Eigenfunction Expansion and Parseval Identity for β-Dirac Operators on the Whole Line

1. Introduction

Quantum analysis is an important and rapidly developing area of modern mathematics. One of its fundamental tools is q-calculus, a branch of mathematical analysis that does not rely on the concept of limits. Consequently, certain functions that are not differentiable in the classical sense may admit q-derivatives. Today, q-calculus has found numerous applications in both mathematics and physics, including relativity theory, basic hyper-geometric functions, orthogonal polynomials, combinatorics, and the calculus of variations. Owing to its broad range of applications and rich theoretical structure, q calculus continues to attract considerable attention from researchers.

This framework extends the classical difference operator and leads naturally to the $\left(q,w\right)$-difference, or Hahn quantum, operator. Unlike classical differentiation, this approach does not require continuity or differentiable assumptions, making it especially suitable for the analysis of discrete and non-smooth systems.

Generalized quantum difference operators and their corresponding inverse operators have become powerful tools in both pure and applied mathematics. Notably, they have found important applications in the calculus of variations [1,2,3,4,5], where they are used to derive necessary optimality conditions for discrete dynamical systems. Furthermore, quantum difference analogues of classical eigenvalue problems have been formulated and investigated within the framework of Sturm–Liouville theory [5,6,7,8,9,10]. These studies have contributed significantly to the development of spectral theory and its applications in quantum calculus.

A significant development in quantum difference calculus was made by Hamza et al. in 2015, who introduced the concepts of the derivative, integral, and the corresponding inverse operator within the framework of a general quantum difference operator theory [11]. In the same year, several fundamental inequalities, including Hölder’s, Minkowski’s, Gronwall’s, Bernoulli’s, and Lyapunov’s inequalities, were extended and adapted to the $\beta$-calculus setting [12], thereby enriching the analytical foundations of the theory.

Subsequently, Hamza et al. established existence and uniqueness results for solutions of general quantum difference equations in [13]. Further contributions were made by Faried and Sherhata, the authors established new results on homogeneous second-order linear general quantum difference equations in [14]. Furthermore, Hamza et al. in [15] introduced generalized exponential, trigonometric, and hyperbolic functions associated with the $\beta$-difference operator, thereby providing essential tools for the analysis of quantum difference equations.

More recently, Cardoso investigated the $\beta$-Sturm-Liouville problem in [16], further demonstrating the applicability of the theory to spectral analysis and operator theory. The growing interest in this area has led to a number of recent studies on $\beta$-operators and their applications, including the works [17,18], which continue to expand the theoretical and practical scope of general quantum calculus.

The Dirac equation, introduced by Dirac in 1928, provides a relativistic description of spin particles such as electrons. By unifying the principles of quantum mechanics and special relativity, it successfully accounts for both the spin and relativistic behavior of elementary particles. Owing to its fundamental importance in mathematical physics, the Dirac equation has been extensively studied in the literature [19,20,21,22].Studies conducted in classical analysis are also seen to be carried out to q-calculus and Hahn’s difference operator.

Several important contributions to the theory of q-Dirac systems have been done, including studies on one-dimensional q-Dirac equations, dissipative and singular eigenfunction expansions for q-Dirac operators and q fractional Dirac-type systems in [23,24,25,26,27]. These works have advanced the spectral theory of quantum difference Dirac operators.

Many concepts and results from classical analysis have been successfully extended to q-calculus and Hahn’s difference calculus. These developments suggest that analogous results can also be established within the framework of $\beta$-calculus. Motivated by this observation, the present study investigates a $\beta$-Dirac problem and derives the corresponding Parseval identity. More specifically, we establish the existence of a spectral function and obtain an expansion theorem in terms of eigenfunctions associated with the problem.

Studies conducted in classical analysis are also seen to be carried out to q-calculus and Hahn’s difference operator, this research shows us it can be applied to $\beta$-calculus. Our main objective is to establish the existence of a spectral function corresponding to the $\beta$-Dirac operator. Based on this spectral function, we develop a spectral representation of the problem and derive the corresponding Parseval identity. Furthermore, we obtain an eigenfunction expansion theorem, showing that suitable functions can be represented in terms of the eigenfunctions of the $\beta$-Dirac system. These results extend several classical concepts of spectral theory to the setting of generalized quantum difference operators and contribute to the growing literature on $\beta$-calculus.

2. Materials and Methods

Before presenting our main results, we first introduce some preliminary facts and notations related to the general quantum (or only $\beta$-) difference operator (see [28,29]). $\beta -$derivative and $\beta -$integral which serve as our primary tools play a fundamental role in this study. Let $T\subseteq \mathbb{R}$ be any interval and assume that $\beta :T\to T$ a strictly increasing and continuous mapping possessing a unique fixed point $s_0\in T$ satisfying

$$\left(\xi -s_0\right)\left(\beta \left(\xi \right)-\xi \right)\le 0$$

(1)

for every $\xi \in T$. Moreover, equality holds if and only if $\xi =s_0$. Note that it is possible to replace the above inequality by $\left(\xi -s_0\right)\left(\beta \left(\xi \right)-\xi \right)\ge 0$ for $\xi \in T$. For each function $\phi :T⟶\mathbb{R}$ is defined $\beta -$difference operator in [11]:

$$D_{\beta }\left[\phi \right]\left(\xi \right):={\displaystyle \frac{\phi \left(\beta \right(\xi \left)\right)-\phi \left(\xi \right)}{\beta \left(\xi \right)-\xi }},\forall \xi \ne s_0$$

and it is to be remarked that the point $\xi =s_0$ in T is the $\beta$-derivative at $s_0$ if

$$D_{\beta }\left[\phi \right]\left(s_0\right)={\phi }^{\prime }\left(s_0\right)$$

exists and belongs to $\mathbb{R}$. It is well known that we can get Hahn’s difference operator when $\beta \left(\xi \right)=q\xi +\omega$, $\omega >0$ for being the fixed point given by $s_0={\displaystyle \frac{\omega }{1-q}}$, and we obtain Jackson difference operator when we take $\beta \left(\xi \right)=q\xi$, for all $q\in (0,1)$.

Let us define ${\beta }^k\left(\xi \right):=(\beta \circ \beta \circ ...\circ \beta )\left(\xi \right)$ for $\xi \in T$ and $k\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ with ${\beta }^0\left(\xi \right):=\xi$. It can be clearly seen that the sequence of functions ${\left\{{\beta }^k\left(\xi \right)\right\}}_{k\in {\mathbb{N}}_0}$ uniformly converges to the constant function ${\beta }^{\ast }\left(\xi \right):=s_0$ on each compact interval $J\subset T$ containing $s_0$ (see also [11]).

