Spectra of open waveguides in periodic media.
We study the essential spectra of formally self-adjoint elliptic systems on doubly periodic planar domains perturbed by a semi-infinite periodic row of foreign inclusions. We show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component, which is the union of the discrete spectra of model problems in the infinite perturbation strip; these model problems arise by an application of the partial Floquet-Bloch-Gelfand transform.
Keywords: Spectral bands, double periodic medium, unbounded periodic perturbation, open waveguide
MSC: 35P05, 47A75, 49R50, 78A50
Composite materials are used extensively in the modern engineering practice. They are mathematically interpreted, modelled and treated by applying the Floquet-Bloch-Gelfand-theory (FBG-theory in the sequel) for elliptic spectral boundary value problems with periodic coefficients in periodic domains. This approach has led to many important theoretical results and applications in topics like homogenization, diffraction in waveguides, band-gap engineering etc. The usual setting for the theory concerns purely periodic media (periodic coefficients, periodic domains), though certain types of perturbations may be allowed as well. In this paper we introduce and examine quite novel type of perturbations of a double-periodic medium with semi-infinite rows of foreign inclusions as depicted in Fig. 1, a), b). The perturbation may influence the essential spectrum of the problem, and the description of this spectrum becomes the main goal of our paper. Such perturbations of periodic lattices have not been the subject of thorough mathematical investigation, but in the physical literature they are related, for example, to defected photonic crystals, cf. [16, Ch. 7].
In the case of homogeneous media, foreign inclusions like infinite or semi-infinite strips are called open waveguides; we accept the same terminology in the periodic case. The physical meaning of the notion is clear, see [5, 6] for acoustics and [16, 17, 21, 22] for similar defects of periodic media in solid state physics and optical systems. Our approach can readily be generalized for different shapes of insertions, see Fig. 3. These types of perturbed periodic media may also appear in applications like composite materials, because of improper manufacturing or also a specific feature created on purpose. For technical simplicity we will deal with particular open waveguides as depicted in Fig. 1, but in Section 4 we describe minor modifications of our approach for shapes like in Fig. 3 as well as other generalizations, including smoothness of coefficients and the boundary.
We will consider a general elliptic spectral problem, see (1.18)–(1.19), in the perturbed domain , Fig. 1. The main result of the paper, Theorem 7 in Section 3.5, contains the following statement: the essential spectrum of the main problem is the union of two subsets, the first of which is the essential spectrum of the problem on the purely periodic domain , see Fig. 2, a), and the second one, denoted by , which is caused by the perturbation of the domain, the open waveguide , see Fig. 1. The subset equals the union of the discrete spectra of a family of model problems on a domain , Fig. 2, b), which is related to . We will discuss the structure of in more detail in Section 1.4, after presenting the notation and definitions.
1.2 Purely periodic medium.
We consider the geometry of the unperturbed perforated plane and the related spectral boundary value problem. Let be the unit open square, and let be an open set in the plane with closure and a smooth boundary of Hölder class , where . Here is a not necessarily connected set describing perforation; neither the case is excluded. We define the periodicity cells (outlined by dotted line in Fig. 1, b)
where is a multi-index and The perforated plane is covered by the periodicity cells (1.1), so it satisfies
where denotes a translation of similarly to (1.1). Assuming to be a domain, we observe that includes the boundary strip
with some and, thus, is a domain and in particular a connected set as well.
We consider the spectral problem
and its variational form
Concerning the notation, by we understand a spectral parameter and is a vector function, realized as a column, so that stands for transposition. The matrix differential operators and in (1.4) and (1.5) take the form
while is the unit outward normal vector on and is a matrix function of size with -smooth entries; is Hermitian, positive definite and 1-periodic in (it is convenient to define the coefficient matrix in the intact plane). Furthermore, is a matrix function of size and linear in , and is the null matrix. The substitution gives an -matrix of first order differential operators with constant complex coefficients, while (respectively, ) is a formally self-adjoint differential matrix operator in divergence form (resp. the corresponding Neumann condition operator); the bar in (1.7) and (1.8) indicates complex conjugation. Finally, is the natural scalar product in the Lebesgue space , and is the Sobolev space with standard norm
The last superscript in the integral identity (1.6) shows the number of components in the test function however, this index is omitted in the notation of norms and scalar products. In this way, the left- and right-hand sides of (1.6) involve scalar products in and , respectively.
