CC BY
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Vladikavkaz Mathematical Journal 2021, Volume 23, Issue 3, P. 16-26
УДК 517.927.4
DOI 10.46698/y9113-7002-9720-u
TITCHMARSH-WEYL THEORY OF THE SINGULAR HAHN-STURM-LIOUVILLE EQUATION
B. P. Allahverdiev1 and H. Tuna2
1 Department of Mathematics, Süleyman Demirel University,
32260 Isparta, Turkey;
2 Department of Mathematics, Mehmet Akif Ersoy University,
15030 Bürdür, Türkey E-mail: bilenderpasaoglu@sdu.edu.tr, hustuna@gmail.com
Dedicated to the 80th anniversary of Stefan Grigorievich Samko
Abstract. In this work, we will consider the singular Hahn-Sturm-Liouville difference equation defined by — -i Dw,qy(x) + v(x)y(x) = Ay(x), x £ (w0, to), where A is a complex parameter, v is
a real-valued continuous function at w0 defined on [w0, to). These type equations are obtained when the ordinary derivative in the classical Sturm-Liouville problem is replaced by the w,q-Hahn difference operator Dw,q. We develop the w,q-analogue of the classical Titchmarsh-Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn-Sturm-Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson-Norlund integral and then we study families of regular Hahn-Sturm-Liouville problems on [w0,q-n], n £ N. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.
Key words: Hahn's Sturm-Liouville equation, limit-circle and limit-point cases, Titchmarsh-Weyl theory. Mathematical Subject Classification (2010): 39A13, 39A70, 34B20, 39A12.
For citation: Allahverdiev, B. P. and Tuna, H. Titchmarsh-Weyl Theory of the Singular Hahn-Sturm-Liouville Equation, Vladikavkaz Math. J., 2021, vol. 23, no. 3, pp. 16-26. DOI: 10.46698/y9113-7002-9720-u.
1. Introduction
The Titchmarsh-Weyl theory is at the heart of the area of ordinary differential operators and deals with the existence of square integrable solutions. This theory has led to important contributions over the years to our understanding of the spectral properties of differential operators. H. Weyl introduced this theory in 1910. In [1], Weyl proved that the singular Sturm-Liouville problem of the type
— (p (x) y' (x))' + q (x) y (x) = A y (x), 0 < x < to,
has a non-trivial square integrable solution. He constructed a sequence of nested circles which converges to a circle or a point and defined the limit-point, limit-circle classification. This theory has been attracting the attention of many researchers; see, for instance, [2-6].
© 2021 Allahverdiev, B. P. and Tuna, H.
The study of the Hahn difference operator first appeared in [7, 8] where the quantum difference operator Dw>q was introduced. Such an operator is known to be a generalization of the forward difference operator and the quantum q-difference operator defined by Jackson [9]. Hahn difference operators have received considerable attention due to their applications in the construction of families of orthogonal polynomials and approximation problems, see [10-14] and the references therein.
There are some papers in the literature dealing with Hahan's difference equations. In [15, 16], Hamza and Ahmed studied the existence and uniqueness of solution for the initial value problems for Hahn's difference equations. Moreover they proved Gronwall's and Bernoulli's inequalities with respect to the Hahn difference operator and investigated the mean value theorems for this calculus. In 2016, Hamza and Makharesh [17] investigated Leibniz's rule and Fubini's theorem associated with Hahn's difference operator. Sitthiwirattham [18] consider the nonlocal boundary value problem for nonlinear Hahn's difference equation. In [19], the regular Hahn-Sturm-Liouville problem
is studied, where w0 ^ x < to, a € C, a^b € R := (-to, to), i = 1,2, and p(-) is a real-valued function defined on [w0, b] and continuous at w0. Annaby et al. [19] define a Hilbert space of w,q-square summable functions. The authors discussed the formulation of the self-adjoint operator and the properties of the eigenvalues and the eigenfunctions. Furthermore, they construct the Green's function and give an eigenfunction expansion theorem.
In this paper, we attempt to study the w,q-analogue of the classical Titchmarsh-Weyl theory.
The paper is organized as follows. In Section 2, we summarize all the necessary definitions and properties of Hahn's difference operator. In Section 3, we formulate the singular Hahn-Sturm-Liouville difference equation and develop the classical Titchmarsh-Weyl theory for such equation.
In this section, our aim is to present some basic concepts concerning the theory of Hahn calculus. For more details, the reader may refer [7, 8, 19, 20]. Throughout the paper, we let q € (0,1) and w > 0.
Define w0 := w/ (1 — q) and let I be a real interval containing w0.
