ∈ [0 , m ( y ) = 0 and m (cid:48) ( y ) = 0. As shown above, m ( y ) = 0 yields m ( y ) = m − ( y ) and so we have ∆ / ∈ ( − ,
1) for ∆ = (cid:2) m ( y ) + m − ( y ) (cid:3) / ∈ ( − , m is real due to ω , k ∈ R . It therefore follows that m ( y ) = m − ( y ) (cid:54) = m ( y ) . According to (50), this contradicts m (cid:48) ( y ) = 0 . (cid:4) The above result together with Eq. (36) provides a simple criterion for a ZWS, whichcomplements Proposition 9. 19 roposition 16
The following statements are equivalent: (i) ( ω, k ) is a ZWS; (ii) m ( y ) =0 for any y. Proof.
Assume (i).
Then M (1 ,
0) = ± I by Proposition 9. Hence by (16) M ( y + 1 , y ) = ± I and so m ≡ , which is (ii). Now assume (ii).
It requires that ∆ = ± ∂ ∆ /∂ ( ω ) = 0 by Eq. (36) . According to Proposition 9, ∆( ω , k ) = ± ,∂ ∆( ω , k ) /∂ ( ω ) = 0 implies that ( ω, k ) is a ZWS, which is (i). (cid:4) Interestingly, the function m ( y ) , whose zeros are the Neumann solutions for the in-terval [ y, y + 1], shares some, but not all, of the properties of m ( y ) . For instance, m ( y )displays the same properties (i) , (ii) stated by Theorem 15 for m ( y ) but it does nothave the property (iii) . The dissimilarity stems from the fact that (50) yields m (cid:48) ( y ) = (cid:0) µ k − ρω (cid:1) ( m − m ) , where, in contrast to (50) , the first factor is not sign-definite.Also the derivatives of ∆( ω , k ) are not expressible via m ( y ) as they are via m ( y ) in(36). As a result, Proposition 16 does not hold for m ( y ) in the sense that while it is truethat m ( y ) = 0 for any y if ( ω, k ) is a ZWS, the inverse statement is not. An immediatecounter-example is the point ω = 0 , k = 0 , where m ( y ) = 0 for any y by (20) but thispoint is not a ZWS; moreover, the model case µ ( y ) /ρ ( y ) = const ≡ c mentioned in § m ≡ ω (0 , k ) = ck (see (28)) which has no ZWS points.Thus, the Dirichlet solution ω D ,n ( k ) for [ y, y + 1] does not depend on y only if ( ω D ,n , k ) isa zero-width stopband, but the same is not generally true for the Neumann solutions. ω n ( K, k ) In this Section, we address the multisheet surface ω n ( K, k ) = (cid:112) ω n ( K, k ) ( ≥
0) which isdefined by Eq. (18), and study the curves in its cuts taken at constant K , constant k andconstant ω . Remark 17
If Eq. (18) with either K or k or ω being fixed defines a differentiable function,then its derivative of any order can be expressed in terms of partial derivatives of ∆( ω , k ) given in (29). Below we examine in detail the first non-zero derivatives. The higher-order ones are easy toobtain in a similar fashion by differentiating (18). It is understood hereafter that ω, k ∈ R . By (18), ω n ( K, k ) = ω n ( − K, k ) = ω n ( K, − k ) which permits confining considerations toRe K (cid:62) , k (cid:62) . ω n ( k ) for fixed K Consider the dependence of ω n ( k ) ≡ ω n ( K, k ) for fixed K , Fig. 2. By Eq. (18), the branches ω n ( k ) are defined as level curves ∆( ω , k ) (= cos K ) = const, which lie in the passbands forfixed K ∈ R ⇔ | ∆ | ≤ K ∈ π Z + i ( R \ ⇔ | ∆ | > Figure 2: (a) (left) The curves ω n ( k ) ≡ ω n ( K, k ) at different fixed K for the profile(21). (b) Sections of the curves for the piecewise constant profile defined by µ = µ =1 , ρ = 1 for y ∈ [0 , /
