Trapped modes in zigzag graphene nanoribbons

We study a scattering on an ultra-low potential in zigzag graphene nanoribbon. Using mathematical framework based on the continuous Dirac model and augumented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies where the continuous spectrum changes its multiplicity and show that the trapped modes may appear for energies slightly less than a threshold and its multiplicity does not exceeds one. We prove that trapped modes do not appear outside the threshold, provided the potential is sufficiently small.


Introduction
The problem of disorder in graphene nanoribbons has been studied extensively. The main purpose of those studies is to eliminate disorder completely and produce pure high-quality graphene nanoribbons [7]. Approaching the goal of graphene nanoribbons free of impurities and other defects one can focus on production of such deliberately for the use in electronic devices. One of desired features for graphene to possess is a possibility of electron localization. Such localization is difficult to achieve due to Klein tunneling [10]. As graphene electrons behave like massless particles they undergo tunneling through barriers. However, due to interference between continuous states of the nanoribbon and a localized state of a disorder, a trapped mode can be produced. There are several types of disorder including short-range and long-range. Impurities such as vacancies and adatoms are classified as short-range type and can be described by a sharp potential, that varies on the scale shorter than graphene lattice constant (0.246 nm) [2]. On the other hand, electric or magnetic field, interactions with substrate, Coulomb charges [13], ripples and wrinkles can lead to long-range disorder described by smooth potential (a Gaussian for example). In the present studies we assume that graphene is free of short-range defects and the potential is modeled as a long range one.
There are two groups of graphene nanoribbons, that differ by the edge type and are called zigzag and armchair [4,5]. In this paper, we give a condition for existence and a choice of ultra-low potential that produces a trapped mode in zigzag graphene nanoribbon. We work within continuous Dirac model, where graphene is isotropic and its electrons dynamics can be described by a system of 4 equations [5]     where ω = E ν F is scaled energy (with E denoting energy and ν F ≈ 10 6 m s Fermi velocity), a potential P is a real-value function with compact support such that sup |P| ≤ 1 and δ is a real-value small parameter.
The number of equation is a consequence of the discrete description of graphene lattice and low energy approximation which leads to the continuous model [5,15]. In the discrete model graphene is described as a composition of two triangular interpenetrating lattices of carbon atoms (called A and B) [5]. Then low energy approximation can be done in a twofold way, close to two different energy minima (called K and K ′ ) in the graphene dispersion relation. Consequently, in the continuous model, we have two waves that describe an electron in any single point of the ribbon (A or B) coupled in two different ways, close to K (first two equations in (1)) or K ′ point (last two equations in (1)). A nanoribbon is modeled as an unit strip Π = (0, 1) × R due to rescaling. The zigzag boundary of the nanoribbon requires one wave (A) to disappear specifically at one edge and the other (B) at the other one [4] u ′ (0, y) = 0 , u(0, y) = 0 , v ′ (1, y) = 0 , v(1, y) = 0 . (2) As our potential P is assumed to be of long-range type, it can be described by a diagonal matrix with equal elements [2]. As neither the potential nor the boundary conditions couples K and K ′ valleys, the system of 4 equations can be split into two systems of 2 equations where only intravalley scattering is allowed. We consider one of them (two last equations in (1)) supplied with the boundary conditions: Energy threshold ω = ω N > 1, N = 2, 3, . . . is defined by one of maxima in zigzag dispersion relation ω −2 = κ −2 sin 2 κ and it reads d dκ κ −2 sin 2 κ = 0 (see Figure 1). The index N defines a threshold energy ω N as it indicates change of the multiplicity of the continuous spectrum from 2N − 3 to 2N − 1 for N ≥ 2. A trapped mode is defined as a vector eigenfunction (from L 2 space) that corresponds to an eigenvalue embedded in the continuous spectrum. The main result of the paper is the following theorem about existence of trapped modes in zigzag graphene nanoribbon for energies close to one of the thresholds that can be chosen arbitrary.
