R E S E A R C H
Open Access
Spectral properties and a Parseval’s equality
in the singular case for
q
-Dirac problem
Seyfollah Mosazadeh
1**Correspondence:
1Department of Pure Mathematics,
Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran
Abstract
This paper is devoted to studying aq-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 singularq-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.
MSC: 39A13; 34L40; 34L05; 33D15
Keywords: q-Dirac problem; singularity; spectral function; Parseval’s equality
1 Introduction
In 1910, Jackson introduced the q-derivative operator, Dq, different from the classical
derivative, and its right-inverse, theq-integration [19,20]. Then theq-calculus was based on these notations. This calculus has a lot of applications in different mathematical ar-eas, such as calculus of variations, orthogonal polynomials, theory of relativity, quan-tum theory and statistical physics (see [1,23,29]). Furthermore, there are several phys-ical models involvingq-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’sq-difference operatorDq(definitions are given in the next section).
Let us consider theq-problem which consists of theq-Dirac system
⎧ ⎨ ⎩
–1qDqY2(x,λ) +p(x)Y1(x,λ) =λY1(x,λ), –Dq–1Y1(x,λ) +r(x)Y2(x,λ) =λY2(x,λ),
0≤x≤a≤ ∞, (1.1)
and the boundary conditions
Y1(0,λ) =Y2(0,λ), (1.2)
Y1
q–n–1,λ=Y2
q–n,λ, n∈N, (1.3)
whereλis the spectral parameter,q∈(0, 1) is fixed,p(x) andr(x) are real-valued functions and continuous at zero, andp(x),r(x)∈L1
q(0,∞).
The limit-point and limit-circle classifications of the singular point,x=∞of theq -difference equation were defined in [8], and using Titchmarsh’s technique, sufficient con-ditions that the singular point is in a limit-point case were given. In the case whenq= 1, i.e.Dq= dxd, 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], aq-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 someq-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 ofq-(basic) Sturm–Liouville problems [6]. Finally, in Sect.4, we prove the existence of a spectral function for singularq-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 requiredq-notations andq-results which will be used throughout the paper. Hereafter,q∈(0, 1) is fixed (for some details, see [6]). We start with theq-shifted factorial forα∈Randn= 1, 2, 3, . . . :
(α;q)n:= n–1
k=0
1 –αqk.
A setA⊆Ris called aq-geometric setifqx∈Afor allx∈A.
Theq-analogous of sine and cosine functions [4,6,13] are defined onCby
sin(x;q) := ∞
m=0
(–1)mqm(m+1)(x(1 –q))2m+1 (q;q)2m+1
,
cos(x;q) := ∞
m=0
(–1)mqm2(x(1 –q))2m
(q;q)2m
.
Let f be a real- or complex-valued function defined on a q-geometric setA. Theq -difference operatorDq,the Jackson q-derivative, is defined by
Dqf(x) :=
f(x) –f(qx)
x–qx , x∈A\{0}.
If 0∈A, theq-derivative at zero is defined by
Dqf(0) := lim m→∞
f(xqm) –f(0)
xqm , x∈A,
if the limit exists and does not depend onx. Hence, forx∈A,
Dq–1f(x) = (Dqf)xq–1. (2.1)
The right-inverse toDq,the Jackson q-integration[20], is given by x
0
f(t)dqt:=x(1 –q)
∞
m=0
qmfxqm, x∈A,
provided that the series converges, and
Dq x
0
f(t)dqt=f(x),
moreover, iflimm→∞f(xqm) =f(0) for allx∈A(in this case, we sayf isq-regular at zero),
then
x
0
Dqf(t)dqt=f(x) –f(0). (2.2)
Hahn [17] defined theq-integration for a functionf over [0,∞) by ∞
0
f(t)dqt= (1 –q)
∞
m=–∞
qmfqm.
Furthermore, the following non-symmetric Leibniz formula holds:
Dq(fg)(x) =g(x)Dqf(x) +f(qx)Dqg(x).
Iff andgareq-regular at zero, we get
a
0
g(x)Dqf(x)dqx= (fg)(a) – (fg)(0) – a
0
f(qx)Dqg(x)dqx.