2) and µ = µ = 12 , ρ = 2 for y ∈ (1 / , Proposition 18 If ω (cid:54) = 0 and ∂ ∆ /∂ ( ω ) (cid:54) = 0 , then d ω n d k = kω n d ω n d ( k ) = − kω n ∂ ∆ /∂ (cid:0) k (cid:1) ∂ ∆ /∂ ( ω ) , (51)21 here by (36), (37) and (38) d ω n d ( k ) = (cid:82) µ ( y ) m ( y )d y (cid:82) ρ ( y ) m ( y )d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ∈ R or K ∈ π Z + i R = (cid:82) µ ( y ) | u n ( y ) | d y (cid:82) ρ ( y ) | u n ( y ) | d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K ∈ R . (52)In addition, d ω d k (cid:12)(cid:12)(cid:12) ω =0 k =0 = (cid:115) (cid:104) µ (cid:105)(cid:104) ρ (cid:105) ; d ω n d k (cid:12)(cid:12)(cid:12) ω (cid:54) =0 k =0 = 0 , d k d ω (cid:12)(cid:12)(cid:12) ω =0 k (cid:54) =0 = 0 . (53)The former equality follows from (19) or else from (52) where m ( y ) and u ( y ) are constantat ω, k = 0 in view of (20). The two other equalities in (53) follow from (51) andd ω n / d (cid:0) k (cid:1) (cid:54) = 0 (note that ω = 0 , k (cid:54) = 0 belongs to the stopband area where (52) applies,see Fig. 2a).For K ∈ R , the excluded case ∂ ∆ /∂ ( ω ) = 0 in (51) is related to ZWS discussed in § ∂ ∆ /∂ ( ω ) at K ∈ R becomes zero then so does ∂ ∆ /∂ (cid:0) k (cid:1) and their simultaneous vanishing implies a ZWS. Barring extraordinary cases mentionedin 3.3.3, ZWS is an intersection point ( ω, k ) zws of two analytic curves ω n ( k ) (as rigorouslyconfirmed in Proposition 19 below), so there exist two derivatives at ( ω, k ) zws . Their valuescan be determined by continuity from either of equations (52) applied in the vicinity of( ω, k ) zws . Note that Eq. (52) is not defined strictly at ( ω, k ) zws (where m ( y ) = 0 ∀ y , seeProposition 16) while Eq. (52) is, provided that u n ( y ) implies two different eigenfunctionsfrom a subspace corresponding to two intersecting curves ω n ( k ) at ( ω, k ) zws . Proposition 19
The curves ω n ( k ) for fixed K ∈ R are monotonically increasing at k > .Proof. The function ω n ( k ) is analytic for any K ∈ R since A K ( k ) is a family of analyticoperators of Kato’s type A [12]. Hence if ∂ω n /∂ ( k ) = 0 for some real k , then there existscomplex (cid:101) k in the vicinity of k for which ω = ω n (cid:0)(cid:101) k (cid:1) is real. But this would mean thatthe operator B K ( ω ) has a complex eigenvalue k equal to (cid:101) k , which is impossible. Thus ω n ( k ) at K ∈ R is a monotonic function. It increases by virtue of (52) . To provide a fullyself-consistent proof within the operator approach, note that (52) can also be obtained byapplying the perturbation theory [15] to A K given by (6), so thatd ω n d( k ) = dd( k ) ( A K u n , u n ) ρ (cid:107) u n (cid:107) ρ = 1 (cid:107) u n (cid:107) ρ (cid:0) d A K d( k ) u n , u n (cid:1) ρ = (cid:82) µ ( y ) | u n ( y ) | d y (cid:82) ρ ( y ) | u n ( y ) | d y . (cid:4) (54)Consider the example plotted in Fig. 2. It demonstrates monotonicity of the curves ω n ( k ) ≡ ω n ( K, k ) at fixed K ∈ R by tracing the cutoff curves at K = 0 , π ( ⇔ | ∆ | = 1)and the curves at K = π/ ⇔ ∆ = 0) within the passbands. Figure 2 also showsthat, by contrast, the curves ω ( k ) ≡ ω ( K, k ) in the stopbands, i.e. at fixed complex22 ∈ π Z + i ( R \
0) ( ⇔ the level curves | ∆ | = const > ∂ ∆ /∂ ( ω ) = 0 (see (51), (52) ). Inany stopband except the lowest one, there exists a pair of curves ω ext ( k ) and k ext ( ω ) , onwhich (cid:12)(cid:12) ∆( ω , k ) (cid:12)(cid:12) = cosh (Im K ) has maxima in ω and in k (in k at k (cid:54) = 0), respectively.Hence each stopband except the lowest must contain looped curves ω ( k ) with a verticaltangent as they cross the curve ω ext ( k ) - unless the latter fully merges with k ext ( ω ) as inthe model case µ ( y ) /ρ ( y ) = const mentioned in § ω ext ( k ) and k ext ( ω ) mayintersect within a given stopband thus indicating a saddle point or an absolute extremum of∆( ω , k ) (the latter is exemplified in Fig. 2b, see the family of closed level curves | ∆ | > ω ext ( k ) and k ext ( ω ) cannot contact the cutoff curves except at the pointof a ZWS (see Fig. 2b), which is always a saddle point of ∆( ω , k ).It is shown in Appendix A3 that the lower bound for the branches ω n ( k ) at K ∈ R is min y ∈ [0 , (cid:112) µ /ρ. In the remainder of this subsection we prove that this bound is alsoa common limit of ω n ( k ) . To do so, it is convenient to introduce the velocity v n = ω n /k. First we specify the derivative of v n ( k ) in order to demonstrate its monotonicity (note thatit is easy to similarly obtain sign-definite derivatives at fixed K ∈ R for any other optionalchoice of the pair of spectral parameters among ω, k and v or s = v − ). Lemma 20