Definition 1 [7, 8]. Let f : I ^ R be a function. The Hahn difference operator is defined
provided that f is differentiable at w0. In this case, we call Dw>qf, the w, q-derivative of f.
Remark 1. The Hahn difference operator unifies two well known operators. When q ^ 1, we get the forward difference operator, which is defined by
q 1D-Wq-l,q-l Dw,q y(x) + v(x)y(x) = A y(x),
aiy(wo) + a2D-uq-1,q-1 y(w o) = 0, biy(b)+ b2D_Wq-1,q-1 y(b) =0,
2. Notation
by
(1)
Aw f (x) :
f(u + x)-f (x) (w + x) — x
x € R.
When w ^ 0, we get the Jackson q-difference operator, which is defined by
(qx) — x
Furthermore, under appropriate conditions, we have
lim f (x) = f ' (x).
In what follows, we present some important properties of the w, q-derivative.
Theorem 1 [20]. Let f, g : I ^ R be w, q-differentiable at x € I and h (x) := w + qx. Then for all x € I we have:
i) Du,q (af + bg) (x) = aDw,qf (x) + bDw,qg (x), a, b € I,
ii) Dw,q (fg) (x) = Dw,q (f (x)) g (x) + f (w + xq) Dw,qg (x),
iii) d ll) (x) = U fo)) 9(x)~f (x) Du,g9 (x) U'9\g) g (x) g (w + xq)
iv) Dw,q (h-1 (x)) = D_Wq-1,q-1f (x) .
The concept of the w, q-integral of the function f can be defined as follows.
Definition 2 (Jackson-Norlund integral [20]). Let f : I ^ R be a function and a, b, w0 € I. We define w, q-integral of the function f from a to b by
b b a
I f (x) dw,q (x) := J f (x) dw,q (x) — J f (x) d^,q (x) ,
where
*w,q v^y •— I J \.U/7 U/w,q vav I J vav ^W,q
WQ WQ
/ 1 n \
/ (x) (x) := ((1 - q) x - w) ^ qra/ L + xqra , x € /
n=0 V q /
x 1 — q
W n=0 v
WQ
provided that the series converges at x = a and x = b. In this case, f is called w, q-integrable on [a, b].
Similarly, one can define the w, q-integration for a function f over (w0, to) by
? œ
/ (x) d^ (x) := ((1 - q) - U) £ QV (w +
W n=0 V q /
WQ
The following properties of w, q-integration can be found in [20].
Theorem 2 [20]. Let f, g : I ^ R be w, q-integrable on I, a, b, c € I, a < c < b and
a, P € R. Then the following formulas hold:
b b b
i) j {af (x) + Pg (x) } (x) = a J f (x) d^,q (x) + P j g (x) d^,q (x),
a a a
a
ii) J f (x) d^,q (x) = 0,
a
b c b
iii) J f (x) d^,q (x) = J f (x) d^,q (x) + J f (x) d^,q (x) ,
a a c
b a
iv) J f (x) d^,q (x) = - J f (x) d^,q (x) .
ab
Next, we present the w, q-integration by parts.
Lemma 1 [20]. Let f,g : I ^ R be w, q-integrable on I, a, b € I, and a < b. Then the following formula holds:
b b
/ f (x) DUtqg (x) dui,q (x) + J g (w + qx) DM,q f (x) d^,q (x) = f (b) g (b) - f (a) g (a).
aa
The next result is the fundamental theorem of Hahn's calculus. Theorem 3 [20]. Let f : I ^ R be continuous at w0. Define
x
F (x) := J f (t) d^,q (t) , x € I.
WQ
Then F is continuous at w0. Moreover, DW,qF(x) exists for every x € I and DW,qF(x) = f (x). Conversely,
b
I DUiqF (x) dw,q (x) = f (b) - f (a).
a
Let L"W,q(w0, to) be the space of all complex-valued functions defined on [wo, to) such that
(oo \ \
J\f (x)\2 dw,qx < TO.
WQ /
The space LW,q(w0, to) is a separable Hilbert space with the inner product
oo
(f, g) := J f (x) g (x) du>q x, f,g € L2 tg(u>0, oo)
WQ
(see [20]).
The w, q-Wronskian of functions y(-), z(-) is defined as
Ww,q (y,z)(x):= y (x) Dw,qz (x) - z (x) Dw,qy (x), x € [wo, to). (2)
3. The Singular Hahn—Sturm—Liouville Equation and Titchmarsh—Weyl Theory
In this section, we introduce the Titchmarsh-Weyl theory of the singular Hahn-Sturm-Liouville difference equation.
