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The spectrum of discrete Dirac operator with a general boundary condition
Advances in Difference Equations volume 2020, Article number: 409 (2020)
Abstract
In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space \(l_{2}(\mathbb{N},\mathbb{C}^{2})\) by the discrete Dirac system
and the general boundary condition
where λ is a spectral parameter, Δ is the forward difference operator, (\(h_{n}\)) is a complex vector sequence such that \(h_{n} = ( h_{n}^{(1)}, h_{n}^{(2)} )\), where \(h_{n}^{(i)} \in l^{1} ( \mathbb{N} ) \cap l^{2} ( \mathbb{N} )\), \(i = 1,2\), and \(h_{0}^{(1)} \ne 0\). Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity.
Introduction
Along with the invention of the Schrödinger equation, the physical scope of mathematical problems connected with the spectra of differential equations with prescribed boundary conditions was enormously enlarged. The types of equations that previously had applications only to mechanical vibrations now were to be used for the description of atoms and molecules. There are important and altogether astonishing applications of the results obtained in the spectral theory of linear operators in Hilbert spaces to scattering theory, inverse problems, and quantum mechanics. For instance, the Hamiltonian of a quantum particle confined to a box involves a choice of boundary conditions at the box ends. Since different choices of boundary conditions imply different physical models, spectral theory of operators with boundary conditions constitues a progressing field of investigation [1, 2].
Let T denote a matrix operator
where \(p_{ik} ( x ) \) (\(i,k= 1,2 \)) are real continuous functions on the interval \([ 0,\pi ]\). Let also \(y(x)\) denote a twocomponent vector function
If
and λ is a parameter, then the equation
is equivalent to a system of two simultaneous firstorder ordinary differential equations
In the case of \(p_{12} ( x ) = p_{21} ( x ) = 0\), \(p_{11} ( x ) =V ( x ) +m\), and \(p_{22} ( x ) =V ( x ) m\), where \(V(x)\) is a potential function, and m is the mass of a particle, system (1.1) is called a stationary onedimensional Dirac system in relativistic quantum theory. Levitan and Sargsjan [1] have introduced some basic concepts regarding the general spectral theory of selfadjoint Sturm–Liouville and Dirac operators and presented a discrete analogue of system (1.1) using the method of finite differences. If the functions \(p_{ik} ( x ) \) (\(i,k= 1,2 \)) are complex valued, then the operator T is called nonselfadjoint. Also, if the operator T is defined on an infinite interval, then it is said to be singular. The structure of the spectrum of the operator T differs drastically in the nonselfadjoint singular case. The basic spectral theory of nonselfadjoint singular secondorder operators consisting of Sturm–Liouville theory was begun by Naimark, whose works initiated a deep study of spectral theory of nonselfadjoint operators [3, 4]. He proved that the spectrum of a nonselfadjoint Sturm–Liouville operator consists of the continuous spectrum, the eigenvalues, and the spectral singularities. He also showed that these eigenvalues and spectral singularities are of finite number with finite multiplicities under certain conditions.
Later developments in this area concerned spectral analysis of the boundary value problems of the differential and discrete operators including Sturm–Liouville, Klein–Gordon, quadratic pencils of Schrödinger and Diractype operators within the context of determination of Jost solution and providing suffcient conditions guaranteeing the finiteness of the eigenvalues and spectral singularities [5–19].
In particular, boundary value problems including the integral boundary condition were first considered by Krall [20, 21]. He extended the work of Naimark [3] by applying a suitable integral boundary condition and generated the ordinary and nonhomogeneous expansion of a Sturm–Liouville operator.
Note that investigation of discrete analogues of ordinary differential operators is an important research area since difference equtions are well suited to find solutions with the aid of computers and can model many contemporary problems arising in control theory, biology, and engineering [7–11].
Let us denote by \(l_{2}(\mathbb{N},\mathbb{C}^{2})\) the Hilbert space of all complex vector sequences y={\left(\begin{array}{c}{y}_{n}^{(1)}\\ {y}_{n}^{(2)}\end{array}\right)}_{n\in \mathbb{N}} with the inner product
Consider the nonselfadjoint singular operator \(L_{0}\) generated in the Hilbert space \(l_{2}(\mathbb{N},\mathbb{C}^{2})\) by the discrete Dirac system
and the boundary condition
where λ is a spectral parameter, Δ is the forward difference operator, and \(p_{n},q_{n} \in \mathbb{C}\). In [9] the integral representation for the Weyl function of \(L_{0}\) and spectral expansion of the operator \(L_{0}\) in terms of principal functions have been investigated in detail. Some generalization problems of the nonselfadjoint discrete Dirac operator have been subject to extensive studies in the literature. For instance, in [13] the general form of the operator \(L_{0}\) has been considered for \(n \in \mathbb{Z}\). Also, some authors investigated the problem with eigenparameterdependent boundary conditions [10, 14, 15].