Let K ∈ R , n ∈ N be fixed. Then v n ( k ) ≡ ω n (cid:0) k (cid:1) /k is a decreasing functionwith derivative d v n d( k ) = − k (cid:82) µ | u (cid:48) n ( y ) | d y (cid:82) ρ | u n ( y ) | d y < , (55) where u n and u (cid:48) n are defined by η ( y ) ≡ ( u, iµ u (cid:48) ) T = M ( y, w taken at ω n (cf. (37)).Proof. Multiply Eq. (2) by u (= u n ), integrate by parts and divide the result by k , toyield v n (cid:90) ρ | u n ( y ) | d y = 1 k (cid:90) µ | u (cid:48) n | d y + (cid:90) µ | u n | d y. (56)Substituting from (56) along with (54) into d ω n / d( k ) = k d v n / d( k ) + v n leads to (55).The same result follows by applying the perturbation theory [15] similarly as in (54),whence d v n / d( k ) = − (( µ u (cid:48) n ) (cid:48) , u n ) ρ /k (cid:107) u n (cid:107) ρ and integrating by parts yields (55). (cid:4) Proposition 21
Let K ∈ R be fixed. Then for any n ∈ N lim k →∞ ω n k = min y ∈ [0 , µ ( y ) ρ ( y ) . (57)23 roof. Rewrite (2) in the form − ( µ u (cid:48) ) (cid:48) + k (cid:18) µ ρ − ω k (cid:19) ρu = 0 . (58)where v = ω /k . For any fixed v ≡ α > min (cid:112) µ /ρ, the coefficient ( µ /ρ ) − v changessign on the interval [0 ,
1] and hence there exist infinitely many distinct values k > . The latter means that any curve v n ( k ) , n ∈ N , intersects theline α ( k ) ≡ α for any α > min (cid:112) µ /ρ. Combining this statement with the above-mentionedfacts that all v n ( k ) are decreasing and have the lower bound min (cid:112) µ /ρ yields (57). (cid:4) It is noteworthy that there is no common limit for a finite spectrum of eigenvalues of adiscrete Schr¨odinger operator with a large potential [14]. K /πω k = 1 ( ) K ω ( ) K ω ( ) K ω ω ω ω ω ω ω (a) K /π ω = 8 k ( ) K k ( ) K k ( ) K k (b)
Figure 3:
The Floquet branches ω n ( K ) ≡ ω n ( K, k ) at fixed k = 1 . (b) Real isofre-quency branches K j ( k ) at fixed ω = 8. The same profile (21) is used. The cutoffvalues of ω in (a) and of k in (b) can be compared with Figs. 1 and 2a. ω n ( K ) for fixed k Consider the function ω n ( K ) ≡ ω n ( K, k ) implicitly defined by Eq. (18): ∆( ω , k ) = cos K at fixed k . Since ω n ( k ) is periodic and even, it suffices to deal with one-half of the Brillouinzone Re K ∈ [0 , π ], see Fig. 3a. For brevity, denote the cutoff values ω n ( πm, k ) of ω n ( K, k )as ω n ( πm, k ) ≡ ω n,m , m = 0 , . (59)Let us indicate the passbands and stopbands of ω n ( K, k ) by Im K = 0 and Im K (cid:54) = 0,respectively (the latter being short for K = πm + i Im K (cid:54) = πm ). Explicit expressions forthe first non-zero derivative of ω n ( K ) readily follow by expanding both sides of (18) andinvoking the formulas for ∂ ∆ /∂ ( ω ) obtained in § for24eal K (see below) can also be obtained by means of perturbation theory [15] applied toan appropriately modified form of (2), (3) with an operator explicitly dependent upon K . Proposition 22
If either (i) Im K = 0 and K (cid:54) = πm (hence ∂ ∆ /ω (cid:54) = 0 by Proposition 6)or (ii) Im K (cid:54) = 0 and ∂ ∆ /∂ω (cid:54) = 0 , then d ω n d K = − sin K ( ∂ ∆ /∂ω ) ω n , (60) where sin K = √ − ∆ and ∂ ∆ /∂ω = 2 ω∂ ∆ /∂ ( ω ) is given by (36 ) or (37) for (i) andby (36) for (ii). If K = πm and ∂ ∆ /∂ω (cid:54) = 0 , then d ω n d K = 0 , d ω n d K = ( − m +1 ( ∂ ∆ /∂ω ) ω n,m , (61) where ω n,m = ω n,m ( k ) are the roots of equation ∆( ω , k ) = ( − m and ∂ ∆ /∂ω is given by(36) or (38) . Consider the special cases where ∂ ∆ /∂ω = 0. Let K = πm and ∂ ∆ /∂ω = 0 at ω (cid:54) = 0 , which implies a cutoff ω n,m