Second result shows that trapped modes may appear only for energy slightly smaller than threshold and that spectrum far from the threshold is free of embedded eigenvalues. Moreover their multiplicity does not exceed one. Theorem 1.2. There exist positive numbers ε 0 and δ 0 , which may depend on N, such that if (i) ω ∈ [ω N , ω N + ε 0 ] and |δ| < δ 0 , then problem (3), (4) has no trapped modes; (ii) ω ∈ [ω N − ε 0 , ω N ) and |δ| < δ 0 , then the multiplicity of a trapped mode to problem (3), (4) does not exceed 1.
Those results come from the analysis of the trapped modes in the K ′ valley (system (3), (4)), however the analysis in the K valley (first two equations in (1) with boundary conditions (2)) is analogous and requires complex conjugation only.
The continuous spectrum of the problem (3), (4) with P = 0 covers the whole real axis and, hence, its eigenvalues, if exist, possess the natural instability, that is, a small perturbation may lead them out from the spectrum and turn into points of complex resonance, cf. [3,21] and the review paper [14]. A few of approaches have been proposed to compensate for this instability and to detect eigenvalues embedded into the continuous spectrum. First of all, a simple but very elegant trick was developed in [8] for scalar problems. Namely, under a symmetry assumption on waveguide's shape an artificial Dirichlet condition is imposed on the mid-hyperplane of the waveguide which shifts the lower bound of the spectrum above and allows to apply the variational or asymptotic method to find out a point in the discrete spectrum of the reduced problem. At the same time, the odd extension of the corresponding eigenfunciton through the Dirichlet hyperplane gives an eigenfunction of the original problem so that it remains to verify that the eigenvalue falls into the original continuous spectrum. In other words, the problem operator is restricted into a subspace where it may get the discrete spectrum which becomes a part of the point spectrum in the complete setting. For vectorial problems the existence of such invariant subspaces usually demand very strong conditions on physical and geometrical properties of waveguides and therefore the trick works rather rarely or needs supplementary ideas, cf. [24,9] and [18]. Unfortunately, the Dirac equations do not possess the necessary properties and we are not able to find a way to apply the trick in our problem.
Another approach accepting formally self-adjoint elliptic systems but employing much more elaborated asymptotic analysis is based on the concept of enforced stability of embedded eigenvalues [20,21,22]. In this way, having an eigenvalue in the continuous spectrum of a waveguide with N open channels for wave propagation one can select a small perturbation of the problem by means of tuning N parameters such that, although the eigenvalue enjoys a perturbation, it remains sitting on the real axis and does not move into the complex plane. It is remarkable that, as it was shown in a different situations [20,19,21,6] and others, it is possible to take as an "initiator" of a trapped mode a particular standing wave at the threshold value of the spectral parameter and by an appropriate choice of the perturbation parameters to construct an eigenvalue which is situated near but only on one side of the threshold so that it belongs to the continuous spectrum. This method was introduced and developed in [20,21,22]. Aiming to apply it for detection of eigenvalues for the zigzag graphene nanoribbon, we unpredictably observed that the corresponding boundary value problem in whole is not elliptic (see Appendix A). As a result, many steps of the detection procedure require serious modifications.
The paper is organized as follows. In Sect. 2 we analyse Dirac model without potential. For each non-zero energy we construct all bounded solutions and identify thresholds where the dimension of the space of such solution changes. We construct also unbounded solutions near threshold and introduce a symplectic form, which will play an important role in the study of the scattering problem. These unbounded solutions are studied in Sect. 2.4, 2.8 and 2.9. In Sect. 2.10, we present a solvability result for non-homogenous problem. In Sect. 3 we add potential to the model, and consider scattering problem with the use of artificial augumented scattering matrix introduced in Sect. 3.2. In Sect. 4 we analyze trapped modes, providing in Sect. 4.1 a necessary and sufficient condition for their existence, from which in Sect. 4.2 we extract the potential description and prove Theorem 1.1. Finally, in the last section, Sect. 4.3, we analyze the multiplicity of trapped modes proving Theorem 1.2.