The $\beta$-integral, the quantum difference inverse operator, is given by

$${\int }_{{\xi }_1}^{{\xi }_2}\psi \left(\xi \right)d_{\beta }\xi ={\int }_{s_0}^{{\xi }_2}\psi \left(\xi \right)d_{\beta }\xi -{\int }_{s_0}^{{\xi }_1}\psi \left(\xi \right)d_{\beta }\xi ,\forall {\xi }_1,{\xi }_2\in T.$$

where

$${\int }_{s_0}^{{\xi }_1}\psi \left(\xi \right)d_{\beta }\xi ={\displaystyle \sum _{k=0}^{\infty }}\left({\beta }^k\left({\xi }_1\right)-{\beta }^{k+1}\left({\xi }_1\right)\right)\psi \left({\beta }^k\left({\xi }_1\right)\right).$$

Provided that the infinite series on the the right side is convergent then the function $\phi \left(\xi \right)$ is said $\beta$-integrable in $[s_0,{\xi }_1]$ see [11,16]. Furthermore, the $\beta$-integration of $\psi$ on $(s_0,\infty )$ is defined in [30] by the formula

$${\int }_{s_0}^{\infty }\psi \left(\xi \right)d_{\beta }\xi =\underset{t\to \infty }{lim}{\int }_{s_0}^t\psi \left(\xi \right)d_{\beta }\xi ,(t\in T),$$

assuming that some converges absolutely. Throughout this study, the functions $\psi$ and $\phi$ will be assumed $\beta -$ integrable on T and $D_{\beta }\psi$, $D_{\beta }\phi$ both continuous at $s_0$. Let ${\xi }_1,{\xi }_2\in T$ where ${\xi }_1<{\xi }_2$, then by using the integration by parts formula for the $\beta -$calculus [11,31] the following equality holds;

$${\int }_{{\xi }_1}^{{\xi }_2}\phi \left(\xi \right)D_{\beta }\psi \left(\xi \right)d_{\beta }\xi =\psi \left({\xi }_2\right)\phi \left({\xi }_2\right)-\psi \left({\xi }_1\right)\phi \left({\xi }_1\right)-{\int }_{{\xi }_1}^{{\xi }_2}\psi \left(\beta \left(\xi \right)\right)D_{\beta }\phi \left(\xi \right)d_{\beta }\xi .$$

Let us establish the basic integral theorem of $\beta -$calculus [11,31] : a function $\psi$ is continuous at $s_0$. Let

$$\Psi \left(t\right):={\int }_{s_0}^t\psi \left(\xi \right)d_{\beta }\xi ,t\in T.$$

Then $\Psi$ is continuous at $s_0$. Furthermore, $D_{\beta }\Psi \left(t\right)$ exists for every $t\in T$ and $D_{\beta }\Psi \left(t\right)=\psi \left(t\right)$. Moreover,

$${\int }_{{\xi }_1}^{{\xi }_2}{D}_{\beta }\Psi \left(\xi \right)d_{\beta }\left(\xi \right)=\Psi \left({\xi }_2\right)-\Psi \left({\xi }_1\right).$$

The separable Hilbert space is defined as follows

$$L_{\beta }^2(s_0,\infty ):=\left\{\psi \mid {\int }_{s_0}^{\infty }{\left|\psi \left(\xi \right)\right|}^2{d}_{\beta }\xi <\infty ,\psi :(s_0,\infty )\to \mathbb{R}\right\}$$

with the norm by

$$\parallel \psi \parallel :={\left({\int }_{s_0}^{\infty }{\left|\psi \left(\xi \right)\right|}^2{d}_{\beta }\xi \right)}^{{\displaystyle \frac{1}{2}}}<\infty ,$$

and given with the inner product as

$${〈\psi ,\phi 〉}_{\beta }=〈\psi ,\phi 〉:={\int }_{s_0}^{\infty }\psi \left(\xi \right)\overline{\phi \left(\xi \right)}{d}_{\beta }\xi ,\psi ,\phi \in L_{\beta }^2\left(s_0,\infty \right)$$

(see [30]).

Finally, in this section we remember the $\beta -$Wronskian of $\psi ,\phi$ on T if

$$W_{\beta }\left[\psi ,\phi \right]\left(\xi \right):=\psi \left(\xi \right)D_{\beta }\phi \left(\xi \right)-\phi \left(\xi \right)D_{\beta }\psi \left(\xi \right)$$

(2)

exists. Here the left expression in (2) is a analogous to the ordinary Wronskian. The reader also see for more information about $\beta -$Wronskian and its properties in [14] (pp. 7–8). Moreover, we recall that the $\beta$-Wronskian of $\psi \left(\xi \right)=\left(\begin{array}{c}{\psi }_1\left(\xi \right)\\ {\psi }_2\left(\xi \right)\end{array}\right)$, $\phi \left(\xi \right)=\left(\begin{array}{c}{\phi }_1\left(\xi \right)\\ {\phi }_2\left(\xi \right)\end{array}\right)$ on T if

$$W_{\beta }\left(\psi ,\phi \right)\left(\xi \right):={\psi }_1\left(\xi \right){\phi }_2\left[D_{\beta }{\beta }^{-1}\left(\xi \right)\right]-{\phi }_1\left(\xi \right){\psi }_2\left[D_{\beta }{\beta }^{-1}\left(\xi \right)\right]$$

(3)

exists. Here the left expression in (2) is a analogous to the ordinary Wronskian.

3. Results

In this chapter, our first goal is defined the $\beta$-Dirac Systems of a form on the general quantum difference operators as follows:

$$\mathcal{L}y\left(\xi \right):=\left\{\begin{array}{c}-D_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{y}_2\left(\xi \right)+p\left(\xi \right)y_1\left(\xi \right),\hfill \\ D_{\beta }{y}_1\left(\xi \right)+r\left(\xi \right)y_2\left(\xi \right),\hfill \end{array}\right.$$

(4)

where $\mathcal{L}y\left(\xi \right)=\lambda y\left(\xi \right),y\left(\xi \right)=\left(\begin{array}{c}{y}_1\left(\xi \right)\\ y_2\left(\xi \right)\end{array}\right)$, p and r are real valued functions defined on $\xi \in T\subset \left(s_0,b\right)$ and continuous at $s_0$ and $p\left(\xi \right),r\left(\xi \right)\in L_{\beta ,loc}^1\left(\mathbb{R}\right)$.