corresponding to a ZWS. Thend ω n d K = ( − m + n +1 (cid:14)(cid:113) ( − m +1 ( ∂ ∆ /∂ω ) ω n,m . (62)Next let Im K (cid:54) = 0 and ∂ ∆ /∂ω = 0 , which defines the point ω ≡ ω ext in a stopband atwhich | ∆( ω ) | = cosh (Im K ) reaches its maximum | ∆ ext | > § K ( ω ) satisfies (d Im K/ d ω ) ω ext = 0 andd Im K d ω = ( − m (cid:0) ∂ ∆ /∂ω (cid:1) ω ext (cid:112) ∆ − < K > . (63)The explicit form of ∂ ∆ /∂ω , which appears in (62), (63) and is negative at m = 0 andpositive at m = 1, is defined by (29). It can be written in the following equivalent forms ∂ ∆ ∂ω = 4 ω ∂ ∆ ∂ ( ω ) = − ω (cid:90) d y (cid:90) y ρ ( y ) ρ ( y ) M ( y + 1 , y ) M ( y, y ) d y = − ω (cid:90) d y (cid:90) y +1 y ρ ( y ) ρ ( y ) M ( y + 1 , y ) M ( y , y )d y (64)= − ω (cid:90) d y (cid:90) ρ ( y ) ρ ( y + y ) M ( y + 1 , y + y ) M ( y + y , y )d y , ∂ ∆ /∂ω = 0 and ω (cid:54) = 0 (i.e. ∂ ∆ /∂ ( ω ) = 0) have been used. Finally, consider thecase ω = 0 , which implies ∂ ∆ /∂ω = 0 , ∂ ∆ /∂ω = 2 ∂ ∆ /∂ ( ω ). If both ω = 0 and k = 0( ⇒ K = 0), then referring to (19), the derivative (62) for m = 1 reduces tod ω d K = 1 (cid:14)(cid:113) (cid:104) ρ (cid:105) (cid:10) µ − (cid:11) . (65)If ω = 0 and k > ⇒ K = i Im K (cid:54) = 0), then (d Im K/ d ω ) ω =0 = 0 and (63) becomesd Im K d ω = 2 (cid:2) ∂ ∆ /∂ ( ω ) (cid:3) ω =0 (cid:112) ∆ (0 , k ) − , (66)where (cid:2) ∂ ∆ /∂ ( ω ) (cid:3) ω =0 < .It is evident from Eq. (60) that the Floquet branches ω n ( K ) for any fixed real k aremonotonic in K ∈ [0 , π ]. For completeness, let us also mention two important results fromthe general theory of Schr¨odinger equation [15, 19, 11] that extend to the case of Eq. (2)with fixed k. These results state that Im K ( ω ) is a convex function and that each branch ω n ( K ) has one and only one inflection point in K ∈ [0 , π ], unless it is the lowest branch ω ( K ) at k = 0 or a branch bounded by a ZWS at either or both cutoffs K = πm, in whichcase there is no inflection points. Note in conclusion that Eqs. (61) and (62) provide anexplicit definition for the near-cutoff asymptotics of branches ω n ( K ) that were analyzed in[7] by a different means (the scaling approach, also extended in [7] to 2D-periodicity). K ( k ) for fixed ω Consider the dependence of K ( k ) = arccos ∆( ω , k ) on k ≥ ω. Let the branches K j ( k ) ∈ [0 , π ] for real K be numbered in the order of increasing k. Since ω n ( k ) ≡ ω n ( K, k )is strictly increasing in k (see Fig. 2), the number of real branches K j ( k ) at any fixed value ω is fully defined by its position with respect to the frequency-cutoff points at k = 0: there isa single real branch K ( k ) for a fixed ω in the interval 0 < ω < ω ( π,
0) ; two real branches K ( k ) , K ( k ) for ω in ω ( π, < ω < ω (0 ,
0) ... etc. Besides, the first real branch K ( k )starts at k = 0 and spans a range [0 , π ) or (0 , π ] iff (cid:12)(cid:12) ∆( ω , (cid:12)(cid:12) < , i.e. iff the given ω isfixed within the passband at k = 0 . For example, the value ω = 8 ∈ ( ω (0 , , ω ( π, K j ( k ) with K ( k ) ∈ [0 , π ) , see Fig. 3b.Denote by k j,m ( ω ) ≡ k j,m , m = 0 , , (67)the roots of equation ∆( ω , k ) = ( − m which define the points at which K j ( k ) = πm andthe given ω is the cutoff; these points k j,m are separated by the stopband intervals | ∆ | > K (cid:54) = 0 . The explicit form of the first derivative of K ( k ) for real or complex K follows from (18) and the formulas for ∂ ∆ /∂ ( k ) in exactly the same way as that ω n ( k ) in § roposition 23 If K (cid:54) = πm and k (cid:54) = 0 , then d K d k = − ∂ ∆ /∂k sin K , (68) where ∂ ∆ /∂k (cid:54) = 0 for real K. If K j ( k ) = πm at k (cid:54) = 0 and ( ∂ ∆ /∂k ) k j,m (cid:54) = 0 , then thelocally defined inverse function k ( K ) satisfies d k d K j = 0 , d k d K j = ( − m +1 ( ∂ ∆ /∂k ) k j,m . (69) If