- Research
- Open access
- Published:
Spectral properties and a Parseval’s equality in the singular case for q-Dirac problem
Advances in Difference Equations volume 2019, Article number: 522 (2019)
Abstract
This paper is devoted to studying a q-analog of the singular Dirac problem. First, we investigate some spectral properties of the problem. Then we prove the existence of a spectral function and establish a Parseval’s equality, for the singular q-Dirac system in a Hilbert space. Although there were given some results for this type of problem, we think that Parseval’s equality has not been studied yet.
1 Introduction
In 1910, Jackson introduced the q-derivative operator, \(D_{q}\), different from the classical derivative, and its right-inverse, the q-integration [19, 20]. Then the q-calculus was based on these notations. This calculus has a lot of applications in different mathematical areas, such as calculus of variations, orthogonal polynomials, theory of relativity, quantum theory and statistical physics (see [1, 23, 29]). Furthermore, there are several physical models involving q-functions, q-derivatives, q-integrals and their related problems [9, 11, 12, 15, 28].
In the present paper, we study an analog of Dirac system when the differential operator is replaced by Jackson’s q-difference operator \(D_{q}\) (definitions are given in the next section). Let us consider the q-problem which consists of the q-Dirac system
and the boundary conditions
where λ is the spectral parameter, \(q\in (0,1)\) is fixed, \(p(x)\) and \(r(x)\) are real-valued functions and continuous at zero, and \(p(x),r(x) \in L^{1}_{q}(0,\infty )\).
The limit-point and limit-circle classifications of the singular point, \(x=\infty \) of the q-difference equation were defined in [8], and using Titchmarsh’s technique, sufficient conditions that the singular point is in a limit-point case were given. In the case when \(q=1\), i.e. \(D_{q}=\frac{d}{dx}\), the Dirac system (1.1) was studied in many works, and for more details we refer the reader to Refs. [10, 14, 16, 22, 25–27, 30]. In [2, 3], a q-analog of one dimensional Dirac system on a finite interval was investigated and the authors studied the existence and uniqueness of its solution, and some spectral properties. Also, the asymptotic formulas for the eigenvalues and the eigenfunctions were obtained in [18].
The paper is organized as follows. In Sect. 2, we introduce some q-notations and results that will be useful in the next sections. In Sect. 3, we study some spectral properties of the q-problem (1.1)–(1.3) by the theory of q-(basic) Sturm–Liouville problems [6]. Finally, in Sect. 4, we prove the existence of a spectral function for singular q-Dirac system (1.1), and a Parseval’s equality is established for vector functions in a Hilbert space.
2 q-notations and results
In this section, we introduce some the required q-notations and q-results which will be used throughout the paper. Hereafter, \(q\in (0,1)\) is fixed (for some details, see [6]). We start with the q-shifted factorial for \(\alpha \in \mathbb{R} \) and \(n=1,2,3, \dots \):
A set \(A \subseteq \mathbb{R} \) is called a q-geometricset if \(qx \in A\) for all \(x\in A\).
The q-analogous of sine and cosine functions [4, 6, 13] are defined on \(\mathbb{C}\) by
Let f be a real- or complex-valued function defined on a q-geometric set A. The q-difference operator \(D_{q}\), the Jacksonq-derivative, is defined by
If \(0\in A\), the q-derivative at zero is defined by
if the limit exists and does not depend on x. Hence, for \(x \in A\),
In the q-derivative, as \(q\rightarrow 1\), the q-derivative is reduced to the classical derivative.
The right-inverse to \(D_{q}\), the Jacksonq-integration [20], is given by
provided that the series converges, and
moreover, if \(\lim_{m\rightarrow \infty } f(xq^{m})=f(0)\) for all \(x\in A\) (in this case, we say f is q-regular at zero), then
Hahn [17] defined the q-integration for a function f over \([0,\infty )\) by
Furthermore, the following non-symmetric Leibniz formula holds:
If f and g are q-regular at zero, we get
The q-Wronskian of \(f(x)\) and \(g(x)\) is defined to be
\(\{ Y_{1},Y_{2} \}\) forms a fundamental set of solutions for (1.1) if their q-Wronskian does not vanish at any point of A.
For more details of q-calculus, we also refer the reader to Refs. [5, 7, 13, 24].