Problem statement
We consider problem (3) without potential (P = 0) supplied with the boundary conditions (4). Our goal is to find solutions to this problem, especially bounded ones and thereby describe continuous spectrum of the operator corresponding to (5), (4). Let us introduce the spaces Then D is a self-adjoint operator in L 2 (Π) × L 2 (Π) with the domain X × Y . We note that for ω = 0 we have There are no non-trivial solutions to (5), (4) for ω = 0. Now, assume that ω = 0, then problem (5), (4) can be written as the system We are looking for non-trivial solutions which are exponential (or possibly power exponential) in y, i.e.
Proof. Multiplying the first equation in (8) by U and integrating in x ∈ (0, 1), we have Taking the imaginary part of this equation, we get 2ℜλ Therefore, if ℑλ = 0, then ℜλ must be positive provided U = 0.
Consider the case λ 2 = ω 2 . Then there exists an exponential solution only for λ = 1 and it has the following form Let λ 2 = ω 2 . Then the solution to (8) is given by where κ satisfies and λ can be evaluated from: One can verify that κ 2 + λ 2 = ω 2 . Relations (10) can be written also as
In what follows we look only at positive ω. If ω is negative then according to the second formula (13), it can be obtained from the corresponding solution (u, v) with positive ω by taking the second component v with minus sign.
Thus for each ω ∈ (0, ∞) there exists a bounded solution to (5), (4) of the form (15). Hence the continuous spectrum of the Dirac operator D is the whole real line, i.e. σ c = R. (7) with non-real wave vector λ

Location of the wave vector λ
The forthcoming analysis, which is based on the Laplace transform of the problem (5), (4) with respect to y, requires a knowledge of location of roots to equation (11), (12) or equivalently of the equation Let us denote the left-hand side of (30) by F (λ), which is analytic with respect to λ. One can verify that We collect required properties of the roots in the following Proposition 1. (i) All roots λ of (30) are simple except of the case ω > 1 and ω is at the threshold -it is a root of (14). In this case the root λ = 1 is double and all the other roots are simple.
which implies the required assertion.

Symplectic form
There is a natural symplectic structure on the set of solutions to problem (5), (4), cf. [17,Ch. 5]. It will play important role in the study of the scattering matrix.
For two solutions w = (u, v) andw = (ũ,ṽ) of the problem (5), (4), let us define the quantity Since where Π a,b = (0, 1) × (a, b), a < b, we see that q a does not depend on a and we will use the notation q for this form.
The form q is sesquilinear and anti-Hermitian hence it is symplectic.

Biorthogonality conditions
Here we discuss the biorthogonality conditions for solutions to (5), (4). Since we are interested mostly in the case when ω = ω ε , where ω −1 ε = ω −1 N + ε , we will consider this case here. We introduce solutions to (5), (4) as follows where τ stands for + or − and j = 1, . . . , N (if j = 1, then only τ = − is admissible). Furthermore, if j = 1, . . . , N − 1, then the functions U τ j and V τ j are given by (16) and in the case j = N they are given by (23). Since where C τ θ jk is a constant, and since the form q is independent of a, we conclude that for j, k = 1, . . . , N − 1 and τ, θ = ±. Let us start with the oscillatory waves. We put Then by (33) In the case ω = ω N , we have , for waves (17), (18).

Biorthogonality conditions for the complex wave vector λ
Let us check if the waves w ε± N fullfil orthogonality conditions. For waves w ± N we have We put Then using (35) together with (25), (26) and (29) we obtian The functions a ε , b ε and c ε are real and analytic, and a ε , b ε , c ε > 0 for small ε > 0.
From the above evaluations we see that waves w ε+ N and w ε− N do not fullfil the biorthogonality conditions. That is why we consider their linear combinations where α ε is unknown constant and N ε 1 and N ε 2 are normalizing factors. Our aim is to fulfil the biorthogonality relations which implies, in particular, that Therefore, using (36), (37) and (38), we get From (40) and (36), we find also the normalization factors N ε 1 and N ε By (25), (26) and (39), we have and where and Then biorthogonality conditions take the form (40).