We assume that is algebraically complete : there exists a such that for any row of homogeneous polynomials in of common degree one can find a row of polynomials satisfying
According to [40, § 3.7.4], this assumption yields the Korn inequality
and, hence, the sesquilinear Hermitian positive form on the left in (1.6) is closed in Moreover, the operator is elliptic and the Neumann boundary operator covers it in the Shapiro-Lopatinskii sense everywhere on (see, e.g., [35, Thm. 1.9]).
We emphasize that the results of the paper remain valid for other types of boundary conditions, in particular, for the Dirichlet conditions, cf. Section 4.1.,. A description of all admissible boundary conditions can be found in  and [35, § 1]. Moreover, the -smoothness of the boundary was assumed in order to have the elementary formula (1.11) for the operator domain and to simplify the technical computations. The results of our paper hold true in the case of uniformly Lipschitz boundaries, but the proofs would require small modifications, cf. Section 4.2.
1.3 Periodic medium with open semi-infinite periodic waveguide.
In this section we describe the geometry of the open waveguide and the full spectral problem. Let be the semi-strip with some (overshaded in Fig. 1, b). In the rectangle we introduce an open set with a smooth boundary and closure We define a semi-infinite row of holes or inclusions as depicted in Fig. 1, a) or b), respectively:
The cell is outlined in Fig. 1, a), by dotted line. We also introduce a smooth Hermitian matrix function in entries of which are supported in and become -periodic in inside the semi-strip ,
and the sum
is assumed to be positive definite in , where
In other words, we make a perturbation of coefficients and boundary inside the semi-strip , see Fig. 1. For instance, one may suppose that is the null matrix and which means filling in all holes inside cf. Fig. 1, b). Vice versa, in the case one perforates the plane with a semi-infinite row of holes, see Fig. 1, a). Even in the case of absence of holes we still call (1.14) the perforated strip, and the perforated plane in (1.2) can also contain no holes.
Notice that the Korn inequality
is still valid and can be derived by summing up inequalities of type (1.10) in the cells with ( excluded) and with
Replacing with (1.15) in (1.7) still gives an elliptic and formally self-adjoint matrix operator The same change in (1.8) yields the Neumann boundary condition operator where is regarded as the outward unit normal on . In the domain (1.16) we consider the spectral problem
and the corresponding integral identity
Since remains as a closed positive Hermitian form in the variational formulation (1.20) of the problem (1.18), (1.19) supplies it with a positive self-adjoint operator in with the differential expression and the domain
The main goal of the paper is to describe the essential component in the spectrum of ,
We emphasize that in general
and moreover, the discrete spectrum of is empty, thus,
We will identify the difference
but leave aside two interesting and important questions. First, we are not able to describe completely the component in a general perturbed problem (1.18), (1.19), although, of course concrete examples of isolated and embedded eigenvalues in can be constructed in scalar problems. Second, the existence or absence of the point spectrum (eigenvalues of infinite multiplicity) in the purely periodic problem (1.4), (1.5) remains unknown; note that this question is answered in the literature only for particular scalar problems (see, e.g., papers [46, 18, 24] and books [43, 28]). Notice that can be included in , but the latter stays unknown, too.