3 Spectral properties of the q-Dirac problem
In this section, we investigate some spectral properties of the q-Dirac problem (1.1)–(1.3). Further, the integral equations corresponding to the solution of (1.1) are presented.
Theorem 3.1
The vector eigenfunctions \(Y(x,\lambda _{1})\), \(Y(x,\lambda _{2})\)corresponding to the different eigenvalues \(\lambda _{1}\), \(\lambda _{2}\)are orthogonal.
Proof
Since the vector eigenfunctions
satisfy the q-system (1.1), then
Multiplying (3.1)–(3.4) by \(Y_{1}(x,\lambda _{2})\), \(Y_{2}(x,\lambda _{2})\), \(-Y_{1}(x,\lambda _{1})\) and \(-Y_{2}(x,\lambda _{1})\), respectively, and applying (2.1) we have
Thus, for each \(n\in \mathbb{N}\),
where T is the transposition sign. This together with (2.2) yields
therefore, from (1.2)–(1.3) we obtain
since \(\lambda _{1}\neq \lambda _{2}\), consequently, \(Y(x,\lambda _{1})\) and \(Y(x,\lambda _{2})\) are orthogonal. □
Theorem 3.2
The eigenvalues of theq-Dirac problem (1.1)–(1.3) are real.
Proof
Suppose, on the contrary that \(\lambda ^{0}\) is a non-real eigenvalue of (1.1)–(1.3), and \(Y(x,\lambda ^{0})\) is the corresponding vector eigenfunction of \(\lambda ^{0}\). Then \(\overline{Y(x,\lambda ^{0})}\) is the corresponding vector eigenfunction of \(\overline{ \lambda ^{0}}\). Hence, it follows from \(\lambda ^{0} \neq \overline{ \lambda ^{0}}\) and (3.5) with \(\lambda _{1}=\lambda ^{0}\), \(\lambda _{2}=\overline{\lambda ^{0}}\) that
i.e., \(Y(x,\lambda ^{0})\equiv 0\). This contradiction proves the theorem. □
For each \(n\in \mathbb{N}\), the characteristic function for the problem (1.1)–(1.3) is defined by
Let \(\zeta (x,\lambda )=(\zeta _{1}(x,\lambda ),\zeta _{2}(x,\lambda ))\) be the unique solution [2] of the q-Dirac system (1.1) under the initial conditions
Obviously, \(\zeta (x,\lambda )\) satisfies (1.2). In the following lemma, we present the integral equations corresponding to the solution \(\zeta (x,\lambda )\).
Lemma 3.3
For the solution \(\zeta (x,\lambda )\), the following integral equations hold:
Proof
For \(p(x)=r(x)=0\), the q-system (1.1) has two solutions
with the q-Wronskian
Therefore, in the case when \(p(x)=r(x)=0\),
is a fundamental set of (1.1). Moreover, a particular solution \(\zeta _{p}(x,\lambda )=(\zeta _{1}(x,\lambda ),\zeta _{2}(x,\lambda ) )\) of the q-system (1.1) may be written as
by q-analog of the method of variation of parameters, where \(v_{1}(x), v_{2}(x)\) are q-regular at zero. Substituting (3.10) into (1.1), we obtain the following system:
and hence by Cramer’s rule and applying (3.8) we have
Since \(D_{q^{-1}} v_{i}(x)=(D_{q}v_{i})(x q^{-1})\), it follows from replacing x by xq in (3.11) and integrating from 0 to x that
Now, from (3.9) and (3.12), we can write the general solution of (1.1) as
Using (3.7), (3.13) and (3.14) we obtain \(c_{1}= \frac{1}{2}\), \(c_{2}=\frac{1}{2q}\), and then the proof is complete. □
In the following theorem, we prove another property of the eigenvalues of (1.1)–(1.3).
Theorem 3.4
The eigenvalues of the problem (1.1)–(1.3) are simple.