k = 0 , then d K d k = (cid:40) , d K d k = − K (cid:2) ∂ ∆ /∂ ( k ) (cid:3) k =0 at K (cid:54) = πm, (cid:113) − m +1 [ ∂ ∆ /∂ ( k )] k =0 at K = πm. (70)Consider the implication of possibly existing ZWS. Assume that a fixed ω is a ZWS forsome k (cid:54) = 0 . This means that K j ( k j,m ) = πm and ( ∂ ∆ /∂k ) k j,m = 0 where k j,m (cid:54) = 0 . Then(69) is altered to d K j d k = ( − m + j (cid:113) ( − m +1 ( ∂ ∆ /∂k ) k j,m . (71)Now assume that a fixed ω is a ZWS at k = 0, i.e. let K = πm and (cid:2) ∂ ∆ /∂ ( k ) (cid:3) k =0 = 0 . Then d K / d k = 0 by (70), andd K d k = (cid:113) − m +1 [ ∂ ∆ /∂ ( k ) ] k =0 . (72)The second-order derivative of ∆ in (71), (72) can be obtained by differentiating (36) inthe same way as in (64). Note that ∂ ∆ /∂ ( k ) also appears in the formula analogous to(63) for d Im K/ d k at the point k ext where d Im K/ d k = 0.Thus, by (69) and (71), all real branches K j ( k ) at fixed ω have vertical tangents at theedge points K j ( k j,m ) = πm, k j,m (cid:54) = 0 (see Fig. 1b), unless the cutoff ω = ω n ( πm, k j,m ) isa ZWS in which case K j ( k ) does not make a right angle with the line K = πm. In turn,by (70) and (72), the real branch K ( k ) has a horizontal tangent at k = 0 , K (cid:54) = π and anon-zero first derivative at k = 0 , K = π , unless ω = ω n ( π,
0) is a ZWS stopband in whichcase the slope of K ( k ) vanishes at k = 0 , K = π . Remark 24
If the cutoff ω = ω n ( π, is not a ZWS, then (i) the curve K ( k ) = K ( − k ) has a kink at k = 0 ; (ii) ∇ ω ( K, k ) = at k = 0 by virtue of (53) and (61). .4 Convexity of the closed isofrequency branch K ( k ) The normal to real isofrequency branches K j ( k ) defines the direction of group velocity ∇ ω ( K, k ) which makes their shape relevant to many physical applications. In particular,negative curvature of an isofrequency curve is known to give rise to rich physical phenomenarelated to wave-energy focussing. Since the function K ( k ) = arccos ∆ with | ∆ | ≤ K ∈ [0 , π ] , no vertical line can cross twice the curve K ( k ); however, this by itselfdoes certainly not preclude a negative curvature. In fact any real branch K j ( k ) , whichextends from K j = 0 to K j = π, has vertical tangents at those edge points and hence musthave at least one inflection between them (unless the exceptional case of ZWS, see § K ( k ) if the reference ω is taken within the passband range at k = 0 and hence K ( k ) does not reach one of theedge points 0 or π. In other words, the situation in question is when K ( k ) extended bysymmetry to any real K, k ≶ closed curve.In the present subsection we address an important case of a relatively low frequency ω which is restricted to the passband below the first cutoff ω ( π,
0) at the edge of theBrillouin zone K = π at k = 0 . For any fixed ω < ω ( π, K ( k ) = arccos ∆( ω , k ) ∈ [0 , π ) that is continuous in the definition domain k ∈ [ − k , , k , ] , where k , is the least root of equation ∆ = 1 (see (67)). According to (91) , ω (cid:112) (cid:104) ρ (cid:105) / (cid:104) µ (cid:105) ≤ k , ( ω ) ≤ ω max y ∈ [0 , (cid:112) ρ ( y ) /µ ( y ) . (73)We will show that K ( k ) is strictly convex. The proof is preceded by a lemma. Lemma 25
For fixed ω < ω ( π, , derivatives of the function ∆( ω , k ) of any order in k are strictly positive at k ≥ .Proof. Let ω = 0 . Then ∆(0 , k ) > k ≥ ∂ ∆(0 , k ) /∂ ( k ) > k ≥ k ) at fixed ω satisfies the conditions of the Laguerre theorem (seeProposition 7). In other words, all zeros of ∂ ∆(0 , k ) /∂ ( k ) lie in k < < ω < ω ( π, . This means that − < ∆( ω , < ∂ ∆( ω , k ) /∂ ( k ) , which is where ∆ ≤ − , still lies in k < . Thus, if ω < ω ( π,
0) then ∂ ∆( ω , k ) /∂ ( k ) > k ≥ ∂ p ∆ /∂ ( k ) p > k ≥ p ≥ . (cid:4) Theorem 26
The curve K ( k ) is convex at any fixed ω such that ω < ω ( π, . Proof.