We recall that $\beta :T⟶T$ is a function satisfying the conditions described in Section 2 being $s_0$ its unique fixed point.

Now, we will consider the boundary value problem (4) with the boundary conditions

$$\begin{array}{c}{y}_1(s_0,\lambda )sin\alpha +y_2(s_0,\lambda )cos\alpha =0,\end{array}$$

(5)

$$\begin{array}{c}{y}_1(b,\lambda )sin\theta +y_2(b,\lambda )cos\theta =0,\alpha ,\theta \in \mathbb{R},.\end{array}$$

(6)

Let ${\lambda }_1,{\lambda }_2,{\lambda }_3,...$ be the eqienvalues and $y_1(\xi ;{\lambda }_1),y_2(\xi ;{\lambda }_2),y_3(\xi ;{\lambda }_3),...$ be the corresponding eigenfunctions of the $\beta$-Dirac system where $y_n(\xi ;{\lambda }_n)=\left(\begin{array}{c}{y}_{n1}(\xi ;{\lambda }_n)\\ y_{n2}(\xi ;{\lambda }_n)\end{array}\right)$ for $n\in \mathbb{N}$. Owing the solutions of this system are linearly independent, it follows that

$$y_n\left(\xi \right)=c_n{\psi }_1(\xi ;{\lambda }_n)+d_n{\psi }_2(\xi ;{\lambda }_n)$$

where ${\psi }_1(\xi ;{\lambda }_n),{\psi }_2(\xi ;{\lambda }_n)$ are the solutions of the $\beta$-Dirac System (4) which satisfy the initial conditions

$${\psi }_{11}(s_0;{\lambda }_n)=1,D_{\beta }{\psi }_{12}(s_0;{\lambda }_n)=0,{\psi }_{21}(s_0;{\lambda }_n)=0,D_{\beta }{\psi }_{22}(s_0;{\lambda }_n)=1$$

(7)

and without loss of generality $\left|c_n\right|\le 1,\left|d_n\right|\le 1.$

Now, let us set

$${\gamma }_n^2:={\int }_{-q_{\beta }^m}^{q_{\beta }^m}{\parallel y_n\left(\xi \right)\parallel }_E^2{d}_{\beta }\xi$$

where

$$q_{\beta }:=\left\{\begin{array}{cc}{\left|s_0\right|}^{-1},\hfill & 0<\left|s_0\right|<1\hfill \\ 1+\delta ,\hfill & s_0=0\vee \left|s_0\right|=1,0<\delta <1\hfill \\ \left|s_0\right|,\hfill & \left|s_0\right|>1.\hfill \end{array}\right..$$

Let $\phi (.)=\left(\begin{array}{c}{\phi }_1(.)\\ {\phi }_2(.)\end{array}\right)\in L_{\beta }^2\left((-q_{\beta }^m,q_{\beta }^m);E\right)$. If we apply the Parseval equality (see [34]) to $\phi \left(\xi \right)$, then we obtain

$$\begin{array}{ccc}\hfill {\int }_{-q_{\beta }^m}^{q_{\beta }^m}{\parallel \phi \left(\xi \right)\parallel }_E^2{d}_{\beta }\xi & =& {\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{{\gamma }_n^2}}{\left\{{\int }_{s_0}^b{〈\phi \left(\xi \right),y_n\left(\xi \right)〉}_{\beta }{d}_{\beta }\xi \right\}}^2\hfill \\ & =& {\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{{\gamma }_n^2}}{\left\{{\int }_{s_0}^b{〈\phi \left(\xi \right),c_n{\psi }_1(\xi ;{\lambda }_n)+d_n{\psi }_2(\xi ;{\lambda }_n)〉}_{\beta }{d}_{\beta }\xi \right\}}^2\hfill \\ & =& {\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{c_n^2}{{\gamma }_n^2}}{\left\{{\int }_{s_0}^b{〈\phi \left(\xi \right),{\psi }_1(\xi ;{\lambda }_n)〉}_{\beta }{d}_{\beta }\xi \right\}}^2+\hfill \\ & & +2{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{c_n{d}_n}{{\gamma }_n^2}}{\displaystyle \prod _{j=1}^2}\left\{{\int }_{s_0}^b{〈\phi \left(\xi \right),{\psi }_j(\xi ;{\lambda }_n)〉}_{\beta }{d}_{\beta }\xi \right\}\hfill \\ & & +{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{d_n^2}{{\gamma }_n^2}}{\left\{{\int }_{s_0}^b{〈\phi \left(\xi \right),{\psi }_2(\xi ;{\lambda }_n)〉}_{\beta }{d}_{\beta }\xi \right\}}^2.\hfill \end{array}$$

(8)

Now we introduce a monotone increasing step function ${\sigma }_{ij}$ on $(-q_{\beta }^m,q_{\beta }^m)$, by

$${\sigma }_{11}\left(\lambda \right)=\left\{\begin{array}{cc}-\sum _{\lambda <{\lambda }_n<0}{\displaystyle \frac{c_n^2}{{\gamma }_n^2}};\hfill & \lambda \le 0\hfill \\ \sum _{0\le {\lambda }_n<\lambda }{\displaystyle \frac{c_n^2}{{\gamma }_n^2}};\hfill & \lambda >0,\hfill \end{array}\right.$$

$${\sigma }_{12}\left(\lambda \right)=\left\{\begin{array}{cc}-\sum _{\lambda <{\lambda }_n<0}{\displaystyle \frac{c_n{d}_n}{{\gamma }_n^2}};\hfill & \lambda \le 0\hfill \\ \sum _{0\le {\lambda }_n<\lambda }{\displaystyle \frac{c_n{d}_n}{{\gamma }_n^2}};\hfill & \lambda >0,\hfill \end{array}\right.$$

$${\sigma }_{21}\left(\lambda \right)={\sigma }_{12}\left(\lambda \right),$$

$${\sigma }_{22}\left(\lambda \right)=\left\{\begin{array}{cc}-\sum _{\lambda <{\lambda }_n<0}{\displaystyle \frac{d_n^2}{{\gamma }_n^2}};\hfill & \lambda \le 0\hfill \\ \sum _{0\le {\lambda }_n<\lambda }{\displaystyle \frac{d_n^2}{{\gamma }_n^2}};\hfill & \lambda >0.\hfill \end{array}\right.$$

From (8), we obtain

$${\int }_{-q_{\beta }^m}^{q_{\beta }^m}{\parallel \phi \left(\xi \right)\parallel }_E^2{d}_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\Phi }_j\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right),(i,j=1,2),$$

(9)

where

$${\Phi }_i\left(\lambda \right)={\int }_{-q_{\beta }^m}^{q_{\beta }^m}{\langle \varphi ,{\psi }_i\rangle }_{\mathcal{H}}{d}_{\beta }\xi (i=1,2).$$

We will obtain the Parseval identity for (4), (5) and (6) from (8) by letting $m\to \infty$.