Consider the singular Hahn-Sturm-Liouville difference equation
r(y):= -q-1D_Wq-i;q-iDu,qy (x) + v (x) y (x) = Xy (x) , x € (wo, to) , (3)
where X is a complex parameter, v is a real-valued continuous function at w0 defined on [w0, to). We note that there exists a unique solution of (4) satisfying the conditions [19]
y (wo) = dl, D_Uq-1 ,q-1 y (wo) = d2,
where d1,d2 € C.
Lemma 2 (see [19]). For any y and z in (w0, to), the following relation holds
x x
J T(y)z du^t - j y T(z) du>qt = [y,z] (x) - [y, z] (w0), (4)
where
[y,z] =9(^,-1,-12) - (D i iy)z
Theorem 4. For each X € C and x € [w0, to), the w, q-Wronskian of any two solution of equation (4) is independent of x.
< Let y and z be two solutions of equation (4). It follows from (4) that
x x
JT(y)-zdUJ^qt - JyT(z) du^t = [y,z] (x) - [y,z](u0)
wo wo
= wu>q (y,z) (h~l (x), A) - Wu>q (y,z) (h~l (coo), A) . Since r(y) = Xy and r(z) = Xz, we have
x
J r(y)z dwqt - j yr(z) dwqt = (X - X) j y(t, X)z(t, X)dw,qt = 0
wo wo wo
= Wwq (y, z) (h- (x) , X) - Wwq (y, z) (h-1 M , X) .
(5)
Then we have Ww,q(y, z)(h-1 (x), X) = Ww,q(y, z)(h-1(w0), X), i. e., the Wronskian is independent of x (x € [wo, to)). >
Now we impose a boundary condition for the solution y of equation (4) as
Dwqy (q-n) sin a + y (q-n) cos a = 0, a € R, n € N := {1, 2,... }. (6)
Let x(x, X) and f(x, X) be the solutions of the equation (4) satisfying
X (w0,X)=sin Z, f (w0,X)=cos Z, D-wq-1,q-1 X (W0, X) = - COS Z, D-wq-1,q-1 f (wo, X) = sin Z,
where 0 ^ Z < n. Since Ww,q(x, f) = 1, the solutions x and f are linearly independent.
x
x
Lemma 3. For x € [w0, to) and A € C, we have
X(x, A) = % (x, A) , <p(x, A) = <p (x, A) . < If x(x, A) is a solution of the equation (4), then we have
—q-1D_Wq- 1q-1 Dw,qx(x, A) + v (x) x(x, A) = Ax(x, A), x € (w0, to). Taking the complex conjugate, we obtain
-Q 1D_UJ„-i„-iDUJ,qx(x,\)+v(x)x(x,\) = \x(x,\), xe(uj0,oo).
By (7), x(x, A) is a solution of
—Q~lD_ujq-iiq-iDUJiqu (x) + v (x) u (x) = Au(x), x € (wo, oo).
But u = x(x,X) is also a solution of the equation (4) with the same conditions (7). By the uniqueness of solutions we get the desired result. >
Lemma 4. If y(x, A) is a solution of equation (4), then we have
b
2ia J \y{x,\)\2dUJ,qx = WUJ,q (y,y) (h~l (b), A) — Wu,q (y,y) (h~l (u0), A) ,
(8)
a = Im A, b > 0, A € C.
< Substituting z(x, A) = y(x, A) in (4), we have (5). Thus, we get (8). > Now using (7) we shall construct a solution $ of (4) as
$(x, A) := x(x, A) + n^(x, A), x € [w0, to),
where n is a constant. If we substitute $ for y in (6) we obtain
(fi + nF) sin a + (Y + n$)cos a = 0, (9)
where
fi = Dw,q x (q_n, A, F = DW)q ^ (q_n, A) , Y = X (q_n,A) , $ = ^ (q_n,A) (n € N).
Then, we get
Y cos a + fi sin a . .
n = —z-:-, 10
$ cos a + F sin a
n is a meromorphic function of A because x(x, A) and <^(x, A) are entire functions of A. Furthermore, since the eigenvalues of the regular problem are real, all poles of n are real and simple. If cos a is replaced by a complex variable z, then we have
T z + Q
V =--• 11
1 $z + F J
It follows from the theory of Mobius transformations [21] that the equation (11) is a one-to-one conformal mapping in z for every A. Hence n describes a circle Cq-n in the complex plane.