1.4 Discussion on the main result
In the case of the purely periodic plane , Fig. 2, a), the spectrum of the problem (1.4)-(1.5) has representation as a union of spectral bands (see (2.15), (2.16), below), which is a well-known consequence of the FBG-theory; we refer here to [27, 28, 45]. Consider for a moment the domain of Fig. 2, b) with foreign inclusions or holes, which form a periodic row, infinite in both directions. Then, the spectrum may be different from the purely periodic case, and we denote by the essential spectrum of the problem on (cf. (3.4), below). Analogously to [17, 21, 22], the increment can be detected by performing the partial FBG-transform in -direction (Section 3.1) and investigating the kernel of the model problem in the perforated strip (separated by dashed lines in Fig. 2, b); cf. (3.11), (3.9) ). This problem depends on the Floquet parameter , and, for certain values of the spectral parameter and it can have a solution in the Sobolev space Such values form the point spectrum of the model problem (3.30) for the operator .
The last set in this formula requires some comments. First, embedded eigenvalues in live inside the essential spectrum , which in turn is contained in (compare (3.32) with (2.15)). Second, depends on and therefore some points of the discrete spectrum may fall into with In any case none of the indicated points in stays in the increment component (1.25) of
We emphasize that, contrary to the case of , the lacking periodicity of the domain prevents a direct use of the partial FBG-transform, hence, the proof of (1.26) requires improved mathematical tools. The new procedure of our paper involves the construction of a parametrix and singular Weyl sequences in order to describe, respectively, the regularity field and the essential spectrum of the operator .
1.5 Cranked and branching open waveguides
Fig. 3 shows open waveguides of the shape of the letters and They appear due to perforation and perturbation of coefficients in overshadowed joints of semi-strips. In Section 4 we explain how the results of Section 3 for the -shaped case, Fig. 1, a), b), can be readily adapted to these cranked and branching open waveguides. Here, skewed branches of the - and -shaped waveguides must maintain the periodicity and thus the tangent of tilt angles has to be rational number. We do not know a formula for the essential spectrum in the irrational case.
We also emphasize that for a clear reason, no relevant perturbation in a disk with radius can affect the essential spectrum of the boundary value problem (1.18), (1.19). Moreover, assume that is a matrix decaying together with its derivatives at infinity as , and that the coefficient matrix (1.15)
still keeps the above mentioned basic properties of . This replacement of the coefficient matrix does not change .
1.6 Structure of the paper
In Section 2 we recall generally known information on the purely periodic case which will be used later on. The main interest is focused on the model problem (2.11) in the periodicity cell , which is obtained using the FBG-transform . The open periodic semi-infinite waveguide will be considered in Section 3, where we apply the partial -transform to formulate another model problem (3.30) in the perforated infinite strip with periodicity conditions on its lateral sides.
The spectra of those two model problems form the essential spectrum of the original problem (1.20). To verify the corresponding formulas (3.38) and (2.16), (3.39) we first present two types of singular Weyl sequences for and on the other hand construct a right parametrix for the formally self-adjoint problem (1.18), (1.19). This is the most involved part of our paper. To that end, we follow  and also [39, §3.4], and study an operator family for a second model problem in the weighted Sobolev spaces (Kondratiev spaces) leading to important conclusions on exponential decay properties of the solutions in the strip Finally, we glue the parametrix (3.46) from solutions of the model problems with the help of appropriate cut-off functions. The parametrix enables to prove that, for any outside the union of sets (2.16) and (3.39), the operator of the inhomogeneous problem (1.18), (1.19), cf. (3.40), is Fredholm in the Sobolev-Slobodetskii spaces; therefore such points form the intersection of the regularity field of with the semi-axis This completes the proof of Theorem 7, the central assertion in the paper.
We start the last section of the paper by describing several concrete problems in acoustics, elasticity and piezoelectricity, to which our theory may apply. However, as mentioned above, the original exact formulation of the spectral problem was simplified in several aspects, so we comment in the next subsections on certain supplementary issues in order to obtain more generality for further interesting physical applications. We finish the paper with Section 4.5, where we present small modifications of the parametrix to be applied to semi-bounded open periodic waveguides in the shape of the letters and as in Fig. 3.