In this paragraph, it will be explained by reminding what the concepts of bounded and total variation mean: The function $\phi$ is said to be of bounded variation on interval $\left[a,b\right]$ if and only if there exists a positive constant M such that

$${\displaystyle \sum _{k=1}^n}|\phi \left({\xi }_k\right)-\phi \left({\xi }_{k-1}\right)|\le M$$

for all finite partitions $\mathbb{P}=\left\{{\xi }_0,{\xi }_1,...,{\xi }_n\right\}$ of $\left[a,b\right]$.

If $\phi :\left[a,b\right]\to \mathbb{R}$ is of bounded variation on $\left[a,b\right]$, then the total variation of $\psi$ on $\left[a,b\right]$ is defined to be

$$V_a^b\left\{\phi \right\}:=sup{\displaystyle \sum _{k=1}^n}|\phi \left({\xi }_k\right)-\phi \left({\xi }_{k-1}\right)|,$$

where the supremum is taken over all partitions of $\left[a,b\right]$ (see [32]).

Before giving the main theorem, we establish the following elementary facts.

Lemma 1.

Let η be any positive real number. There exists a positive constant number ${\rm Y}={\rm Y}\left(\eta \right)$ not depending on $q_{\beta }^m$ such that the inequality

$$V_{-\eta }^{\eta }\left\{{\sigma }_{ij}\left(\lambda \right)\right\}<{\rm Y}\left(i,j=1,2\right)$$

(10)

holds.

Proof. 

From (7)

$$D_{\beta }^{(j-1)}{\psi }_{ij}\left(s_0;{\lambda }_n\right)={\delta }_{ij}\left(i,j=1,2\right),$$

where ${\delta }_{ij}$ is the Kronecker delta. Thus, there exists a $k>0$ such that

$$|D_{\beta }^{(j-1)}{\psi }_{ij}(s_0;{\lambda }_n)-{\delta }_{ij}|<\epsilon ,\epsilon >0,\left|\lambda \right|<\eta ,\xi \in \left[s_0,k\right].$$

(11)

Let ${\phi }_k\left(\xi \right)$ be a nonnegative function such that ${\phi }_k\left(\xi \right)$ vanishes outside the interval $\left[s_0,k\right]$ with

$${\int }_{s_0}^k{\phi }_k\left(\xi \right)d_{\beta }\xi =1.$$

(12)

Now, we apply the Parseval equality (9) to $D_{\beta }^{(h-1)}{\phi }_k\left(\xi \right),(h=1,2)$, then we get

$${\int }_{s_0}^k{\left|D_{\beta }^{(h-1)}{\phi }_k\left(\xi \right)\right|}^2{d}_{\beta }\xi ={\int }_{-\eta }^{\eta }{\displaystyle \sum _{i,j=1}^2}{\Phi }_{i_h}\left(\lambda \right){\Phi }_{j_h}\left(\lambda \right)d{\sigma }_{ij,\beta }\left(\lambda \right),$$

where

$${\Phi }_{i_h}\left(\lambda \right)={\int }_{s_0}^k{D}_{\beta }^{(h-1)}{\phi }_k\left(\xi \right){\psi }_{ij}(\xi ,{\lambda }_n)d_{\beta }\xi =\pm {\int }_{s_0}^k{\phi }_k\left(\xi \right)D_{\beta }^{(h-1)}{\tau }_{ij}(\xi ,{\lambda }_n)d_{\beta }\xi .$$

Using (11) and (12), we obtain

$$\left|{\Phi }_{ih}\left(\lambda \right)-{\delta }_{ih}\right|<\epsilon ,i,h=1,2,\left|\lambda \right|<\eta .$$

(13)

Now if we reapply the Parseval equality (9) to ${\phi }_k\left(\xi \right),\left(h=1,2\right)$, then we get

$${\int }_{s_0}^k{\left|D_{\beta }^{(h-1)}{\phi }_k\left(\xi \right)\right|}^2{d}_{\beta }\xi \ge {\int }_{-\eta }^{\eta }{\displaystyle \sum _{i,j=1}^2}\left({\delta }_{ih}-\epsilon \right)\left({\delta }_{jh}-\epsilon \right)\left|d{\sigma }_{ij,\beta }\left(\lambda \right)\right|.$$

(14)

If we take $h=1$ in (14), we have

$$\begin{array}{cc}\hfill {\int }_{s_0}^k{\left|{\phi }_k\left(\xi \right)\right|}^2{d}_{\beta }\xi & \ge {\left(1-\epsilon \right)}^2{\int }_{-\eta }^{\eta }\left|d{\sigma }_{11,\beta }\left(\lambda \right)\right|+\epsilon \left(1+\epsilon \right){\int }_{-\eta }^{\eta }\left|d{\sigma }_{12,\beta }\left(\lambda \right)\right|+\hfill \\ \hfill & +\epsilon \left(1+\epsilon \right){\int }_{-\eta }^{\eta }\left|d{\sigma }_{21,\beta }\left(\lambda \right)\right|+{\epsilon }^2{\int }_{-\eta }^{\eta }\left|d{\sigma }_{22,\beta }\left(\lambda \right)\right|\hfill \\ \hfill & ={(1-\epsilon )}^2\left({\sigma }_{11,\beta }\left(\eta \right)-{\sigma }_{11,\beta }(-\eta )\right)+2\epsilon (1+\epsilon )V_{-\eta }^{\eta }\left\{{\sigma }_{12,\beta }\left(\lambda \right)\right\}+\hfill \\ \hfill & +{\epsilon }^2\left({\sigma }_{22,\beta }\left(\eta \right)-{\sigma }_{22,\beta }(-\eta )\right).\hfill \end{array}$$