The task is now to find the center and the radius of Cq-n (n € N). Let us denote by On rn the center of the circles Cq-n (n € N) and its radius, respectively. Theorem 5. Let X € C, a = Im X = 0. Then, we have
On (X) = -
W„,g (x,lp) (q~n, A) Wu>q(<p,lp) (q~n, A):
/ \ _1
rn (X) = ¡2a J \f(x,X)\2 d^,q xl (n € N).
^ WQ '
(12)
(13)
< On(X) (n € N) is the symmetric point at to. Let z' and z" are in the z-plane such that
n (X,z') = to, n (\,z") =On (X).
Then z' and z" must be symmetric with respect to the real axis of the z—plane, i. e., z' = z". But n(X, z') = to if and only if
Dw,qf(q_n, X)
z = —
f(q_n ,X)
Hence, we have
On (X)= n[X, -
Dw,q f (
^(<?_ra' A) } <P A) + DUA<p (q-, A)
X (q~n, A) DUtq<p {q~n, A) - <p {q~n, A) D^qX (q~n, A) y (q~n, A) DWyqip (q~n, A) - ip {q~n, A) DUtq(p (q~n, A)
WUtq(<p,lp) (q~n, A))
(n € N).
It is evident that rn(X) is the distance between the center of Cq-n and the point n(X, 0) on Cq-n(X). It follows from Ww,q(X,f)(q_n) = 1 (n € N) that
rn (X) =
Du,qx(q~n, A) Wu>q(X,v)(q-n,X)
(x,f)(q_n,X)
Du>q<p (q~n,\) Wu,q (X) lp)(Q~ni A)
Du,g<p (q~n, A) Wu,g(<p,<p) (q~n, A)
WUtg A)
1
\wu,g A) I
(n €
By virtue of Lemma 4, we conclude that
q-n
Wu>g (Lp,Tp)(q~n, A) = 2ia J |^(x,A)|2 d^.
WQ
Thus, we get
q-n
\Wu,g (<p,<p)(q-n,\)\=2\<r\ j |^(x,A)|2 d( which proves the theorem. >
x.
WQ
_n
Now, our next concern will be the behavior of the family of circles {Cq-n(A)} (n € N). Theorem 6. Let A = p + ia. If Im A = a > 0, then the upper half-plane is associated with the exterior of the circle Cq-n (n € N). < It follows from (2) that
( Du,,g<fi (q~n, A) ] = 1 [ Du>q<p (q~n, A) Du,q<p {q~n_, A) \ m\ ip (q~n, A) J ip (q~n, A) <p{q~n,\) J
= 1 W^foy) (g~ra, A) = _ 1 (g~ra, A)
2' (<?"«, A)|2 2' lp (q-n} A)|2
q-n
= , , 1 .n|2 / |^,A)|2dw,qx>0 (neN),
wo
i. e., if a > 0, the exterior of Cq-n (A) is mapped onto the upper half-plane of the z-plane. > Now, we can prove the following result.
Theorem 7. Let Im A = a > 0. Then n lies on the circles Cq-n (n € N) if and only if
q-n
/i 12 Im n
\x(x,X)+ r]Lp(x,X)\ dUJ,qx = (neN), (14)
wo
and n belongs to the interior of Cq-n if and only if
/i |2 Im n
\x(x, A) + r]ip(x, A) du qx <- (neN)
a
wo
< Let n € C. Then, we have
Wu^a (x + W) X + W) (wo, A) = (x, X) (wo, A) + (ip, x) (w0, A) + (w0, A) + |r?|2 W7^ (y?,^) (w0, A) = -r] + rj = —2i Im rj.
It follows from Lemma 4 that
(15)
2<T J \x(x, X)+r]Lp(x,X)\2dUJiqx = i (W^x + ^X + w) (<?""", A)+2iImr?) (neN). (16)
wo
From Theorem 6, if Im z < 0, then n is inside Cq-n (n € N) for v > 0. By (11), we obtain
fi + nF Yz + fi
z = ——-—, T]
Y + + F'
fi = Dw,qX (q_n, A) , F = Dw,q^ (q_n, A) , Y = x (q_n,A) , $ = ^ (q_n,A) (n € N).
Hence,
v 7 \ Y + r?$ Y + r?$J |Y + r?$|2 V
q-n
Then, Im z < 0 if and only if
iWg(x + + w){q-n, A) >0 (neN). (17)
By virtue of (16) and (17), we conclude that
q-n
/i 12 Im n
\x(x,X)+r](p(x,X)\ du,qx <(tie N). (18)
WQ
On the other hand, n is on the circle Cq-n if and only if Im z = 0. Therefore, we have
wu>g(x + v<p,x + w)(q~n,A)=o (neN). (19)
Substituting (19) in (16), we obtain the equality (14). >
Theorem 8. Let Im A = a > 0. The circles Cq-n are nested as n ^ to.
