Titchmarsh-Weyl theory of the singular Hahn-Sturm-Liouville equation

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) = ^y (x), 0 to,

2.    Notation

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].

The study of the Hahn difference operator first appeared in [7, 8] where the quantum difference operator D _ω,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 q 1D ^q ,q Du,qy(x) + v(x)y(x) = N(x), aiy(wo) + a2D-Uq-1,q-1 у(шо) = 0, biy(b) + b2D-шq-1,q-1 y(b) = 0, is studied, where wo ^ x < то, a E C, ai,bi E R := (-to, to), i = 1, 2, and p(-) is a realvalued function defined on [wo, b] and continuous at wo. Annaby et al. [19] define a Hilbert space of ω, q-square summable functions. The authors discussed the formulation of the selfadjoint 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 ω, 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 E (0,1) and w >  0 .

Define w o := w/ (1 — q) and let I be a real interval containing w q .

Definition 1 [7, 8]. Let f : I ^ R be a function. The Hahn difference operator is defined by

f (w + qx) — f (x)

D _ш,q f (x) = <

w + (q — 1)x ^f ‘ ^(w o ),

x = wo, x = wo,

provided that f is differentiable at ω 0 . In this case, we call D _ω,q f , the ω, 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

А _ш f (x) :=

H^ ^^- fx) (w + x) — x

x E R .

When ш ^ 0 , we get the Jackson q -difference operator, which is defined by

D q f (x):= ^f ( qx) " ^{f ( X )} , x = 0.

(qx) — x

Furthermore, under appropriate conditions, we have lim Dш,q f (x) = f ‘ (x).

q→ 1 ,

ω→ 0

In what follows, we present some important properties of the ω, q -derivative.

Theorem 1 [20] . Let f,g : I ^ R be ш, q-differentiable at x G I and h (x) := ш + qx. Then for all x G I we have:

• i)    D u,q (af + bg) (x) = aD ^,q f (x) + bD wq g (x), a, b G I,

• ii)    D u,q ^(fg)^(x) = ^D _ш,q ^{(f (x))} g ^{(x) + f (ш +} xq) ^D _ш,q g ^(x),

  iii) D _ш,q f (x) =

  
  

^D w,q ^{(f (x))} g ^{(x) — f (x) D} ^,q g ^(x) g (x) g (ш + xq)

iv) D _ш,q (h ¹ (x)) = D -_Шq- 1 ,q- 1 f (x) .

The concept of the ω, 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, ш о G I . We define ш, q -integral of the function f from a to b by

b

/ f (x)

d

ω,q

b

(x) := у

f (x) d _ш,q

a

(x) —

f (x) d _ш,q (x) ,

ω 0

where x∞

/ f (x) d _ш,q (x) := ((1 — q) x — ш) ^ q n f ( ш

ω ₀                                  n =0

1 —

q

n

1 — q

+ xqn) ,

x G I,

provided that the series converges at x = a and x = b . In this case, f is called ш, q -integrable on [a, b].

Similarly, one can define the ш, q -integration for a function f over (ш о , то ) by

∞ j f (x) du,q(x) := ((1 — q)

ω 0

—

ω

∞

)E q'f ш

n=0

1 — q ⁿ

1 — q

+ qn) .

The following properties of ω, q -integration can be found in [20].

Theorem 2 [20] . Let f, g : I ^ R be ш, q-integrable on I, a, b, c G I , a < c < b and

• a, в e R . Then the following formulas hold:

bb

b

• i)    j {af (x) + вд (x) } d ^,q (x) = a j f (x) d _ш,q (x) + в j g (x) d ^,q (x),

a a

• ii)    j f (x) d^,q (x) = 0, a

b

a

a

c

b

• iii)    j f (x) d _ш,q (x) = j f (x) d _ш,q (x) + j f (x) d _ш,q (x),

a

b

• iv)    j f (x) d _ш,q (x) =

a

a

c

a j f (x) dш,q (x) .

b

Next, we present the ω, q -integration by parts.