Properties of coefficients (43) and (44)
Accoriding to definitions (43) and (44), one can check that In the next proposition, we collect some more properties of coefficients (43) and (44), which will play an important role in the sequel.
The following quantity will play an important role where the last equality is borrowed from Proposition 2. By (50), the absolute value of d is equal to 1. Moreover the function d(ε) depends on ε ∈ [0, ε 0 ] and by definitions of α 1 and α 2 , see (43) and (44) We note also that and where .

Non-homogeneous problem
Here, we consider the non-homogeneous problem supplied with the boundary conditions u(0, y) = 0 and v(1, y) = 0, y ∈ R.
Despite the problem, which we are dealing with is not elliptic the proof of this assertion is more or less standard and we present it in Appendix C.
In what follows we assume that an integer N ≥ 2 is fixed, ω N = κ 2 N + 1, where κ N is defined in Sect. 2.3, and ω = ω ε = ω N /(1 + εω N ). Then we have the following waves . . , N − 1 are oscillatory and w ± N are of exponential growth.
Theorem 2.2. Let γ = γ N and ε N be the same positive numbers as in Proposition 1 and also (g, h) ∈ L + γ (Π) ∩ L − γ (Π). Denote by (u ± , v ± ) ∈ X ± γ × Y ± γ the solution of problem (55), (56), (57), which exist according to Theorem 2.1. Then where Proof of this Theorem is presented in Appendix D. Results similar to Theorems 2.1 and 2.2 are well-known for elliptic problems and can be found, for example, in [11], [12] and [17], but we remind that our problem is not elliptic, see Appendix A.
2 For the simplicity of the notation, we will be writing L ± σ (Π) for both spaces of functions and vectors. Here for example we write L ± σ (Π) instead of L ± σ (Π) × L ± σ (Π). This notation will be applied to the other spaces introduced later as well.

Problem statement
Here we examine the problem with a potential, prove solvability results and asymptotics formulas for solutions. Consider the nanoribbon with a potential: where D is the same as in (5), P = P(x, y) is a bounded, continuous and realvalued function with compact support in Π and δ is a small parameter. We assume in what follows that where R 0 is a fixed positive number. We assume that positive numbers γ and ε N are fixed such that ε ∈ [0, ε N ] and γ = γ N is from Proposition 1(iii), i.e.
Since the norm of the multiplication operator δP in L 2 (Π) is less than δ we derive from Theorem 2.1 the following assertion is isomorphism for |δ| ≤ δ 0 , where δ 0 is a positive constant depending on the norm on the inverse operator (D − ω ε I) −1 : L ± γ (Π) → L ± γ (Π).
We define two operators acting in the introduced spaces Some important properties of these operator are collected in the following Theorem 3.2. The operators A ± γ are Fredholm and dim kerA + where N is the same as in (63).
Proof. By Theorem 3.1, the operator (64) is isomorphic. This implies that the operator A − γ is surjective for such δ and its index and the dimension of his kernel does not depend on δ and hence equals 2N − 1 as it is in the case δ = 0.
Let us derive an asymptotic formula for the solution to the perturbed problem (60), (61).
Proof. We write (65) as According to Theorem 2.2 solutions to (67) are w ∓ : Applying relation (59) to (68), then multiplying obtained equations by χ ± respectively and summing them up, we get (69) can be written in the form (66) with R ∈ H γ + .