Since

$$V_{-\eta }^{\eta }\left\{{\sigma }_{12,\beta }\left(\lambda \right)\right\}\le {\displaystyle \frac{1}{2}}\left[{\sigma }_{11,\beta }\left(\eta \right)-{\sigma }_{11,\beta }(-\eta )+{\sigma }_{22,\beta }\left(\eta \right)-{\sigma }_{22,\beta }(-\eta )\right],$$

(15)

we get

$$\begin{array}{cc}\hfill {\int }_{s_0}^k{\left|{\phi }_k\left(\xi \right)\right|}^2{d}_{\beta }\xi & \ge (2{\epsilon }^2-3\epsilon +1)\left[{\sigma }_{11,\beta }\left(\eta \right)-{\sigma }_{11,\beta }(-\eta )\right]+\hfill \\ \hfill & +\epsilon (2\epsilon -1)\left[{\sigma }_{22,\beta }\left(\eta \right)-{\sigma }_{22,\beta }(-\eta )\right].\hfill \end{array}$$

(16)

Putting $h=2$ in (14), we get

$$\begin{array}{cc}\hfill {\int }_{s_0}^k{\left|D_{\beta }{\phi }_k\left(\xi \right)\right|}^2{d}_{\beta }\xi & \ge (2{\epsilon }^2-3\epsilon +1)\left[{\sigma }_{22,\beta }\left(\eta \right)-{\sigma }_{22,\beta }(-\eta )\right]+\hfill \\ \hfill & +\epsilon (2\epsilon -1)\left[{\sigma }_{11,\beta }\left(\eta \right)-{\sigma }_{11,\beta }(-\eta )\right].\hfill \end{array}$$

(17)

If we add the inequalities (16) and (17), then we get

$$\begin{array}{cc}\hfill {\int }_{s_0}^k{\left|{\phi }_k\left(\xi \right)\right|}^2{d}_{\beta }\xi & +{\int }_{s_0}^k{\left|D_{\beta }{\phi }_k\left(\xi \right)\right|}^2{d}_{\beta }\xi \ge \hfill \\ \hfill & \ge {\left(2\epsilon -1\right)}^2\left[{\sigma }_{11,\beta }\left(\eta \right)-{\sigma }_{11,\beta }(-\eta )+{\sigma }_{22,\beta }\left(\eta \right)-{\sigma }_{22,\beta }(-\eta )\right].\hfill \end{array}$$

Therefore the assertion of the lemma for the functions ${\sigma }_{11,\beta }(-\eta )$ and ${\sigma }_{22,\beta }(-\eta )$ follows from their monotonicity, whereas we get the result of the lemma for the function ${\sigma }_{12,\beta }(-\eta )$ by (15). □

Now it is worth to mention the so-called Helly’s selection theorem in [32]:

Suppose that $\left\{{\phi }_n\right\}$ is a uniformly bounded sequence of functions with uniformly bounded variation on the interval $\left[a,b\right]$. Then there exist a sub-sequence $\left\{{\phi }_{n_k}\right\}$ that convergence everywhere on $\left[a,b\right]$ to a function $\phi$ of bounded variation. Moreover for every continuous function $\psi$ on $\left[a,b\right]$,

$$\underset{n\to \infty }{lim}{\int }_a^b\psi \left(\lambda \right)d{\phi }_n\left(\lambda \right)={\int }_a^b\psi \left(\lambda \right)d\phi \left(\lambda \right).$$

Let us set forth the Hilbert space $\mathcal{H}:=L_{\beta }^2(-q_{\beta }^m,q_{\beta }^m)$ by giving the inner product

$${〈\psi ,\phi 〉}_{\mathcal{H}}:={\int }_{-q_{\beta }^m}^{q_{\beta }^m}\psi \left(\xi \right)\overline{\phi \left(\xi \right)}{d}_{\beta }\xi .$$

Let $\sigma$ be a non-decreasing function on $(-q_{\beta }^m,q_{\beta }^m)$. We denote all measurable real functions of Hilbert space by $L_{\sigma }^2(-q_{\beta }^m,q_{\beta }^m)$ which holds

$${\int }_{-q_{\beta }^m}^{q_{\beta }^m}{\phi }^2\left(\lambda \right)d\sigma \left(\lambda \right)<\infty ,$$

with the inner product

$${〈\psi ,\phi 〉}_{\sigma }:={\int }_{-q_{\beta }^m}^{q_{\beta }^m}\psi \left(\lambda \right)\phi \left(\lambda \right)d\sigma \left(\lambda \right).$$

We now state our main result of the study as follows.

Theorem 1.

Let $\phi (.)\in L_{\beta }^2(-q_{\beta }^m,q_{\beta }^m)$. Then, there exist monotonic functions ${\sigma }_{11}\left(\lambda \right)$ and ${\sigma }_{22}\left(\lambda \right)$, which are bounded over every finite interval, and a function ${\sigma }_{12}\left(\lambda \right)$, which is of bounded variation over every finite interval with the property

$${\int }_{-\infty }^{\infty }{\phi }^2\left(\xi \right)d_{\beta }\left(\xi \right)={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\Phi }_j\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right),$$

(18)

where

$${\Phi }_i\left(\lambda \right)=\underset{m\to \infty }{lim}{\int }_{-q_{\beta }^m}^{q_{\beta }^m}\phi \left(\xi \right){\psi }_i(\xi ;\lambda )d_{\beta }\xi .$$

We note that the function $\sigma ={\left\{{\sigma }_{ij}\right\}}_{i,j=1}^2,({\sigma }_{12}={\sigma }_{21})$ is called spectral function for the equation (4).

Proof. 

Assume that the function ${\phi }_m\left(\xi \right)$ satisfies the following conditions:

(a) ${\phi }_m\left(\xi \right)$ vanishes identically zero outside the interval $\left[-q_{\beta }^m,q_{\beta }^m\right]$ where $q_{\beta }^n<q_{\beta }^m$.

(b) Both ${\phi }_m\left(\xi \right)$ and $D_{\beta }{\phi }_m\left(\xi \right)$ are $\beta$-regular at $s_0$.

(c) ${\phi }_m\left(\xi \right)$ satisfyies the boundary conditions (5)-(6).