Proof
The eigenvalues of (1.1)–(1.3) are the zeros of \(\Delta _{n}(\lambda )\). From (3.6) we have
Now, let \(\lambda =\lambda ^{0}\) be a double eigenvalue of (1.1)–(1.3) with the corresponding vector eigenfunction \(Y(x,\lambda ^{0})\). Then \(\Delta _{n}(\lambda ^{0})=0\) and \(\frac{ \partial \Delta _{n}}{\partial \lambda }(\lambda ^{0})=0\), i.e., for \(n\in \mathbb{N}\), the system
has the nontrivial solution \((a,b)=(1,-1)\). Hence,
On the other hand, differentiating (1.1) with respect to λ, we get
Multiplying (1.1) and (3.16) by \(\frac{\partial Y_{1}}{ \partial \lambda }\), \(\frac{\partial Y_{2}}{\partial \lambda }\), \(-Y_{1}\) and \(-Y_{2}\), respectively, and applying (2.1), we obtain
Therefore, integrating with respect to x from 0 to \(q^{-n}\), with applying (2.2), we have
According to Lemma 3.3, \(\frac{\partial Y_{1}}{\partial \lambda } (0, \lambda ^{0})= \frac{\partial Y_{2}}{\partial \lambda } (0, \lambda ^{0}) =0\). Taking this and (3.15) into the left-side of (3.17), we obtain
Consequently, \(Y_{1}(x, \lambda ^{0})=Y_{2}(x, \lambda ^{0}) \equiv 0\), i.e. \(Y(x, \lambda ^{0}) \equiv 0\). Thus, we arrive at the contradiction. The proof is complete. □
4 Spectral function and Parseval’s equality
Let \(\lambda _{m,n}\), \(m\geq 0\), \(n\in \mathbb{N}\), be the eigenvalues of the q-Dirac problem (1.1)–(1.3) (i.e. the roots of \(\Delta _{n}(\lambda )\)) with the corresponding eigenfunctions
If \(f(x)=(f_{1} (x), f_{2}(x))\) is a vector function, \(f_{1},f_{2} \in L^{2}_{q}(0,q^{-n})\), \(n\in \mathbb{N}\), \(Y_{m,n,i}(x )=Y_{i}(x, \lambda _{m,n})\), \(i=1,2\), and
then from [7] we have
Denote the non-decreasing step function \(\rho _{n}\) by
Therefore, (4.1) can be written as
where \(F_{i,n} (\lambda )=\int _{0}^{q^{-n}} f_{i} (x) Y_{i}(x,\lambda ) \,d_{q}x \), \(i=1,2 \).
Lemma 4.1
For any positiveτ, the following inequality holds:
Proof
Since \(\zeta _{1}(x,\lambda )\) and \(\zeta _{2}(x,\lambda )\) are continuous at zero, it follows from (3.7) that there is a positive number τ and nearby zero such that
Denote the vector function \({}_{\tau }f(x)=({}_{\tau }f_{1}(x), {}_{ \tau }f_{2}(x))\) by
Then, from (4.2) and (4.4), we obtain
and we arrive at (4.3). □
The following lemmas were proved in [21].
Lemma 4.2
Let \(\{ v_{n} \}_{n=1}^{\infty }\)be a uniformly bounded sequence of real non-decreasing function ofλon a finite interval \([a,b]\). Then there exist a subsequence \(\{ v_{n_{k}} \}_{k=1}^{ \infty }\)and a non-decreasing functionvsuch that, for \(\lambda \in [a,b]\), \(\lim_{k\rightarrow \infty } v_{n_{k}}(\lambda )=v(\lambda )\).
Lemma 4.3
Assume \(\{ v_{n} \}_{n=1}^{\infty }\)is a real uniformly bounded sequence of real non-decreasing function ofλon a finite interval \([a,b]\), and suppose for \(\lambda \in [a,b]\), \(\lim_{n\rightarrow \infty } v_{n}(\lambda )=v(\lambda )\). Ifgis any continuous function ofλon \([a,b]\), then \(\lim_{n\rightarrow \infty } \int _{a}^{b} g(\lambda ) \,dv_{n}(\lambda )= \int _{a}^{b} g(\lambda ) \,dv(\lambda )\).
Now, let ρ be any non-decreasing function of λ on the interval \((-\infty, \infty )\). We define by \(L^{2}_{\rho }(-\infty, \infty ) \times L^{2}_{\rho }(-\infty, \infty )\) the Hilbert space of all vector functions \(g=(g_{1},g_{2}): (-\infty, \infty )\times (- \infty, \infty ) \rightarrow \mathbb{R}\) which \(g_{1},g_{2}\) are measurable with respect to the Lebesgue–Stieltjes measure defined by ρ, such that \(\int _{-\infty }^{\infty } g_{i}(\lambda ) \,d\rho ( \lambda ) < \infty \), \(i=1,2\), with inner product
In the following theorem, we prove the main result of this section.