Augumented scattering matrix
Scattering matrix is our main tool for the identification of trapped modes [20,21,22]. Using q-form, we define incoming/outgoing waves. Scattering matrix is defined via coefficients in this combination of waves. It is important to point out that this matrix is often called augumented as it contains coefficients of the waves exponentially growing at infinity as well. Finally, by the end of the section we define a space with separated asymptotics and check that it produces a unique solution to the perturbed problem. Let Q R (w,w) = q R (w,w) − q −R (w,w).
If w = (u, v) andw = (ũ,ṽ) are solutions to (5) for |y| ≥ R 0 then using Green's formula one can show that this form is independent of R, R ≥ R 0 . We introduce two sets of waves with cutoff close to ±∞, which we will call outgoing and incoming (for physical interpretation see Appendix B) and with k = 2, . . . , N for the sign + and k = 1, . . . , N for the sign −. The reason for introducing this sets of waves is the following property where k, j = 2, . . . , N, τ, θ = ± and k, j = 1, τ, θ = −. Moreover, Thus the sign of the Q-form distinguish among W and V waves.
In the next lemma we give a description of the kernel of the operator A − γ , which will be used in the definition of the scattering matrix.
Proof. Let z ∈ ker A − γ . Since applying Theorem 2.2, we get Multiplying these equalities by χ ± and summing them up, we get We write the last relation in the form and consider the map which is linear and is denoted by J . Let us show that it's kernel is trivial. Indeed, if all C 2 jτ vanish, then where Π R = (−R, R) × (0, 1). Hence C 1 jτ = 0, which leads to z ∈ H + γ and therefore z = 0. This shows that the mapping J is invertible and we obtain existence of a basis in the form (74) together with uniqueness of coefficients S jθ kτ . The matrix of coefficients S jθ kτ in (74) is called the scattering matrix (see the footnote on the previous page concerning k = 1 and j = 1).

Block notation
We shall use a vector notation

Equation (74) in the vector form reads
here both vectors z and r has 2N − 1 elements. Matrix S = S(ε, δ) is written blockwise

Relations (72) and (73) take the form
where I is the identity matrix and O is the null matrix. Proof. Since z τ k satisfies the homogeneous equation (61), by using the Green formula one can show that Q(z, z) = 0. Therefore Consider the non-homogeneous problem (65) with f ∈ L + γ (Π). This problem has a solution w ∈ H − γ which admits the asymptotic representation which is an rearrangement of the representation (66) This motivate the following definition of the space H out γ consisting of vector functions w ∈ H − γ which admits the asymptotic representation (77) with C 2 jτ = 0. The norm in this space is defined by Now, we note that the kernel in Theorem 3.4 can be equivalently spanned by Z τ k ∈ H + γ , where the incoming and outgoing waves were interchanged (compare with (74)) andS is a scattering matrix corresponding to that exchange.
where the constant c is independent of ε ∈ [0, ε 0 ] and |δ| ≤ δ 0 . Moreover, Proof. Existence. According to Theorem 3.3 there exists a solution to (65) of the form (77). Subtracting a linear combination of the elements z ± j , we obtain a solution from H out γ .
Uniqueness is proved in the same way as the isomorphism property of the mapping J in Theorem 3.4.
One can check that the solution to this problem is given by the folowing Neumann series which represents an analytic function with respect to ε and δ. Furthermore, According to (76) and (84) which implies (85).

Necessary and sufficient conditions for the existence of trapped modes solutions
In this section we present a necessary and sufficient condition for existence of a trapped mode in terms of the scattering matrix. As before we consider problem (60), (61) assuming that |δ| ≤ δ 0 and ε ∈ (0, ε 0 ]. Proof. If w ∈ H 0 is a solution to (61) then certainly w ∈ ker A −γ and hence 5 where a = (a • , a † ) ∈ C 2N −2 and V, W and r are the vector functions from the representation of the kernel of A −γ in (3.3). Using the splitting of vectors and the scattering matrix in • and † components we write the above relation as The first term in the right-hand side contains waves oscillating at ±∞ and to guarantee vanishing of this term there we must require a • = 0. Since r vanish at ±∞ the requirement w ∈ H 0 is equivalent to the following demand: Using that S is unitary and a • = 0, we get Since |d| = 1 this implies a † S †• = 0 and the relation (86) takes the form Taking into account representations (41) and (42) and equating the coefficients for increasing exponents at ±∞ we arrive at the relations where a † = (a 1 , a 2 ). Due to the definition of d, this is equivalent to a † (S † † +dΥ) = 0 and then expression (87) decays exponentially at ±∞.