2 Spectrum of the purely periodic problem
The spectrum of the operator in the purely periodic domain can be studied with the help of the FBG-transform. This will lead to the formula (2.16), the main object of Section 2. The FBG-transform is defined by
and it establishes the isometric isomorphism
and is the Lebesgue space of abstract functions in with values in Banach space and the norm
Moreover, the mapping
is an isomorphism, too. Here, is the subspace of vector functions satisfying the periodicity conditions
In what follows we shorten the notation to
The inverse FBG-transform is given by
2.2 The model problem on the periodicity cell
Owing to (2.1), we have
together with the periodicity conditions (2.6) on the exterior part of the boundary of the cell The variational formulation of problem (2.9), (2.10), (2.6) amounts to finding a number and a non-trivial vector function such that
In view of the compact embedding in the bounded domain the spectrum of the variational problem (2.11) and boundary value problem (2.9), (2.10), (2.6) is discrete and forms the unbounded monotone sequence
where multiplicities are counted. According to the general results of the perturbation theory for linear operators111A quadratic pencil easily reduces to a linear non-self-adjoint spectral family., the functions
remains an eigenpair of the same problem but at where is the unit vector of the -axis.
The above mentioned properties of functions (2.13) ensure that the spectral bands
are bounded connected closed segments in The formula
2.3 Unique solution of the inhomogeneous problem
As regards to (2.16), we now prove the inclusion . Let us consider the boundary value problem
with the data
and a fixed parameter such that
In (2.18), stands for the Sobolev-Slobodetskii space of traces with the intrinsic norm
This norm is equivalent to the following one:
Here, is the arc length element on and while can be fixed arbitrarily.
Clearly, the mapping
with the periodicity conditions (2.6). The right-hand sides meet the estimate
while the necessary information about the Sobolev-Slobodetskii space is provided by the isomorphisms (2.2), (2.4) and the formula (2.8) for derivatives. By the assumption (2.19), the problem (2.22), (2.6) has for any a unique solution denoted by
and estimated as follows:
for the unique solution of the problem (2.17) with fixed parameter (2.19). Thus, mapping (2.21) with this is indeed an isomorphism, which in particular means that belongs to the regularity field of the operator in (1.11). This coincides with the resolvent set of because the discrete spectrum is evidently empty.
2.4 The singular Weyl sequence
We next show that . Let us assume
satisfies the differential equations (1.4) and the boundary conditions (1.5), although it does of course not fall into the Sobolev space However, (2.28) is useful for constructing a singular sequence in for the operator at the point namely a sequence with the following properties:
2 weakly in as
To define the entries of this sequence, we introduce the plateau function
where is a cut-off function such that
and is taken from (1.3); therefore, in the vicinity of each component of the boundary the two-dimensional plateau function
becomes a constant, either or
The above specification of shows that
The property 1 clearly holds true. Furthermore, the weak convergence to in 2 occurs at least along a subsequence of indices , because, by (2.29) and (2.31), supp as It remains to verify the property 3 Recalling (2.32) and (2.28), we have
with and At the same time, we obtain
where stands for the commutator of the differential operator (1.7) and the multiplication operator with Owing to definition (2.29)-(2.31), the plateau function (2.31) varies only inside the union of four rectangles of size and the common area (see the overshaded frame in Fig. 4, a) ). Hence, due to periodicity in (2.28), we arrive at the inequality
as well as the property 3.
3 Spectrum of the open waveguide in periodic medium
3.1 Partial Floquet-Bloch-Gelfand-transform
Our aim is to apply the partial FBG-transform to detect the effect of the open waveguide to the essential spectrum of the problem (1.18)–(1.19). Due to the lack of periodicity, this cannot be done directly in , hence, we introduce and study in Sections 3.1–3.3 the problem in the domain , Fig. 2, b). The results of Section 3.3 will be applied in Sections 3.4–3.5 to the original problem, which leads to the proof of the main result, Theorem 7.