Using the Parseval identity (9) to the function ${\psi }_m\left(\xi \right)$ we obtain;

$${\int }_{-q_{\beta }^m}^{q_{\beta }^m}{\phi }_m^2\left(\xi \right)d_{\beta }\xi ={\displaystyle \sum _{k=-\infty }^{\infty }}{\displaystyle \frac{1}{{\gamma }_k^2}}{\left\{{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)y_k\left(\xi \right)d_{\beta }\xi \right\}}^2.$$

(19)

Then, by integrating bu parts, we obtain

$$\begin{array}{cc}\hfill {\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)y_k\left(\xi \right)d_{\beta }\xi & ={\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)\mathcal{L}{y}_k\left(\xi \right)d_{\beta }\xi \hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}\left(\begin{array}{cc}{\phi }_{m1}& {\phi }_{m2}\end{array}\right)\left(\begin{array}{c}-D_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{y}_{k2}+p{y}_{k1}\\ D_{\beta }{y}_{k1}+r{y}_{k2}\end{array}\right)d_{\beta }\xi \hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_{m1}[-D_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{y}_{k2}+p{y}_{k1}]d_{\beta }\xi +\hfill \\ \hfill & +{\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_{m2}[D_{\beta }{y}_{k1}+r{y}_{k2}]d_{\beta }\xi \hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}\left[-y_{k2}{D}_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{\phi }_{m1}+p{\phi }_{m1}{y}_{k1}\right]d_{\beta }\xi +\hfill \\ \hfill & +{\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}\left[y_{k1}{D}_{\beta }{\phi }_{m2}+r{\phi }_{m2}{y}_{k2}\right]d_{\beta }\xi ,\hfill \end{array}$$

where ${\phi }_{mi}:={\phi }_{mi}\left(\xi \right),y_{ki}:=y_{ki}\left(\xi \right),\left(i=1,2\right),p:=p\left(\xi \right),r:=r\left(\xi \right).$ Thus, we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{\left|{\lambda }_k\right|\ge s}\frac{1}{{\gamma }_k^2}{\left\{{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)y_k\left(\xi \right)d_{\beta }\xi \right\}}^2\le }\\ & \le & \sum _{\left|{\lambda }_k\right|\ge s}{\displaystyle \frac{1}{{\gamma }_k^2}}\left\{{\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}(-y_{k2}{D}_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{\phi }_{m1}+p{\phi }_{m1}{y}_{k1})d_{\beta }\xi +\right.\hfill \\ & & +{\left.{\displaystyle \frac{1}{{\lambda }_k}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}(y_{k1}{D}_{\beta }{\phi }_{m2}+r{\phi }_{m2}{y}_{k2})d_{\beta }\xi \right\}}^2\hfill \\ & \le & {\displaystyle \frac{1}{s^2}}{\displaystyle \sum _{k=-\infty }^{\infty }}{\displaystyle \frac{1}{{\gamma }_k^2}}\left\{{\int }_{-q_{\beta }^n}^{q_{\beta }^n}(-y_{k2}{D}_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{\phi }_{m1}+p{\phi }_{m1}{y}_{k1})d_{\beta }\xi +\right.\hfill \\ & & +{\left.{\int }_{-q_{\beta }^n}^{q_{\beta }^n}(y_{k1}{D}_{\beta }{\phi }_{m2}+r{\phi }_{m2}{y}_{k2})d_{\beta }\xi \right\}}^2\hfill \\ & =& {\displaystyle \frac{1}{s^2}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\left[(-D_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{\phi }_{m1}+p{\phi }_{m1})+(D_{\beta }{\phi }_{m2}+r{\phi }_{m2})\right]}^2{d}_{\beta }\xi .\hfill \end{array}$$

Using (19), we obtain

$$\begin{array}{c}\hfill {\displaystyle \left|{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)y_k\left(\xi \right)d_{\beta }\xi -\sum _{-\tau \le {\lambda }_k\le \tau }\frac{1}{{\gamma }_k^2}{\left\{{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)y_k\left(\xi \right)d_{\beta }\xi \right\}}^2\right|\le }\\ & \le & {\displaystyle \frac{1}{s^2}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\left[(-D_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{\phi }_{m1}+p{\phi }_{m1})+(D_{\beta }{\phi }_{m2}+r{\phi }_{m2})\right]}^2{d}_{\beta }\xi .\hfill \end{array}$$

Furthermore, we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{-\tau \le {\lambda }_k\le \tau }\frac{1}{{\gamma }_k^2}{\left\{{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)y_k\left(\xi \right)d_{\beta }\xi \right\}}^2=}\\ & =& \sum _{-\tau \le {\lambda }_k\le \tau }{\displaystyle \frac{1}{{\gamma }_k^2}}{\left\{{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right)\left(c_k{\phi }_1(\xi ;{\lambda }_k)+d_k{\phi }_2(\xi ;{\lambda }_k)\right)d_{\beta }\xi \right\}}^2\hfill \\ & =& {\int }_{-\tau }^{\tau }{\displaystyle \sum _{i,j=1}^2}{\Phi }_{im}\left(\lambda \right){\Phi }_{jm}\left(\lambda \right)d{\sigma }_{ij,\beta }\left(\lambda \right)\hfill \end{array}$$

where

$${\Phi }_{im}\left(\lambda \right)={\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m\left(\xi \right){\psi }_i(\xi ;\lambda )d_{\beta }\xi (i=1,2).$$

Consequently, we get

$$\begin{array}{c}\hfill {\displaystyle \left|{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m^2\left(\xi \right)d_{\beta }\xi -{\int }_{-\tau }^{\tau }{\displaystyle \sum _{i,j=1}^2}{\Phi }_{im}\left(\lambda \right){\Phi }_{jm}\left(\lambda \right)d{\sigma }_{ij,\beta }\left(\lambda \right)\right|\le }\\ & \le & {\displaystyle \frac{1}{{\tau }^2}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\left[(-D_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{\phi }_{m1}+p{\phi }_{m1})+(D_{\beta }{\phi }_{m2}+r{\phi }_{m2})\right]}^2{d}_{\beta }\xi .\hfill \end{array}$$

(20)

By Lemma 1 and appliying Helly’s selection theorem we can find subsequence $\left\{-q_{\beta }^{n_k}\right\}$ and $\left\{q_{\beta }^{n_k}\right\}$ such that the function ${\sigma }_{ij,\beta }\left(\lambda \right)$ converges to monotone function ${\sigma }_{ij}\left(\lambda \right)$. Passing to the limit with along the subsequemces $\left\{-q_{\beta }^{n_k}\right\}$ and $\left\{q_{\beta }^{n_k}\right\}$ in (20), we obtain

$$\begin{array}{c}\hfill {\displaystyle \left|{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m^2\left(\xi \right)d_{\beta }\xi -{\int }_{-\tau }^{\tau }{\displaystyle \sum _{i,j=1}^2}{\Phi }_{im}\left(\lambda \right){\Phi }_{jm}\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right)\right|\le }\\ & \le & {\displaystyle \frac{1}{{\tau }^2}}{\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\left[(-D_{\beta }{\beta }^{-1}{D}_{{\beta }^{-1}}{\phi }_{m1}+p{\phi }_{m1})+(D_{\beta }{\phi }_{m2}+r{\phi }_{m2})\right]}^2{d}_{\beta }\xi .\hfill \end{array}$$