Proof of Theorem 1.1
To prove Theorem 1.1 it is sufficient to construct a potential P (subject to certain conditions) which produces a trapped mode. According to Theorem 4.1, we must find a solution to the equation To analyse this equation, we write where σ is a real number close to 0 and as according to (85) s is of order δ, then a newly introduced matrix s is of order 1. To get a relation between σ and ε, we can use (52) which gives We will seek for P and small δ > 0 that fulfil the relations Since SS * = S * S = I, we have that s • † = 0, s N + N − = 0 and Thus, (88) becomes Since the norm of these vectors is 1 and both of them close to 1 this equation is equivalent to Now to solve this equation, we fix δ = sin σ, that according to the expansion (90) gives δ = √ εC d (1 + O(ε)) and (91) becomes Let us proceed and write equations (90) and (92) as a system, using the following asymptotic formula which follows from (85) and (89). We obtain the system of 4(2N − 3) + 3 equations and To change those equations from vector to scalar notation, we introduce set of 4(2N − 3) + 3 indices:

Now we write (93) as
with a vector M = {M α } α∈I , a vector Φ = {Φ α } α∈I with the elements a matrix A ={A β α } α,β∈I with elements given by a vector η = {η α } α∈I with real unknown coefficients and a vector µ = {µ α } α∈I that depends on δ and η analytically (analyticity follows form Theorem 3.6). Now our goal is to solve system (97) with respect to η. We will reach it in three steps. First, we eliminate constant −2 in the right-hand side in (97) by an appropriate choice of function Φ. Secondly, we choose functions {Ψ α } α∈I in such a way that A is unit and our system becomes nothing but η = f (η) (with a certian small function f ) and by Banach Fixed Point Theorem is solvable. The choice of function Φ is the following Φ α = 0, α = (N, +, +, ℜ); Φ α = −2, α = (N, +, +, ℜ) and it is possible due to the following Lemma.
is non zero and that follows from direct calculation By Lemma 2 all the multiplicands of Φ in (98) are linearly independent. It follows that it is possible to choose Φ so that (99) holds and equations (97) is Now we set matrix A to be unit, that is its elements fulfil the conditions Again using Lemma 2, it is possible to choose functions {Ψ α } α∈I so that the conditions (107) are fulfilled and (106) reads Now, as δ is small, the operator on the right hand side of equation (108) is a contraction operator, moreover µ is analytic in δ and η so Banach Fixed Point Theorem assures that equation (108) is solvable for η.
A numerical example of a potential (leading therm Φ) that produces a trapped mode is and is sketched in Figure 5.