Lemma 1 [20] . Let f, g : I ^ R be w, q-integrable on I, a, b G I, and a < b. Then the following formula holds:

bb j f (x) Dш,qg (x) dш,q (x) + У g (w + qx) D^qf (x) dш,q (x) = f (b) g (b) - f (a) g (a) .

The next result is the fundamental theorem of Hahn’s calculus.

Theorem 3 [20] . Let f : I ^ R be continuous at w o . Define

x

F (x):= j f (t) d _ш,q (t) , x e I.

ω 0

Then F is continuous at w o . Moreover, D w,q F(x) exists for every x G I and D w,q F(x) = f (x) . Conversely,

b j D ,qF (x) dш,q (x) = f (b) - f (a) .

a

Let L ^ q (w o , тс ) be the space of all complex-valued functions defined on [w o , тс ) such that

∞ 2

ll^f II := [ j I f ^(x) | ^{2 d}_Ш,q ^x j < ^тс-

ω 0

The space L^ q (w o , тс ) is a separable Hilbert space with the inner product

∞

^(f, g) ^:= j ^{f (x)} g^(x) d ^,q ^x, ^f,g ^e L ² ,q ^(w o ,^тс)

ω 0

(see [20]).

The w, q -Wronskian of functions y( - ) , z( - ) is defined as

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 г (y) := -q

¹ D -_Шq -1, q - 1 D u,q У (x) + V (x) У (x) = Ay (x) , x G (w q , to) ,

where A is a complex parameter, v is a real-valued continuous function at w o defined on [w q , to). We note that there exists a unique solution of (4) satisfying the conditions [19]

y (w0) = di,   D-Шq-1 ,q-1 y (w0) = d2, where d1 , d2 ∈ C.

Lemma 2 (see [19]). For any y and z in L‘2J,q(wo, to), the following relation holds xx j P(y) z dш,qt - j y r(z) dш,qt = [y,z] (x) - [y,z] (wo),

ω 0                  ω 0

where

^[y^,z] = У ^(D -_Шq - 1 ,q - 1 ^{z) - (D} -_Шq - 1 ,q - 1 У) z

Theorem 4. For each A G C and x G [w q , to), the w, q-Wronskian of any two solution of equation (4) is independent of x .

<1 Let y and z be two solutions of equation (4). It follows from (4) that xx j г(y)zdш,qt - j yP(z) dш,qt = [y,z] (x) - [y,z](wo)

ω 0                 ω 0

= W ^,q (y, z) (h ^-1 (x) , A) - W _ш,q (y, z) (h ^-1 (w o ), A) .

Since Г(у) = Ay and F(z) = Az , we have

x j r(y)zd.,q

ω 0

xx

-1 - j yr(z) d _ш,q t = (A - A) j y(t, A)z(t, A)d _ш,q t = 0

ω 0

= W ^,q (y,z) (h ^-1 (x) ,A)

-

ω 0

W ^,q (y, z) (h ^-1 (w o ) , A) .

Then we have W _ш,_q (y, z)(h ^-1 (x), A) = W _ш,_q (y, z)(h ^-1 (w o ), A) , i. e., the Wronskian is independent of x (x G [w q , to)). >

Now we impose a boundary condition for the solution y of equation (4) as

D ш,_q y (q n ) sin a + y (q ⁿ ) cos a = 0, a G R , n G N := { 1, 2,... } .

Let x(x, A) and ^(x, A) be the solutions of the equation (4) satisfying

X (w o ,A)=sin Z, у (w o ,A)=cos Z,

^D ^q , q X ^(w 0 , ^A) = ^{- cos Z}, ^D

-

wq~1,q~1 У ^(w 0 ^,A) = ^sin C

where 0 C Z < n . Since W _ш,_q (x, ^) = 1 , the solutions x and ^ are linearly independent.