The second derivative of K ( k ) is K (cid:48)(cid:48) ( k ) = − (cid:0) − ∆ (cid:1) − / h, h ( k ) ≡ ∆ (cid:16) ∂ ∆ ∂k (cid:17) + (cid:0) − ∆ (cid:1) ∂ ∆ ∂k , (74)where − < ∆ < k ∈ ( − k , , k , ), see (67) . Note that ∂ ∆ /∂k = 0 at k =0. Let ω < ω ( π, h (0) = (cid:0) − ∆ (cid:1) ∂ ∆ /∂k > h (cid:48) ( k ) = ( ∂ ∆ /∂k ) +28 − ∆ (cid:1) ∂ ∆ /∂k > . Due to h (0) > h (cid:48) ( k ) > k > , it follows that h ( k ) > k > . Hence K (cid:48)(cid:48) ( k ) < − k , , k , ].Thus, K ( k ) is convex. (cid:4) The obtained result sets an important benchmark against any artefacts of approximateanalytical and/or numerical modelling of the first isofrequency curve K ( k ) = arccos ∆ , which are possible as a result of truncating series for arccos or for ∆ = tr M (1 ,
0) (see(12)). Figure 4 demonstrates an example where an approximate computation of K ( k )produces a spurious concavity. In this regard we note that Figure 1 of [20], which issketch of the generic relation between K and k for fixed but small ω , incorrectly gives thesuggestion that concavities can occur. k x K − − (a) π k x K − − (b) π Figure 4: (a) The approximate and (b) the exact first isofrequency curve K ( k ) =arccos (cid:2) tr M (1 , (cid:3) at fixed ω (= 3 . < ω ( π,
0) for a periodically piecewise constantprofile defined by µ = 1 , µ = 0 . , ρ = 0 . y ∈ [0 , /
2) and µ = 0 . , µ = 0 . ,ρ = 0 .
19 at y ∈ (1 / , . The monodromy matrix (12), which in this case is M (1 ,
0) =(exp Q ) (exp Q ) with Q j defined by (10) , is computed via the series of the co-factorexponentials, keeping four terms for each of them in the case (a) and 30 terms in thecase (b). In conclusion, a remark is in order concerning the high-frequency case where the firstisofrequency branch K ( k ) defined in k ∈ [ − k , , k , ] is accompanied by the higher-orderbranches K j ≥ ( k ) . In general, K ( k ) should stay convex and K j ≥ ( k ) should have notmore than a single inflection point. However, it seems possible to construct a theoreticalexample, though quite peculiar, of a periodic profile, for which the above is not true. Acknowledgements
The authors thank Prof. E. Korotyaev for helpful discussions. AKK acknowledges supportfrom the University Bordeaux 1 (project AP-2011).29 ppendix
A1. Properties of the operators A K and B K It is evident that the operator A K defined in (6) is symmetric for k , K ∈ R , i.e.( A K u, v ) ρ = − (cid:90) (cid:0) µ u (cid:48) (cid:1) (cid:48) v ∗ d y + k (cid:90) µ uv ∗ d y = (cid:90) µ u (cid:48) v (cid:48)∗ d y + k (cid:90) µ uv ∗ d y = − (cid:90) (cid:0) µ v (cid:48) (cid:1) (cid:48)∗ u d y + k (cid:90) µ uv ∗ d y = ( u, A K v ) ρ , (75)using the identities µ u (cid:48) v ∗ | = µ v (cid:48) u ∗ | = 0 which follow from the boundary condition (7)on u, v ∈ D K iff K is real. The proof of the symmetry of B K for ω , K ∈ R is the same.We now demonstrate that A K and B K are self-adjoint with discrete spectra σ ( A K ) = (cid:8) ω n (cid:9) ∞ and σ ( B K ) = (cid:8) k n (cid:9) ∞ corresponding to complete sets of eigenfunctions (as statedin § R K,λ = (cid:0) A K − ω (cid:1) − or R K,λ = (cid:0) B K − k (cid:1) − , where λ implies ω or k . In order to do so considerthe equivalent equations (cid:0) A K − ω (cid:1) u = g, ω / ∈ σ ( A K ) (cid:0) B K − k (cid:1) u = g, k / ∈ σ ( B K ) with u ( y ) ∈ D K , g ( y ) ∈ L ρ,µ [0 , , (76)which can be recast as η (cid:48) ( y ) − Q ( y ) η ( y ) = γ ( y ) with γ ( y ) = (cid:18) if ( y ) (cid:19) , η (1) = e iK η (0) , (77)where f = − iρg for A K , f = iµ g for B K , and η , Q are defined in (7), (10), respectively.The solution to (77) is a superposition of its partial solution η p with the solution η ( y ) ofthe corresponding homogeneous equation: η ( y ) = η p ( y ) + η ( y ) , η p ( y ) = (cid:90) y M ( y, ς ) γ ( ς ) d ς, η ( y ) = M ( y, η (0) . (78)The vector η (0) is found from the quasi-periodic boundary condition that yields η p (1) + η (1) = e iK η (0) . Thus η ( y ) = (cid:90) G ( y, ς ) γ ( ς ) d ς with G ( y, ς ) = M ( y, ς ) H ( y − ς ) − M ( y, (cid:2) M (1 , − e iK I (cid:3) − M (1 , ς ) , (79)where H ( y − ς ) is the Heaviside function and e iK is not an eigenvalue of M (1 ,