As $\tau \to \infty$, we get

$${\int }_{-q_{\beta }^n}^{q_{\beta }^n}{\phi }_m^2\left(\xi \right)d_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_{im}\left(\lambda \right){\Phi }_{jm}\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right).$$

Let $\phi (.)\in L_{\beta }^2\left(\mathbb{R}\right)$. Choose functions $\left\{{\phi }_{\kappa }\left(\xi \right)\right\}$ satisfying conditions (a)-(c) and such that

$$\underset{\kappa \to \infty }{lim}{\int }_{-\infty }^{\infty }{\left(\phi \left(\xi \right)-{\phi }_{\kappa }\left(\xi \right)\right)}^2{d}_{\beta }\xi =0.$$

Let

$${\Phi }_{i\kappa }\left(\lambda \right)={\int }_{-\infty }^{\infty }{\phi }_{\kappa }\left(\xi \right){\psi }_i(\xi ;\lambda )d_{\beta }\xi ,(i=1,2).$$

Then, we have

$${\int }_{-\infty }^{\infty }{\phi }_{\kappa }^2\left(\xi \right)d_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_{i\kappa }\left(\lambda \right){\Phi }_{j\kappa }\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right).$$

Since

$${\int }_{-\infty }^{\infty }{\left({\phi }_{{\kappa }_1}\left(\xi \right)-{\phi }_{{\kappa }_2}\left(\xi \right)\right)}^2{d}_{\beta }\xi \to 0\left({\kappa }_1,{\kappa }_2\to \infty \right).$$

We have

$${\int }_{-\infty }^{\infty }\left({\Phi }_{i{\kappa }_1}\left(\lambda \right){\Phi }_{j{\kappa }_1}\left(\lambda \right)-{\Phi }_{i{\kappa }_2}\left(\lambda \right){\Phi }_{j{\kappa }_2}\left(\lambda \right)\right)d{\sigma }_{ij}\left(\lambda \right)=$$

$$={\int }_{-\infty }^{\infty }{\left({\phi }_{{\kappa }_1}\left(\xi \right)-{\phi }_{{\kappa }_2}\left(\xi \right)\right)}^2{d}_{\beta }\xi \to 0\left({\kappa }_1,{\kappa }_2\to \infty \right).$$

Therefore, there is a limit function $\Phi$ that satisfies

$${\int }_{-\infty }^{\infty }{\phi }^2\left(\xi \right)d_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\Phi }_j\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right),$$

since $L_{\sigma }^2\left(\mathbb{R}\right)$ is complete.

Now we will show that the sequence $\left\{K_{\kappa }\right\}$

$$K_{\kappa }\left(\lambda \right)={\int }_{-q_{\beta }^n}^{q_{\beta }^n}\left\{{\phi }_1\left(\xi \right){\psi }_1(\xi ;\lambda )+{\phi }_2\left(\xi \right){\psi }_2(\xi ;\lambda )\right\}{d}_{\beta }\xi$$

converges as $\kappa \to \infty$ to $\Phi$ in the metric space $L_{\sigma }^2\left(\mathbb{R}\right)$. Let ${\phi }^{\ast }$ be another function in $L_{\beta }^2\left(\mathbb{R}\right)$. Similarly , ${\Phi }^{\ast }\left(\lambda \right)$ is defined by ${\phi }^{\ast }$. It is obvious that

$${\int }_{s_0}^{\infty }{\left(\phi \left(\xi \right)-{\phi }^{\ast }\left(\xi \right)\right)}^2{d}_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}\left\{\left({\Phi }_i\left(\lambda \right)-{\Phi }_i^{\ast }\left(\lambda \right)\right)\left({\Phi }_j\left(\lambda \right)-{\Phi }_j^{\ast }\left(\lambda \right)\right)\right\}d{\sigma }_{ij}\left(\lambda \right).$$

Let

$${\phi }^{\ast }\left(\xi \right)=\left\{\begin{array}{cc}\phi \left(\xi \right);\hfill & \xi \in \left[-q_{\beta }^{\kappa },q_{\beta }^{\kappa }\right]\hfill \\ 0;\hfill & otherwise\hfill \end{array}\right.,$$

then, we have

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}\left\{\left({\Phi }_i\left(\lambda \right)-K_{\kappa i}\left(\lambda \right)\right)\left({\Phi }_j\left(\lambda \right)-K_{\kappa j}\left(\lambda \right)\right)\right\}d{\sigma }_{ij}\left(\lambda \right)\hfill \\ \hfill & ={\int }_{-\infty }^{-q_{\beta }^{\kappa }}{\phi }^2\left(\xi \right)d_{\beta }\xi +{\int }_{q_{\beta }^{\kappa }}^{\infty }{\phi }^2\left(\xi \right)d_{\beta }\xi \to 0(\kappa \to \infty ),\hfill \end{array}$$

which proves that $\left\{K_{\kappa }\right\}$ converges to $\varphi$ in $L_{\sigma }^2\left(\mathbb{R}\right)$ as $\kappa \to \infty$. □

Theorem 2.

Assume that the functions $\phi (.)$ and ${\phi }^{\ast }(.)$ are belongs to $L_{\beta }^2\left(\mathbb{R}\right)$, also $\Phi \left(\lambda \right)$ and ${\Phi }^{\ast }\left(\lambda \right)$ denote their Fourier transforms. Then, we have

$${\int }_{-\infty }^{\infty }\phi \left(\xi \right){\phi }^{\ast }\left(\xi \right)d_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\Phi }_j^{\ast }\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right),$$

which is referred that the generalized Parseval equality.

Proof. 