Lemma 3. For x E [w q , to) and A E C, we have

x(x,A)= x (^x,A) ,   V^{( x,a )} = V (^x,A) •

<1 If x(x, A) is a solution of the equation (4), then we have

• - q ^-1 D -^q - i ,q - i D u,q x(x, A) + v (x) x(x, A) = Ax(x, a),   x € (w q , to).

Taking the complex conjugate, we obtain

• - q ^-1 D -_Шq-1 ,q-i D u,q x(x, A) + v (x) x(x, A) = A x(x, a), x € (w q , to).

By (7), x(x, A) is a solution of

• — q ^-1 D _-шq- 1 ,_q- i D ш,_q u (x) + v (x) u (x) = Au (x), x E (w q , to).

But u = x(x, A) 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,A) | ² d _ш,q x

ω 0

= W ^,q ⁽y,y) (^h 1 ^(b) ,^A)

^— W ^,q ⁽y,y) (^h 1 ^(w 0 ) ^,A) ,

a = Im A, b > 0, A E 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 ϑ of (4) as

^(x, A) := x(x, A) + nv(x, A), x E [wq, to), where η is a constant. If we substitute ϑ for y in (6) we obtain

(Q + n F ) sin a + (Y + пФ) cos a = 0,                        (9)

where

Q = DX (q -n , A) , F = 'V (q -n , A) , Y = x (q -n ,A) , Ф = v (q -n ,A) (n E N ).

Then, we get

Y cos a + Q sin a ⁿ = ^—           । г ■     ,

Ф cos a + F sin a

П is a meromorphic function of A because x(x, A) and v(x, A) are entire functions of A. Furthermore, since the eigenvalues of the regular problem are real, all poles of η are real and simple. If cos a is replaced by a complex variable z, then we have

Yz + Q Фz + F .

It follows from the theory of M¨obius transformations [21] that the equation (11) is a one-to-one conformal mapping in z for every λ . Hence η describes a circle C _q - n in the complex plane.

The task is now to find the center and the radius of C q -n (n E N ) . Let us denote by © n , r n the center of the circles C q-n ( n E N ) and its radius, respectively.

Theorem 5. Let A E C, a = Im A = 0. Then, we have

W^pYi^Kq-Pl n ⁽ )     W ^,q (y,V)(q -ⁿ ,A),

q-n rn (A)=(2a У |y(x,A)|2 dш,q xj

ω 0

(n E N ).

<1 © _n (A) (n E N) is the symmetric point at to . Let z ‘ and z" are in the z -plane such that П (A,z ‘ ) = to,    n (A, z") = © n (A).

Then z‘ and z" must be symmetric with respect to the real axis of the z—plane, i. e., z‘ = z’’. But n(A, z‘) = to if and only if z′

D _ш,q y(q ^-n , A) y⁽q ^{-n ,A)} ‘

Hence, we have

D ω,q ϕ y ⁽q

(q ^-n ,AH ₌

^-n ,A) )

x (q ⁿ , A) ( y ⁽q ^{-n ,A)} (

D^,q^{q n ,A ) A । ТА   —(_П-^

^^n^   + ^D ^,q ^{X (q}  , ^A)

D M,q ^ ( ^{q -} n ,A⁾ A । ТА   _т(_п-П X'l

^ < q^A )       ^{D ш},q ^{y (q} ,^A)

x ⁽q ^{n ,A) D} _ш,q y^q ^{n ,A}) y ⁽q ^-n ,^A) D u,q y (q ^-n ,^A)

y (q ⁿ ,^A) D -q x⁽q n ,^A) y (q ^-n ,^A) D u,q y ⁽q ^-n ,^A)

r

W -⁽X,y⁾⁽q n ,^A) z _№ W (y, y)(q ^-n , A)) ^{( E} )'

It is evident that rn(A) is the distance between the center of Cq-n and the point n(A, 0) on Cq-n(A). It follows from Wш,q(x,y)(q-n) = 1 (n E N) that rn (A) =