0) for thegiven ω / ∈ σ ( A K ), k / ∈ σ ( B K ). It can be checked that the Green-function tensor G ( y, ς )30atisfies the identity G ( y, ς ) = − TG + ( ς, y ) T , so that its right off-diagonal componentsatisfies G ( y, ς ) = − G ∗ ( y, ς ). By (79) , u = R K,λ g = (cid:90) G ( y, ς ; λ ) f ( ς ) d ς, where G ( y, ς ; λ ) = iG ( y, ς ) . (80)It is seen that the resolvent R K,λ is an integral (bounded) self-adjoint operator generatedby a piecewise continuous kernel. The symmetry ( R K,λ g, v ) = ( g, R K,λ v ) follows for any v ∈ D K from G ( y, ς ; λ ) = G ∗ ( ς, y ; λ ) or else from the symmetry of A K , B K . Thus R K,λ satisfies the Hilbert-Schmidt theorem and A K , B K therefore possess the above-mentionedproperties. A2. Bounds of the function ∆( ω , k ) The far-reaching properties of the analytic function ∆( ω , k ) stated in Proposition 7 followby applying Laguerre’s theorem to ∆( ω ) at any fixed k and to ∆( k ) at any fixed ω .A function satisfying Laguerre’s theorem must be an entire function of order of growthless than 2. Verification of this condition for ∆( ω , k ) requires its uniform estimationin C . The WKB asymptotic expansion (see § ω ) and ∆( k ) for, respectively, any k and ω are entire functions of order of growth . The derivation consists of two Lemmasin which the following auxiliary notation is used: f max ≡ max f ( y ) , f min ≡ min f ( y ) for f ( y ) = ρ ( y ) , µ , ( y ) and y ∈ [0 , Lemma 27
For any ω, k ∈ C , (cid:12)(cid:12) ∆( ω , k ) (cid:12)(cid:12) ≤ cosh (cid:113) µ −
11 min (cid:0) µ | k | + ρ max | ω | (cid:1) . (81) Proof . For any 2 × A with the entries ( a ..a ) , define | A | as | A | = (cid:18) | a | | a || a | | a | (cid:19) (82)and note that | (cid:81) n A n | ≤ (cid:81) n | A n | where the entrywise inequality is understood. Recall that (cid:98)(cid:82) appearing in (12) implies a product integral and is an exponential when the integrandmatrix is constant. Hence it follows from (10) , (12) and (17) that (cid:12)(cid:12) ∆( ω , k ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) tr (cid:98)(cid:82) [ I + Q ( y )d y ] (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) tr (cid:98)(cid:82) (cid:20) I + i (cid:18) − µ − ( y ) µ ( y ) k − ρ ( y ) ω (cid:19) d y (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ tr (cid:98)(cid:82) (cid:20) I + i (cid:18) µ −
11 min µ | k | + ρ max | ω | (cid:19) d y (cid:21) = cosh (cid:113) µ | k | + ρ max | ω | µ (cid:4) . (83)31he inequality (83) confirms that ∆( ω ) and ∆( k ) are entire functions with order ofgrowth not greater than in each argument. Next we demonstrate that ∆ for certain ω , k grows no slower than an exponential of power of ω and/or k . This will enableus to conclude that the order of growth of ∆( ω ) and ∆( k ) is precisely . Lemma 28
For ω , k ∈ R , (cid:12)(cid:12) ∆( ω , k ) (cid:12)(cid:12) ≥ cosh (cid:113) µ −
11 max (cid:0) µ k − ρ max ω (cid:1) for k ≥ µ −
12 min ρ max ω . (84) Proof . First introduce a class M of 2 × M = (cid:26)(cid:18) a − ia ia a (cid:19)(cid:27) , a j ≥ , j = 1 .. . (85)For two matrices A and B from M , we say that A ≥ M B iff a j ≥ b j for any j = 1 ..