In this paper, we consider the operator L generated in the Hilbert space \(l_{2}(\mathbb{N},\mathbb{C}^{2})\) by the nonselfadjoint discrete Dirac equation (1.2) and boundary condition
where (\(h_{n}\)) is complex vector sequence such that \(h_{n} = ( h_{n}^{(1)}, h_{n}^{(2)} )\), \(h_{n}^{(i)} \in l^{1} ( \mathbb{N} ) \cap l^{2} ( \mathbb{N} )\), \(i = 1,2\), \(h_{0}^{(1)} \ne 0\). Clearly, \(L_{0}\) is a particular case of L for \(h_{n} = ( 0, 0 )\), \(n \in \mathbb{N} = \{ 1,2,\ldots \} \). Differently from other studies, rather than considering an eigenparameter dependent boundary condition, we generalize the boundary condition (1.3) by using the orthogonality properties of \(( y_{n} )\) with respect to vectors \(( h_{n} )\). Therefore the conditions required for the finiteness of the eigenvalues and spectral singularities of the operator L differ from the studies mentioned. Thus this paper presents the results in a more general and different approach.
The main objective of this paper is investigating the quantitative properties of the spectrum of the operator L. We apply and adopt the Naimark and Pavlov conditions on the potential and examine the eigenvalues and spectral singularities of the operator L using the boundary uniqueness theorems of analytic functions.
Although the tools we use in this paper are basicly functional analysis techniques, the paper may lay the groundwork for future studies concerning the topics in direct and inverse problems, scattering theory, and applied physics.
The paper contains three sections. The first two are introductory, surveying all necessary results of the BVP (1.2)–(1.4). The last section focuses on the quantitative properties of the spectrum of the operator L.
Jost solution of the operator L
We will assume that
It is known from [9] that equation (1.2) has the solution
and
for \(\lambda = 2\sin \frac{z}{2}\), {E}_{2}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right), {K}_{nm}=\left(\begin{array}{cc}{K}_{nm}^{11}& {K}_{nm}^{12}\\ {K}_{nm}^{21}& {K}_{nm}^{22}\end{array}\right), \(z \in \overline{\mathbb{C}}_{ +} \). Note that the expressions \(K_{nm}^{ij}\), \(i,j = 1,2\), can be written uniquely in terms of \(\{ p_{n} \} _{n \in \mathbb{N}}\) and \(\{ q_{n} \} _{n \in \mathbb{N}}\). Moreover, the inequality
is satisfied for \(i,j = 1,2\), where \([ \vert \frac{m}{2} \vert ]\) is the integer part of \(\frac{m}{2}\), and \(C > 0\) is a constant. Hence \(f_{n}(z)\) is analytic in \(\mathbb{C}_{+}:=\{z \in \mathbb{C}: \operatorname{Im} z>0\}\) and continuous in \(\overline{\mathbb{C}}_{+}:=\{z \in \mathbb{C}: \operatorname{Im} z\geq 0\}\). The function \(f_{n}(z)\) is called the Jost solution of equation (1.2). Also, the following asymptotics hold [9]:
Let \(\varphi _{n}(z)\) be a solution of (1.2) subject to the initial conditions
where
Then φ is an entire function, and
The Wronskian of two solutions
of (1.2) is defined by
Using the usual definition of Wronskian, we have
Let us define the semistrips \(P_{0}: = \{ z:z \in \mathbb{C}, = x + iy,0 \le x < 4\pi ,y > 0 \} \) and \(P = P_{0} \cup [ 0,4\pi )\).
Let us define
and also the functions,
For all \(z \in P\) and \(f_{0}^{(1)} ( z ) \ne 0\), the Green’s function of the operator L is obtained by standard techniques as
where
and
Obviously, for \Omega ={\Omega}_{n}=\left(\begin{array}{c}{\Omega}_{n}^{(1)}\\ {\Omega}_{n}^{(2)}\end{array}\right)\in {l}^{2}(\mathbb{N},{\mathbb{C}}^{2}),
is the resolvent of the operator L.
It is also clear that \(N(z)\) is the Jost function of the operator L defined by using the Jost solution and boundary condition (1.4). The determination of Jost solutions plays an important role in spectral theory of discrete and differential operators. We refer the reader to books [1–4] for further details, which explain how this single function contains all the information about the spectrum of operators.
Eigenvalues and spectral singularities of L
Let us denote the set of eigenvalues and spectral singularities of the operator L by \(\sigma _{d}\) and \(\sigma _{ss}\), respectively. From (2.5)–(2.7) and the definition of the eigenvalues and spectral singularities we have
Let us define the sets
We also denote the set of all limit points of \(M_{1}\) and \(M_{2}\) by \(M_{3}\) and \(M_{4}\), respectively, and the set of all zeros in P of \(N(z)\) with infinite multiplicity by \(M_{5}\). It then also follows that
and the linear Lebesgue measures of \(M_{2}\), \(M_{3}\), \(M_{4}\), and \(M_{5}\) are zero. From the continuity of all derivatives of \(N(z)\) on the real axis we have
It is convenient to rewrite the sets of eigenvalues and spectral singularities of L as
Theorem 3.1
Under conditions (2.1) and\(h_{n}^{(i)} \in l^{1} ( \mathbb{N} ) \cap l^{2} ( \mathbb{N} )\), \(i = 1,2\), we have:

(i)
The set of eigenvalues ofLis bounded and countable, and its limit points lie in\([  2,2 ]\).