< Let us consider another point k such that q-k < q-n. Using (16) we may write
q-k q-n
/i 12 i i 12 Im n
|x(x, A) + r]Lp(x, A) I du,qx < / |x(x, A) + r]Lp(x, A) I dW;qx < ——.
wo wo
This implies that the point n must be inside the circle Cq-k. >
Corollary 1. The circles Cq-n may converge either to a circle or a point as n ^ to. Definition 3. If Co is a point, then the equation (4) is said to be in the limit-point case. Similarly, if Co is a circle, then the equation (4) is said to be in the limit-circle case.
Theorem 9. Let n be a point lying on or inside the limiting circle Co, and ^(x, A) := X(x, A) + n^(x, A), Im A = a > 0, be the solution of (4). Then A) € LW,q(w0, to), i. e.,
oo
J |x(x, A) + n^(x, A)|2 dw,qx < to.
wo
We note that n is called a Titchmarsh-Weyl function, and ^(x, A) is called a Weyl solution of the equation (4).
< Let n be a point lying on or inside the limiting circle Co. Then we have
q-n
/i 12 Im n
|x(x, A) + rj<p(x, X)\ du,qx < (neN). (20)
WQ
Since the right-hand side of (20) is independent of the point q n, we may pass to the limit as n ^ to. Thus, we get
J |^(x, A)|24,;qx < lmT}
which completes the proof. >
a
WQ
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Received December 30, 2020 BlLENDER P. ALLAHVERDIEV
Department of Mathematics, Süleyman Demirel University,
32260 Isparta, Turkey,
Professor
E-mail: bilenderpasaoglu@sdu. edu.tr
https://orcid.org/0000-0002-9315-4652
HusEYiN Tuna
Department of Mathematics, Mehmet Akif Ersoy University,
15030 Burdur, Turkey,
Professor
E-mail: hustuna@gmail. com
https://orcid.org/0000-0001-7240-8687
Владикавказский математический журнал 2021, Том 23, Выпуск 3, С. 16-26
ТЕОРИЯ ТИТЧМАРША - ВЕЙЛЯ СИНГУЛЯРНОГО УРАВНЕНИЯ ХАНА - ШТУРМА - ЛИУВИЛЛЯ
Аллахвердиев Б. П.1, Туна Х.2
1 Университет имени Сулеймана Демиреля, Турция, 32260, Испарта;
2 Университет Мехмета Акифа Эрсоя, Турция, 15030, Бурдур E-mail: bilenderpasaoglu@sdu.edu.tr, hustuna@gmail.com
Посвящается 80-летию профессора Стефана Григорьевича Самко
Аннотация. В этой работе рассматривается сингулярное разностное уравнение Хана — Штурма — Лиувилля, определяемый уравнением —q-1D_^q-i,q-iD^,qy(x) + v(x)y(x) = Ay(x), x € (w0, то), где A — комплексный параметр, v — вещественнозначная функция, определенная на [w0, то) и непрерывная в точке w0. Такого вида уравнения возникают, когда обычную производную в классической задаче Штурма — Лиувилля заменяется на (ш,д)-Хан разностным оператором D^,q. Развивается (ш,д)-аналог классической теории Титчмарша — Вейля для таких уравнений. Другими словами, изучается существование квадратично интегрируемое решение сингулярного уравнения Хана — Штурма — Лиувилля. Сначала определяется подходящее гильбертово пространство в терминах интеграла Джексона — Нерлунда. Затем изучаются семейства регулярных задач Хана — Штурма — Лиувилля на [w0,q-n], n € N. Далее, определяется семейство окружностей, сходящейся либо к точке, либо к кругу. Тем самым, в исчислении Хана возникают случаи предельной точки или предельной окружности, используя технику Титчмарша.
Ключевые слова: уравнение Хана — Штурма — Лиувилля, предельная окружность и предельная точка, теория Титчмарша — Вейля.
Mathematical Subject Classification (2010): 39A13, 39A70, 34B20, 39A12.
Образец цитирования: Allahverdiev, B. P. and Tuna, H. Titchmarsh-Weyl Theory of the Singular Hahn-Sturm-Liouville Equation // Владикавк. мат. журн.—2021.—Т. 23, № 3.—C. 16-26 (in English). DOI: 10.46698/y9113-7002-9720-u.