Theq-Wronskian off(x) andg(x) is defined to be
Wq(f,g)(x) :=f(x)Dqg(x) –g(x)Dqf(x), x∈A.
{Y1,Y2}forms a fundamental set of solutions for (1.1) if theirq-Wronskian does not vanish at any point ofA.
For more details ofq-calculus, we also refer the reader to Refs. [5,7,13,24]. 3 Spectral properties of theq-Dirac problem
In this section, we investigate some spectral properties of theq-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,λ1), Y(x,λ2)corresponding to the different
eigenvaluesλ1,λ2are orthogonal.
Proof Since the vector eigenfunctions
satisfy theq-system (1.1), then 1
qDqY2(x,λ1) +p(x)Y1(x,λ1) =λ1Y1(x,λ1), (3.1)
–Dq–1Y1(x,λ1) +r(x)Y2(x,λ1) =λ1Y2(x,λ1), (3.2)
1
qDqY2(x,λ2) +p(x)Y1(x,λ2) =λ2Y1(x,λ2), (3.3)
–Dq–1Y1(x,λ2) +r(x)Y2(x,λ2) =λ2Y2(x,λ2). (3.4)
Multiplying (3.1)–(3.4) byY1(x,λ2),Y2(x,λ2), –Y1(x,λ1) and –Y2(x,λ1), respectively, and applying (2.1) we have
Dq
Y2(x,λ1)Y1
xq–1,λ2
–Y1
xq–1,λ1
Y2(x,λ2)
= (λ1–λ2)
Y1(x,λ1)Y1(x,λ2) +Y2(x,λ1)Y2(x,λ2)
.
Thus, for eachn∈N,
q–n
0
Dq
Y2(x,λ1)Y1
xq–1,λ2
–Y1
xq–1,λ1
Y2(x,λ2)
dqx
= (λ1–λ2)
q–n
0
Y(x,λ1)YT(x,λ2)dqx,
whereTis the transposition sign. This together with (2.2) yields
Y2(x,λ1)Y1
xq–1,λ2
–Y1
xq–1,λ1
Y2(x,λ2)qx–=0n
= (λ1–λ2)
q–n
0
Y(x,λ1)YT(x,λ2)dqx,
therefore, from (1.2)–(1.3) we obtain
(λ1–λ2)
q–n
0
Y(x,λ1)YT(x,λ2)dqx= 0, ∀n∈N, (3.5)
sinceλ1=λ2, consequently,Y(x,λ1) andY(x,λ2) are orthogonal.
Theorem 3.2 The eigenvalues of the q-Dirac problem(1.1)–(1.3)are real.
Proof Suppose, on the contrary thatλ0is a non-real eigenvalue of (1.1)–(1.3), andY(x,λ0) is the corresponding vector eigenfunction ofλ0. ThenY(x,λ0) is the corresponding vector eigenfunction ofλ0. Hence, it follows fromλ0=λ0and (3.5) withλ
1=λ0,λ2=λ0that
q–n
0
Y1
x,λ02+Y2
x,λ02dqx= 0,
For eachn∈N, the characteristic function for the problem (1.1)–(1.3) is defined by
n(λ) :=Y1
q–n–1,λ–Y2
q–n,λ. (3.6)
Letζ(x,λ) = (ζ1(x,λ),ζ2(x,λ)) be the unique solution [2] of theq-Dirac system (1.1) under the initial conditions
ζ1(0,λ) = 1 =ζ2(0,λ). (3.7)
Obviously,ζ(x,λ) satisfies (1.2). In the following lemma, we present the integral equa-tions corresponding to the solutionζ(x,λ).
Lemma 3.3 For the solutionζ(x,λ),the following integral equations hold:
ζ1(x,λ)
=cos(λ√qx;q) –√1
qsin(λ
√
qx;q)
+√q
x
0
cos(λ√qt;q)sin(λ√qx;q) –sin(λ√qt;q)cos(λ√qx;q)r(t)ζ1(t,λ)dqt
–
x
0
cos(λqt;q)cos(λ√qx;q) +√qsin(λqt;q)sin(λ√qx;q)p(qt)ζ2(qt,λ)dqt,
ζ2(x,λ)
=qsin(λx;q) +cos(λx;q)
–q
x
0
cos(λx;q)cos(λ√qt;q) +√qsin(λx;q)sin(λ√qt;q)r(t)ζ1(t,λ)dqt
+q
x
0
cos(λx;q)sin(λqt;q) –sin(λx;q)cos(λqt;q)p(qt)ζ2(qt,λ)dqt.