(ii)
\(\sigma _{ss} \subset [  2,2 ]\), \(\sigma _{ss} = \overline{\sigma _{ss}}\), and\(\mu (\sigma _{ss}) = 0\), whereμstands for the linear Lebesgue measure.
Proof
From (2.3) and (2.4) we have the analyticity of \(N(z)\) in the upper halfplane and the continuity of \(N(z)\) on the real axis. For \(\beta (z): = e^{  i\frac{z}{2}}N(z)\), we have the asymptotics
Note that \(\beta (z)\) and \(N(z)\) have the same zeros except at infinity. Using (3.1), (3.2), and (3.4) and boundary uniqueness theorems of analytic functions [22], we arrive at (i) and (ii). □
Definition 3.1
The multiplicity of a zero of \(N(z)\) in the region P is introduced as the multiplicity of the corresponding eigenvalue or spectral singularity of the operator L.
Now let us consider the condition
Theorem 3.2
Under condition (3.5), the operatorLhas a finite number of eigenvalues and spectral singularities, and each of them is of finite multiplicity.
Proof
From (2.3) and (3.5) we observe that \(N(z)\) has analytic continuation to the halfplane \(\operatorname{Im} z > \frac{  \varepsilon }{2}\). Since \(N(z)\) is a 4πperiodic function, the limit points of its zeros in P cannot lie in \([ 0,4\pi )\). Hence, using Theorem 3.1, we obtain the finiteness of eigenvalues and spectral singularities of L. □
Note that condition (3.5), which is also known as Naimark’s condition in the literature, ensures the analytic continuation of \(N(z)\) from the real axis to the lower halfplane.
Now we will consider the Pavlov condition
which is weaker than (3.5). Clearly, the function \(N(z)\) is analytic in the upper halfplane and infinitely differentiable on the real axis. It is essential to notice at this point that \(N(z)\) has no analytic continuation from the real axis to the lower halfplane. For this reason, we need to use a different method to investigate the finiteness of the eigenvalues and spectral singularities of L. We will benefit from the following lemma.
Lemma 3.3
([9])
Suppose that the 4πperiodic functionξis analytic in the open halfplane, all of its derivatives are continuous in the closed upper halfplane, and
If the setGwith linear Lebesgue measure zero is the set of all zeros of the functionξwith infinite multiplicity inP, and
where\(t(s) = \inf_{k}\frac{\eta _{k}s^{k}}{k!}\), \(k \in \mathbb{N} \cup \{ 0 \} \), \(\mu ( G_{s} )\)is the Lebesgue measure of thesneighborhood ofG, and\(\omega \in ( 0,4\pi )\)is an arbitrary constant, then\(\xi \equiv 0\).
Theorem 3.4
Assume that (3.6) holds. Then\(M_{5} = \emptyset \).
Proof
Under conditions (3.6), (2.2), (2.3), and (2.4), we obtain that
where
and \(C > 0\) is a constant. We have the following estimate:
where D and d are constants depending C, ε, and β.
Applying the previous lemma to our case, we get that
where \(t(s) = \inf_{k}\frac{\eta _{k}s^{k}}{k!}\), \(k \in \mathbb{N} \cup \{ 0 \} \), \(\mu (M_{5,s})\) is the Lebesgue measure of the sneighborhood of \(M_{5}\) and \(\eta _{k}\) is defined by (3.8).
Now we have
Since \(\frac{1  \beta }{\beta } \ge 1\), (3.11) holds for arbitrary s if and only if \(\mu (M_{5,s}) = 0\) or \(M_{5} = \emptyset \). □
Theorem 3.5
If condition (3.6) is satisfied, then the operatorLhas a finite number of eigenvalues and spectral singularities, and each of them is of finite multiplicity.
Proof
We have to show that the function \(N(z)\) has a finite number of zeros with finite multiplicities in P. From (3.3) and the previous theorem we obtain that \(M_{3} = M_{4} = \emptyset \). Hence the bounded sets \(M_{1}\) and \(M_{2}\) have no accumulation points, that is, \(N(z)\) has only a finite number of zeros in P. Since \(M_{5} = \emptyset \), these zeros are of finite multiplicity. □
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Coskun, N., Yokus, N. The spectrum of discrete Dirac operator with a general boundary condition. Adv Differ Equ 2020, 409 (2020). https://doi.org/10.1186/s13662020028512
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DOI: https://doi.org/10.1186/s13662020028512
MSC
 39A10
 39A12
 47A10
 47A75
Keywords
 Eigenparameter
 Spectral analysis
 Eigenvalues
 Spectral singularities
 Discrete equation
 Dirac equation