It is evident that $\Phi \mp {\Phi }^{\ast }$ are transforms of $\phi \mp {\phi }^{\ast }$. Therefore, we have

$${\int }_{-\infty }^{\infty }{\left(\phi \left(\xi \right)+{\phi }^{\ast }\left(\xi \right)\right)}^2{d}_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}\left\{\left({\Phi }_i\left(\lambda \right)+{\Phi }_i^{\ast }\left(\lambda \right)\right)\left({\Phi }_j\left(\lambda \right)+{\Phi }_j^{\ast }\left(\lambda \right)\right)\right\}d{\sigma }_{ij}\left(\lambda \right)$$

and

$${\int }_{-\infty }^{\infty }{\left(\phi \left(\xi \right)-{\phi }^{\ast }\left(\xi \right)\right)}^2{d}_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}\left\{\left({\Phi }_i\left(\lambda \right)-{\Phi }_i^{\ast }\left(\lambda \right)\right)\left({\Phi }_j\left(\lambda \right)-{\Phi }_j^{\ast }\left(\lambda \right)\right)\right\}d{\sigma }_{ij}\left(\lambda \right).$$

Taking the difference of these two equalities we get the desired result. □

Theorem 3.

Let $\phi (.)\in L_{\beta }^2\left(\mathbb{R}\right)$. Then, the integrals

$${\int }_{-\infty }^{\infty }{\Phi }_i\left(\lambda \right){\psi }_j(\xi ;\lambda )d{\sigma }_{ij}\left(\lambda \right)(i,j=1,2)$$

converge in $L_{\beta }^2\left(\mathbb{R}\right)$. Consequently, we get

$$\phi \left(\xi \right)={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\psi }_j(\xi ;\lambda )d{\sigma }_{ij}\left(\lambda \right)$$

which is known as the expansion theorem.

Proof. 

Take any function ${\phi }_s\in L_{\beta }^2\left(\mathbb{R}\right)$ and any positive number s, an set

$${\phi }_s\left(\xi \right)={\int }_{-s}^s{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\psi }_j(\xi ;\lambda )d{\sigma }_{ij}\left(\lambda \right).$$

Let ${\phi }^{\ast }(.)\in L_{\beta }^2\left(\mathbb{R}\right)$ be a vector valued function on vanishing outside the finite interval $\left[-q_{\beta }^{\tau },q_{\beta }^{\tau }\right]$, where $q_{\beta }^n<q_{\beta }^{\tau }$. Thus, we obtain

$$\begin{array}{cc}\hfill & {\int }_{-q_{\beta }^{\tau }}^{q_{\beta }^{\tau }}{\phi }_s\left(\xi \right){\phi }^{\ast }\left(\xi \right)d_{\beta }\xi ={\int }_{-q_{\beta }^{\tau }}^{q_{\beta }^{\tau }}\left({\int }_{-s}^s{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\psi }_j(\xi ;\lambda )\right){\phi }^{\ast }\left(\xi \right)d_{\beta }\xi \hfill \\ \hfill & ={\int }_{-s}^s{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right)\left\{{\int }_{-q_{\beta }^{\tau }}^{q_{\beta }^{\tau }}{\phi }^{\ast }\left(\xi \right){\psi }_j(\xi ;\lambda )d_{\beta }\xi \right\}d{\sigma }_{ij}\left(\lambda \right)\hfill \\ \hfill & ={\int }_{-s}^s{\Phi }_i\left(\lambda \right){\Phi }_j^{\ast }\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right).\hfill \end{array}$$

(21)

From Theorem 2, we get

$${\int }_{-\infty }^{\infty }\phi \left(\xi \right){\phi }^{\ast }\left(\xi \right)d_{\beta }\xi ={\int }_{-\infty }^{\infty }{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\Phi }_j^{\ast }\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right)$$

(22)

By (21) and (22), we have

$${\int }_{-\infty }^{\infty }\left(\phi \left(\xi \right)-{\phi }_s\left(\xi \right)\right){\phi }^{\ast }\left(\xi \right)d_{\beta }\xi ={\int }_{\left|\lambda \right|>s}{\displaystyle \sum _{i,j=1}^2}{\Phi }_i\left(\lambda \right){\Phi }_j^{\ast }\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right),$$

using the Cauchy-Schwarz inequality, we obtain

$${\int }_{-\infty }^{\infty }\left(\phi \left(\xi \right)-{\phi }_s\left(\xi \right)\right){\phi }^{\ast }\left(\xi \right)d_{\beta }\xi \le {\displaystyle \sum _{i,j=1}^2}\sqrt{{\int }_{\left|\lambda \right|>s}{\Phi }_i{\left(\lambda \right)}^{2}d{\sigma }_{ij}\left(\lambda \right)}\sqrt{{\int }_{\left|\lambda \right|>s}{\Phi }_j^{\ast }{\left(\lambda \right)}^{2}d{\sigma }_{ij}\left(\lambda \right)}.$$

We apply the inequality to the function

$${\phi }^{\ast }\left(\xi \right)=\left\{\begin{array}{cc}\phi \left(\xi \right)-{\phi }_s\left(\xi \right);\hfill & \xi \in \left[-q_{\beta }^s,q_{\beta }^s\right]\hfill \\ 0;\hfill & otherwise,\hfill \end{array}\right.,$$

then, we get

$${\int }_{-\infty }^{\infty }{\left(\phi \left(\xi \right)-{\phi }_s\left(\xi \right)\right)}^2{d}_{\beta }\xi \le {\displaystyle \sum _{i,j=1}^2}{\int }_{\left|\lambda \right|>s}{\Phi }_i\left(\lambda \right){\Phi }_j\left(\lambda \right)d{\sigma }_{ij}\left(\lambda \right).$$

Taking the limit as $s\to$∞ we obtain the expansion result. □

4. Discussion

Prior to the present work, several important results concerning q-Dirac and $\beta$-difference operators have been established in the literature. In particular, eigenfunction expansions, spectral properties, and resolvent operators for various classes of q-Dirac systems have been investigated extensively. Motivated by these developments, this study considers a $\beta$- Dirac problem on the whole line within the framework of generalized quantum difference calculus. The main contribution of this work is the establishment of a spectral function associated with the considered $\beta$-Dirac operator. Using this spectral function, we derive a Parseval-type identity and obtain an eigenfunction expansion theorem. These results extend classical spectral analysis techniques to the setting of $\beta$-Dirac systems and provide a foundation for further investigations of inverse problems, spectral representations, and quantum equations. The obtained expansion formula demonstrates the completeness of the eigenfunctions and highlights their role in representing elements of the underlying Hilbert space. Consequently, the results presented here contribute to the growing spectral theory of generalized quantum operators and their applications. Future research may focus on inverse spectral problems, scattering theory, and non-self-adjoint $\beta$-Dirac systems, where the spectral function established in this paper can serve as a fundamental analytical tool.