Theorem 4.4
For theq-Dirac problem (1.1)–(1.3), there exists a non-decreasing function \(\rho (\lambda )\)on the interval \((-\infty, \infty )\)such that satisfies the following property:
If \(f=(f_{1},f_{2})\)is a vector function, \(f_{i}\in L^{2}_{q }(0,q ^{-n})\), \(i=1,2\), then there exists a function \(F=(F_{1},F_{2}) \in L^{2}_{\rho }(-\infty, \infty ) \times L^{2}_{\rho }(-\infty, \infty )\)such that
and the Parseval’s equality holds:
The function ρ is called the spectral function for the q-problem (1.1)–(1.3).
Proof
Assume that the vector function \(f_{\eta }(x)=(f_{\eta,1}(x),f_{ \eta,2}(x))\) satisfies the following conditions:
- (1)
\(f_{\eta }(0)=(1,1)\);
- (2)
\(f_{\eta }(x)\) vanishes outside \([0,q^{-\eta }]\times [0,q ^{-\eta }]\), \(q^{-\eta } < q^{-n }\);
- (3)
\(f_{\eta,i}(x)\) and \(D_{q} f_{\eta,i}(x)\), \(i=1,2\), are q-regular at zero.
According to (4.2), we can write
where
Since \(Y(x,\lambda )=(Y_{1}(x,\lambda ),Y_{2}(x,\lambda ))\) satisfies the q-system (1.1), we have
Denote
Since \(f_{\eta } (x)\) vanishes in a neighborhood of \((q^{-n }, q^{-n })\), and \(f_{\eta } (0)=Y(0,\lambda )=(1,1)\), using q-integration by parts we get
Applying (4.2), we have, for any \(\tau >0\),
where
Hence, from (4.2) we obtain
Similarly, we have
where
Therefore, it follows from (4.5)–(4.7) that
On the other hand, according to Lemma 4.1, the set \(\{ \rho _{n}( \lambda ) \}\) is bounded. Thus, by Lemmas 4.2 and 4.3, there is a subsequence \(\{ n_{k} \}\) such that \(\{ \rho _{n_{k}}(\lambda ) \}\) converges to a monotone function \(\rho (\lambda )\). Passing to the limit with respect to \(\{ n_{k} \}\) in (4.8), we obtain
So,
as \(\tau \rightarrow \infty \). Now, let \(f=(f_{1},f_{2})\) be an arbitrary vector function in \(L^{2}_{q}(0,\infty )\times L^{2}_{q}(0, \infty ) \). We know that there exists a sequence \(\{ f_{\eta }(x)= (f _{\eta,1}(x),f_{\eta,2}(x) ) \}\) satisfying the conditions (1)–(3) such that
Then
where \(F_{\eta,i} (\lambda )=\int _{0}^{\infty } f_{\eta,i} (x) Y _{i}(x,\lambda )\,d_{q}x\). Since for \(i=1,2\),
as \(\eta _{1}, \eta _{2}\rightarrow \infty \), we get
as \(\eta _{1},\eta _{2}\rightarrow \infty \). This is means that there is a limit vector function \(F=(F_{1},F_{2})\) such that by the completeness of the space \(L^{2}_{\rho }(-\infty,\infty )\times L^{2}_{\rho }(- \infty,\infty ) \),
Now, it remains to show that the function \(\widetilde{F}_{\eta }( \lambda ):=(\widetilde{F}_{\eta,1},\widetilde{F}_{\eta,2})\) with
as \(\eta \rightarrow \infty \), converges to \(F=(F_{1},F_{2})\) in \(L^{2}_{\rho }(-\infty,\infty )\times L^{2}_{\rho }(-\infty,\infty ) \). For this purpose, assume that \(s=(s_{1},s_{2})\) is another function in \(L^{2}_{q}(0,\infty )\times L^{2}_{q}(0,\infty ) \), and by a similar argument, \(S(\lambda )\) is defined by s. Clearly,
For \(i=1,2\), set
Then
as \(\eta \rightarrow \infty \). Consequently, \(\widetilde{F}_{\eta }\) converges to F in \(L^{2}_{\rho }(-\infty,\infty )\times L^{2}_{ \rho }(-\infty,\infty ) \) as \(\eta \rightarrow \infty \). This completes the proof. □
References
Aldwoah, K.A., Malinowska, A.B., Torres, D.F.M.: The power quantum calculus and variational problems. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 19, 93–116 (2012)
Allahverdiev, B.P., Tuna, H.: One dimensional q-Dirac equation. Math. Methods Appl. Sci. 1, 7287–7306 (2017)