Proof Forp(x) =r(x) = 0, theq-system (1.1) has two solutions
ψ1(x,λ) =cos(λ√qx;q),qsin(λx;q),
ψ2(x,λ) =–√qsin(λ√qx;q),qcos(λx;q),
with theq-Wronskian
Wq(ψ1,ψ2)(x,λ) =q. (3.8)
Therefore, in the case whenp(x) =r(x) = 0,
ζg(x,λ) =
c1cos(λ√qx;q) –c2√qsin(λ√qx;q)
c1qsin(λx;q) +c2qcos(λx;q)
T
is a fundamental set of (1.1). Moreover, a particular solutionζp(x,λ) = (ζ1(x,λ),ζ2(x,λ)) of
theq-system (1.1) may be written as
⎧ ⎨ ⎩
ζ1(x,λ) =v1(x)cos(λ√qx;q) –v2(x)√qsin(λ√qx;q), ζ2(x,λ) =v1(x)qsin(λx;q) +v2(x)qcos(λx;q),
(3.10)
byq-analog of the method of variation of parameters, wherev1(x),v2(x) areq-regular at zero. Substituting (3.10) into (1.1), we obtain the following system:
⎧ ⎨ ⎩
cos(λq–12x;q)D
q–1v1(x) –√qsin(λq– 1 2x;q)D
q–1v2(x) = –qp(x)ζ2(x,λ), qsin(λx;q)Dq–1v1(x) +qcos(λx;q)Dq–1v2(x) = –qr(xq–1)ζ1(x,λ),
and hence by Cramer’s rule and applying (3.8) we have
⎧ ⎨ ⎩
Dq–1v1(x) = –qp(x)cos(λx;q)ζ2(x,λ) –r(xq–1)√qsin(λq– 1
2x;q)ζ1(xq–1,λ), Dq–1v2(x) =qp(x)sin(λx;q)ζ2(x,λ) –r(xq–1)cos(λq–
1
2x;q)ζ1(xq–1,λ).
(3.11)
SinceDq–1vi(x) = (Dqvi)(xq–1), it follows from replacingxbyxqin (3.11) and integrating
from 0 toxthat
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
v1(x) =c1–√q
x
0r(t)sin(λ
√
qt;q)ζ1(t,λ)dqt
–q0xp(qt)cos(λqt;q)ζ2(qt,λ)dqt,
v2(x) =c2–
x
0r(t)cos(λ
√
qt;q)ζ1(t,λ)dqt
+q0xp(qt)sin(λqt;q)ζ2(qt,λ)dqt.
(3.12)
Now, from (3.9) and (3.12), we can write the general solution of (1.1) as
ζ1(x,λ) = 2c1cos(λ√qx;q) – 2c2√qsin(λ√qx;q)
+√q
x
0
cos(λ√qt;q)sin(λ√qx;q)
–sin(λ√qt;q)cos(λ√qx;q)r(t)ζ1(t,λ)dqt
–
x
0
cos(λqt;q)cos(λ√qx;q)
+√qsin(λqt;q)sin(λ√qx;q)p(qt)ζ2(qt,λ)dqt, (3.13)
ζ2(x,λ) = 2c1qsin(λx;q) + 2c2qcos(λx;q)
–q
x
0
cos(λ√qt;q)cos(λx;q) +√qsin(λ√qt;q)sin(λx;q)r(t)ζ1(t,λ)dqt
+q
x
0
sin(λqt;q)cos(λx;q)
–cos(λqt;q)sin(λx;q)p(qt)ζ2(qt,λ)dqt. (3.14)
In the following theorem, we prove another property of the eigenvalues of (1.1)–(1.3).
