Submitted:
04 August 2026
Posted:
05 August 2026
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Abstract
In the present work, we investigate a β-Dirac problem on the whole-line. Our primary objectives are threefold: to demonstrate the existence of a spectral function, to deduce the associated Parseval identity, and to formulate an eigenfunction expansion theorem that characterizes the behavior of the problem under consideration.
Keywords:
general quantum operator
; β-difference operator
; β-derivative
; β-integral
; β-Dirac system
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 -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 -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 -difference operator, thereby providing essential tools for the analysis of quantum difference equations.
More recently, Cardoso investigated the -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 -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 -calculus. Motivated by this observation, the present study investigates a -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 -calculus. Our main objective is to establish the existence of a spectral function corresponding to the -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 -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 -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 -) difference operator (see [28,29]). derivative and integral which serve as our primary tools play a fundamental role in this study. Let be any interval and assume that a strictly increasing and continuous mapping possessing a unique fixed point satisfying
for every . Moreover, equality holds if and only if . Note that it is possible to replace the above inequality by for . For each function is defined difference operator in [11]:
and it is to be remarked that the point in T is the -derivative at if
exists and belongs to . It is well known that we can get Hahn’s difference operator when , for being the fixed point given by , and we obtain Jackson difference operator when we take , for all .
Let us define for and with . It can be clearly seen that the sequence of functions uniformly converges to the constant function on each compact interval containing (see also [11]).
The -integral, the quantum difference inverse operator, is given by
where
Provided that the infinite series on the the right side is convergent then the function is said -integrable in see [11,16]. Furthermore, the -integration of on is defined in [30] by the formula
assuming that some converges absolutely. Throughout this study, the functions and will be assumed integrable on T and , both continuous at . Let where , then by using the integration by parts formula for the calculus [11,31] the following equality holds;
Let us establish the basic integral theorem of calculus [11,31] : a function is continuous at . Let
Then is continuous at . Furthermore, exists for every and . Moreover,
The separable Hilbert space is defined as follows
with the norm by
and given with the inner product as
(see [30]).
Finally, in this section we remember the Wronskian of on T if
exists. Here the left expression in (2) is a analogous to the ordinary Wronskian. The reader also see for more information about Wronskian and its properties in [14] (pp. 7–8). Moreover, we recall that the -Wronskian of , on T if
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 -Dirac Systems of a form on the general quantum difference operators as follows:
where , p and r are real valued functions defined on and continuous at and .
We recall that is a function satisfying the conditions described in Section 2 being its unique fixed point.
Let be the eqienvalues and be the corresponding eigenfunctions of the -Dirac system where for . Owing the solutions of this system are linearly independent, it follows that
where are the solutions of the -Dirac System (4) which satisfy the initial conditions
and without loss of generality
Now, let us set
where
Let . If we apply the Parseval equality (see [34]) to , then we obtain
Now we introduce a monotone increasing step function on , by
From (8), we obtain
where
We will obtain the Parseval identity for (4), (5) and (6) from (8) by letting .
In this paragraph, it will be explained by reminding what the concepts of bounded and total variation mean: The function is said to be of bounded variation on interval if and only if there exists a positive constant M such that
for all finite partitions of .
If is of bounded variation on , then the total variation of on is defined to be
where the supremum is taken over all partitions of (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 not depending on such that the inequality
holds.
Proof.
From (7)
where is the Kronecker delta. Thus, there exists a such that
Let be a nonnegative function such that vanishes outside the interval with
Now, we apply the Parseval equality (9) to , then we get
where
Using (11) and (12), we obtain
Now if we reapply the Parseval equality (9) to , then we get
If we take in (14), we have
Since
we get
Putting in (14), we get
If we add the inequalities (16) and (17), then we get
Therefore the assertion of the lemma for the functions and follows from their monotonicity, whereas we get the result of the lemma for the function by (15). □
Now it is worth to mention the so-called Helly’s selection theorem in [32]:
Suppose that is a uniformly bounded sequence of functions with uniformly bounded variation on the interval . Then there exist a sub-sequence that convergence everywhere on to a function of bounded variation. Moreover for every continuous function on ,
Let us set forth the Hilbert space by giving the inner product
Let be a non-decreasing function on . We denote all measurable real functions of Hilbert space by which holds
with the inner product
We now state our main result of the study as follows.
Theorem 1.
Let . Then, there exist monotonic functions and , which are bounded over every finite interval, and a function , which is of bounded variation over every finite interval with the property
where
We note that the function is called spectral function for the equation (4).
Proof.
Assume that the function satisfies the following conditions:
(a) vanishes identically zero outside the interval where .
(b) Both and are -regular at .
(c) satisfyies the boundary conditions (5)-(6).
Using the Parseval identity (9) to the function we obtain;
Then, by integrating bu parts, we obtain
where Thus, we have
Using (19), we obtain
Furthermore, we have
where
Consequently, we get
By Lemma 1 and appliying Helly’s selection theorem we can find subsequence and such that the function converges to monotone function . Passing to the limit with along the subsequemces and in (20), we obtain
As , we get
Let . Choose functions satisfying conditions (a)-(c) and such that
Let
Then, we have
Since
We have
Therefore, there is a limit function that satisfies
since is complete.
Now we will show that the sequence
converges as to in the metric space . Let be another function in . Similarly , is defined by . It is obvious that
Let
then, we have
which proves that converges to in as . □
Theorem 2.
Assume that the functions and are belongs to , also and denote their Fourier transforms. Then, we have
which is referred that the generalized Parseval equality.
Proof.
It is evident that are transforms of . Therefore, we have
and
Taking the difference of these two equalities we get the desired result. □
Theorem 3.
Let . Then, the integrals
converge in . Consequently, we get
which is known as the expansion theorem.
Proof.
Take any function and any positive number s, an set
Let be a vector valued function on vanishing outside the finite interval , where . Thus, we obtain
From Theorem 2, we get
By (21) and (22), we have
using the Cauchy-Schwarz inequality, we obtain
We apply the inequality to the function
then, we get
Taking the limit as ∞ we obtain the expansion result. □
4. Discussion
Prior to the present work, several important results concerning q-Dirac and -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 - 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 -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 -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 -Dirac systems, where the spectral function established in this paper can serve as a fundamental analytical tool.
Author Contributions
Conceptualization, N.P.K, C.K and O.A; methodology, N.P.K, C.K and O.A; formal analysis, N.P.K, C.K and O.A; investigation, N.P.K, C.K and O.A; writing—original draft preparation, N.P.K, C.K and O.A; writing—review and editing, N.P.K, C.K and O.A. All authors have read and agreed to the published version of the manuscript.
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