Proof of Theorem 1.2
This section is devoted to the proof of the second main result formulated in Theorem 1.2 in the introduction. It concerns the multiplicity of trapped modes and states that (i) there are no trapped modes solutions for energies slightly larger than threshold, (ii) multiplicity of trapped modes with energies slightly larger than threshold does not exceed 1 and (iii) the spectrum far from the threshold is free of trapped modes. Consider problem (60), (61). As previously, P is a continuous potential with compact support and subject to (62).
(ii) There exist at least one trapped mode and it can be constructed through conditions given in the previous section, Sect. 4.2. Assume now that there are two trapped modes. According to Theorem 3.3 and Proposition 1 (iii), which assets that there are exactly two solutions w = (u, v) = e −iλy (U, V) with complex λ in the strip {|ℑλ| ≤ γ}, it follows that the trapped mode is of the form with R j ∈ H + γ , j = 1, 2. Now, consider the following linear combination of trapped modes (109) which is a solution to problem (61) as well. From Theorem 3.1, it follows that operator D + (δP − ω)I : is an isomorphism and hence w 3 = 0. From (110) we get w 1 = Cw 2 .
(iii) First we choose γ such that the strip |ℑλ| ≤ γ contains only real roots of (30) for all ω described in the Proposition (iii). then we note that supremum with respect to such ω of the quantity sup ℑλ=±γ ||(D −ω) −1 || L 2 (Π)→L 2 (Π) is bounded then reasoning as in (i), we obtain the estimate for δ 1 .
where D α = D α (∂ x , ∂ y ) for α = 0, v, u. We have det D α (−iη, −iξ) = |η| 2 + |ξ| 2 and, hence, the operator matrix (111) is elliptic with α = 0, u, v. However, it is also necessary to verify the Shapiro-Lopatinskii condition at the both sides of the strip Π. For example, for the right edge 0 × R of the nanoribbon the Cauchy problem must have only one solution decaying as y → ∞.
A similar calculation shows that the Cauchy problem serving for the left edge of the nanoribbon gets the necessary property for the case α = v only.
Reviewing the situation, we see that any of three ADN-tables is suit inside Π but none serves simultaneously at both sides of the nanoribbon. This means that our problem is not included into the standard elliptic theory.
It also should be mentioned that, if there exists an ADN-table fitting everywhere in Π and on ∂Π, then according to [17,Ch. 5], the numbers of incoming and outgoing waves must coincide in each outlet to infinity. The latter, as we have verified in (70) and (71) is not true.

B Mandelstam radiation condition
Here we want to clarify the division of waves in two classes outgoing/incoming accoriding to the appearance of the ±i in (34). To do so, we employ the Mandelstam radiation conditions which define classification into outgoing and incoming waves by the direction of the energy transfer [16,23,25].
Let us write initial system (5) in the form Energy transfer from area Ω is defined as Using relations (112), (113) and performing partial integration we get where Γ is the boundary of the domain Ω. Consider energy transfer along the nanoribbon (along the y-axis) from −∞ to +∞, that is choose (n x , n y ) = (0, 1), then the last formula is equal to where the last equality comes from the the definition of q-form (31). Accordingly the energy transfer along the nanoribbon is proportional to iq, which is ±1 for q = ∓i. It follows that the value of q-form defines direction of wave propagation, namely q = −i describes waves propagating from from −∞ to +∞ and q = +i those from +∞ to −∞. This leads to the definition of outgoing/incoming waves (70), (71) as those traveling to ±∞ and from ±∞ .
Let us obtain estimates for u and v.
As a consequence, we get Now, from trigonometric function properties κ cos κx − λ sin κx = iτ cosh τ x − iλ sinh τ x Therefore, where D = D(∂ x , ∂ y ) was defined in (5). Now choose a positive value ρ sufficiently large so that all eigenvalues λ τ j described in Proposition 1 (iii) are contained in the set {λ ∈ C : |ℑλ| < γ, |ℜλ| < ρ} ( Figure 6). This is possible according to Proposition 1. Applying Cauchy's formula, we get The first integral on the right hand side tends to (u − , v − )with ρ → ∞. Moreover the last two integrals tend to zero for smooth functions (g, h) with compact support. It is enough to prove the theorem for such functions as they are dense in L + γ (Π) ∩ L − γ (Π). The residua in (122) belong to the kernel of (D(∂ x , ∂ y )−ωI). Therefore the last sum is linear combination of solutions w τ k , with τ = ± and k = 1, . . . , N and we obtain (59) with certian coefficinets. Now, we want to find expressions for those coefficients.
Let us define a smooth function η − = η − (y) such that η − (y) = 1 in the neighbourhood of −∞ and η − (y) = 0 in the neighbourhood of +∞. Using the biortog-onality conditions for functions w ∓ j in (34) we get Now note that Applying integration by parts, follows In a similar way we obtain This furnishes the assertion.