Now, letλ=λ0be a double eigenvalue of (1.1)–(1.3) with the corresponding vector eigen-functionY(x,λ0). Then
On the other hand, differentiating (1.1) with respect toλ, we get
⎧
Therefore, integrating with respect toxfrom 0 toq–n, with applying (2.2), we have
4 Spectral function and Parseval’s equality
Letλm,n,m≥0,n∈N, be the eigenvalues of theq-Dirac problem (1.1)–(1.3) (i.e. the roots
ofn(λ)) with the corresponding eigenfunctions
Ym,n(x) =Y(x,λm,n) =
Denote the non-decreasing step functionρnby
ρn(λ) =
Therefore, (4.1) can be written as
q–n
Lemma 4.1 For any positiveτ,the following inequality holds:
positive numberτand nearby zero such that
Then, from (4.2) and (4.4), we obtain
τ
0
τf12(x) +τf22(x)
dqx=
2 τ =
∞
–∞ 2
i=1
τ
0 1
τζi(x,λ)dqx
2
dρn(λ)
≥ τ
–τ
2
i=1
1 τ
τ
0
ζi(x,λ)dqx 2
dρn(λ)
>1 2
ρn(τ) –ρn(–τ)
,
and we arrive at (4.3). The following lemmas were proved in [21].
Lemma 4.2 Let{vn}∞n=1be a uniformly bounded sequence of real non-decreasing function ofλon a finite interval[a,b].Then there exist a subsequence{vnk}∞k=1and a non-decreasing
function v such that,forλ∈[a,b],limk→∞vnk(λ) =v(λ).
Lemma 4.3 Assume{vn}∞n=1is a real uniformly bounded sequence of real non-decreasing function ofλon a finite interval[a,b],and suppose forλ∈[a,b],limn→∞vn(λ) =v(λ).If g
is any continuous function ofλon[a,b],thenlimn→∞abg(λ)dvn(λ) = b
a g(λ)dv(λ).
Now, letρbe any non-decreasing function ofλon the interval (–∞,∞). We define by
L2ρ(–∞,∞)×L2ρ(–∞,∞) the Hilbert space of all vector functionsg= (g1,g2) : (–∞,∞)×
(–∞,∞)→Rwhichg1,g2are measurable with respect to the Lebesgue–Stieltjes measure defined byρ, such that–∞∞ gi(λ)dρ(λ) <∞,i= 1, 2, with inner product
g,hρ:=
∞
–∞
g1(λ)h1(λ) +g2(λ)h2(λ)
dρ(λ).
In the following theorem, we prove the main result of this section.
Theorem 4.4 For the q-Dirac problem(1.1)–(1.3),there exists a non-decreasing function
ρ(λ)on the interval(–∞,∞)such that satisfies the following property:
If f = (f1,f2)is a vector function,fi∈L2q(0,q–n),i= 1, 2,then there exists a function F=
(F1,F2)∈L2ρ(–∞,∞)×L2ρ(–∞,∞)such that
lim n→∞
∞
–∞
F1(λ) +F2(λ) –
q–n
0
f1(x)Y1(x,λ) +f2(x)Y2(x,λ)dqx
dρ(λ) = 0,
and the Parseval’s equality holds:
∞
0
f12(x) +f22(x)dqx=
∞
–∞
F12(λ) +F22(λ)dρ(λ).
The functionρis calledthe spectral functionfor theq-problem (1.1)–(1.3).
(1) fη(0) = (1, 1);
integration by parts we get
+
Therefore, it follows from (4.5)–(4.7) that
So, the completeness of the spaceL2
Then
∞
–∞
Fi(λ) –Fη,i(λ) 2
dρ(λ) = ∞
q–ηf
2
i (x)dqx, i= 1, 2,
asη→ ∞. Consequently,Fηconverges toFinL2ρ(–∞,∞)×L2ρ(–∞,∞) asη→ ∞. This
completes the proof.
Acknowledgements
The author gratefully acknowledges that this research is partially supported by the University of Kashan under grant number 682482/10.
Funding
Not applicable.
Availability of data and materials
Not applicable.
Competing interests
The author declares that they have no competing interests.
Author’s contributions
Author read and approved the final manuscript.
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Received: 1 September 2019 Accepted: 10 December 2019
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