Allahverdiev, B.P., Tuna, H.: Dissipative q-Dirac operator with general boundary conditions. Quaest. Math. 41, 239–255 (2018)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Annaby, M.H.: q-Type sampling theorems. Results Math. 44, 214–225 (2003)
Annaby, M.H., Mansour, Z.S.: Basic Sturm–Liouville problems. J. Phys. A, Math. Gen. 38, 3775–3797 (2005)
Annaby, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Springer, Berlin (2012)
Annaby, M.H., Mansour, Z.S., Soliman, I.A.: q-Titchmarsh–Weyl theory: series expansion. Nagoya Math. J. 205, 67–118 (2012)
Chung, K., Chung, W., Nam, S., Kang, H.: New q-derivative and q-logarithm. Int. J. Theor. Phys. 33, 2019–2029 (1994)
Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. A, Math. Phys. Eng. Sci. 117, 610–624 (1928)
Floreanini, R., Vinet, L.: A model for the continuous q-ultraspherical polynomials. J. Math. Phys. 36, 3800–3813 (1995)
Floreanini, R., Vinet, L.: More on the q-oscillator algebra and q-orthogonal polynomials. J. Phys. A, Math. Gen. 28, 287–293 (1995)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, New York (1990)
Gasymov, M.G., Levitan, B.M.: The inverse problem for a Dirac system. Dokl. Akad. Nauk SSSR 167, 967–970 (1966)
Gray, R.W., Nelson, C.A.: A completeness relation for the q-analogue coherent states by q-integration. J. Phys. A, Math. Gen. 23, 945–950 (1990)
Gulsen, T., Yilmaz, E., Koyunbakan, H.: Inverse nodal problem for p-Laplacian Dirac system. Math. Methods Appl. Sci. 40, 2329–2335 (2016)
Hahn, W.: Beiträge zur Theorie der Heineschen Reihen. Math. Nachr. 2, 340–379 (1949) (in German)
Hira, F.: Eigenvalues and eigenfunctions of q-Dirac system. Preprint
Jackson, F.H.: q-Difference equations. Am. J. Math. 32, 305–314 (1910)
Jackson, F.H.: On q-definite integrals. Quart. J. Pure Appl. Math. 41, 193–203 (1910)
Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover, New York (1970). Translated by R.A. Silverman
Levitan, B.M., Sargsjan, I.S.: Sturm–Liouville and Dirac Operators. Nauka, Moscow (1988) (in Russian); English transl, Kluwer Academic Publishers, Dordrecht, 1991
Malinowska, A.B., Torres, D.F.M.: The Hahn quantum variational calculus. J. Optim. Theory Appl. 147, 419–442 (2010)
Mansour, Z.S.: q-difference equations. MSc Thesis, Faculty of Science, Cairo University (2001)
Panakhov, E.S.: Inverse problem for Dirac system in two partially settled spectrum. Vinity 3304, 1–29 (1981)
Panakhov, E.S., Yilmaz, E., Koyunbakan, H.: Inverse nodal problem for Dirac operator. World Appl. Sci. J. 11, 906–911 (2010)
Prats, F., Toll, J.S.: Construction of the Dirac equation central potential from phase shifts and bound states. Phys. Rev. 113, 363–370 (1959)
Suslov, S.K.: Another addition theorem for the q-exponential function. J. Phys. A, Math. Gen. 33, 375–380 (2000)
Tariboon, J., Ntouyas, S.K.: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013, 282 (2013)
Wei, Z., Guo, Y., Wei, G.: Incomplete inverse spectral and nodal problems for Dirac operator. Adv. Differ. Equ. 2015, 188 (2015)
Acknowledgements
The author gratefully acknowledges that this research is partially supported by the University of Kashan under grant number 682482/10.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
Author read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mosazadeh, S. Spectral properties and a Parseval’s equality in the singular case for q-Dirac problem. Adv Differ Equ 2019, 522 (2019). https://doi.org/10.1186/s13662-019-2464-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-019-2464-y