4. If A ∈ M and B ∈ M then AB ∈ M also. Therefore, if A k , B k ∈ M and A k ≥ M B k for any k = 1 ..n then A .. A n ≥ M B .. B n and tr ( A .. A n ) ≥ tr ( B .. B n ) (which is easy tocheck for n = 2 and is therefore valid for any n ). We note from (10) that µ k ≥ ρ max ω implies I + Q ( y )d y ∈ M for any y ∈ [0 ,
1] and d y >
0; moreover, I + Q ( y )d y ≥ M I + i (cid:18) − µ −
11 max µ k − ρ max ω (cid:19) d y (86)and consequently∆( ω , k ) = 12 tr (cid:98)(cid:90) [ I + Q ( y )d y ] ≥
12 tr (cid:98)(cid:90) (cid:20) I + i (cid:18) − µ −
11 max µ k − ρ max ω (cid:19) d y (cid:21) = cosh (cid:115) µ k − ρ max ω µ . (cid:4) (87) A3. Bounds of the first eigenvalue ω ( K, k ) Proposition 29
For K ∈ [ − π, π ] and k ∈ R , the first eigenvalue ω ( K, k ) is bounded asfollows k min y ∈ [0 , µ ( y ) ρ ( y ) ≤ ω ( K, k ) ≤ (cid:104) µ (cid:105)(cid:104) ρ (cid:105) K + (cid:104) µ (cid:105)(cid:104) ρ (cid:105) k . (88) Proof.
Let u ∈ D K with the unit norm (cid:107) u (cid:107) ρ = 1 be the eigenfunction vector of A K corresponding to the eigenvalue ω . Then ω = ( A K u , u ) ρ = (cid:90) µ (cid:12)(cid:12) u (cid:48) (cid:12)(cid:12) d y + k (cid:90) µ | u | d y ≥ k (cid:90) µ ρ ρ | u | d y ≥ k min y ∈ [0 , µ ρ . (89)32n equivalent proof of the lower bound (89) follows by noting that the initial equation (2)yields zero as the sum of the positive operator − ( µ u (cid:48) ) (cid:48) and the operator multiplying u by (cid:0) k µ − ω ρ (cid:1) , implying that the latter factor must be negative. In order to obtain theupper bound, introduce the function v ( y ) = (cid:104) ρ (cid:105) e iKy such that v ( y ) ∈ D K and (cid:107) v (cid:107) ρ = 1.Hence ω as a minimal eigenvalue of A K satisfies ω = inf u ∈ D K , (cid:107) u (cid:107) ρ =1 ( A K u , u ) ρ ≤ ( A K v, v ) ρ = (cid:104) µ (cid:105)(cid:104) ρ (cid:105) K + (cid:104) µ (cid:105)(cid:104) ρ (cid:105) k . (cid:4) (90) Corollary 30
The bounds of the first cutoff at the centre and the edge of the Brillouinzone are, respectively, k min y ∈ [0 , (cid:115) µ ( y ) ρ ( y ) ≤ ω (0 , k ) ≤ k (cid:115) (cid:104) µ (cid:105)(cid:104) ρ (cid:105) ; ω (0 , k ) < ω ( π, k ) ≤ (cid:115) (cid:104) µ (cid:105)(cid:104) ρ (cid:105) π + (cid:104) µ (cid:105)(cid:104) ρ (cid:105) k . (91)As stated in Proposition 21, the lower bound (88) of ω ( K, k ) and hence of all curves ω n ( K, k ) for K ∈ R is also their limit at k → ∞ . Note that ω (0 , k ) ≥ ω N , ( k ) by (24),where ω N , ( k ) is the lowest branch of solutions of the Neumann problem for y ∈ [0 , k → ∞ as ω (0 , k ). In this regard, recall themodel example µ ( y ) /ρ ( y ) = const ≡ c (see § ω (0 , k ) = ω N , ( k ) = ck mergetogether with their upper and lower bounds. By (91) , unless ω (0 , k ) is a straight line, ithas an inflection point (and so does ω N , ( k )). Furthermore, the case of constant ρ, µ , isan elementary example of the equality of the upper bound in (88) and (91) . References [1]
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