D u,q X ⁽q n ^,A)    W),q ⁽x^,y)⁽q ^{n ,A)}

r

D u,q У ⁽q ^{n ,A)}    W),q ⁽^,^⁾⁽q n ^,A)

D -,, y(q ^{n ,A})W ^,q ⁽x,y⁾⁽q ^{n ,A)} D ^,q y ⁽q ^-n ,^A) W ^,q ⁽y, V) ⁽q ^-n ,^A)

= W ^,q ⁽x,y⁾⁽q ^{n ,A)} W ^,q (y^Kq-M)

By virtue of Lemma 4, we conclude that

| W ^,q ⁽У,У )^ n ^,A)|

(n E N ).

q ^{- n}

W ^,q (y,y)(q n ,A) = 2ia j | y(x,A) | ² d _ш,q x.

ω 0

Thus, we get q-n

|W w,q (y^Kq -nP = ² И j | y^{(x,A)| 2 d} ^,q x,

ω 0

which proves the theorem. ⊲

Now, our next concern will be the behavior of the family of circles { C _q - n (A) } ( n E N).

Theorem 6. Let A = p + iCT. If Im A = ст > 0, then the upper half-plane is associated with the exterior of the circle C _q-n (n E N ).

⊳ It follows from (2) that

Im| D ,^2 1A)U1 i L 1 у (q ^-n ,A) J 2 |

' у (q ^-n ,A) + 'V^A) ^ v ⁽q ^{-n ,A)}        у (q ^-n ,A) J

1 .

= 2 i

W u,q ⁽у^{,у (}q ^{n ,A)} | v ⁽q ^{-n ,A)| 2}

-

1 i W (y,V) (q ^-n , A)

| v ⁽q M^{)| 2}

σ

| V⁽q ^{n ,A)|} 2

q-n j |v(x,A)|2 dш,qx> 0 (n E N),

ω 0

i. e., if ст >  0, the exterior of C _q -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 = ст > 0. Then n lies on the circles Cq-n (n E N) if and only if q-n j |x(x,A)+ nv(x,A)|2 dш,qx =

ω 0

Im n

σ

(n E N ),

and η belongs to the interior of Cq-n if and only if q-n

Im n

σ

(n E N ).

j |x(x,A)+ nv(x,A)| 2 d .,q x<

ω 0

⊳ Let η ∈ C . Then, we have

W w,q (x ⁺ ПУ, X ⁺ ПУ ) ⁽^ 0 , ^A) = W ^q (x,X) ⁽W 0 ,A) + n^W _Ш,q ((^ 0 ,A) + nW u,q (x,V) (^ш 0 , A) + | n | ² W _ш,q (у, у) (^ш 0 , A) = - n + n = - 2i Im n.

It follows from Lemma 4 that q-n

2ст j | x(x,A)+ nv(x, A) ^{| 2} d _ш,q x = ¹ (W _ш,q (x+ nV,X + nv)(q -n ,A) + 2iImn) (n E N ). (16) ω 0

From Theorem 6, if Im z <  0 , then n is inside C _q-n ( n E N) for v > 0 . By (11), we obtain

_ Q + nF z = — y + пФ ,

Q = D .,q X (q ^-n ,A) ,

Yz + Q n =--,

1 Фz + F,

f = d ,. v^-nA),

Y = x(q ^-n ,A), Ф = v(q ^-n ,A) (n E N ).

Hence, i (z — z) = i ^ —

Q + n F y + пФ

q . nF )     . W _ш ,q(x + nv,x + nv) (q ⁿ ,A)

+ =--== г = i-----------------"------- y + пФ J

| Y + пФ | 2

(n E N ).

Then, Im z < 0 if and only if iWq(x + П^,Х + nvHqn,X)> 0 (n € N)-

By virtue of (16) and (17), we conclude that q-n j ^(x,A) + n^(x,A)|2 dш,qx< Imn (n € N).

ω 0

On the other hand, n is on the circle C q -n if and only if Im z = 0. Therefore, we have

W^,q (x + nv, X + ПУ )(q-n,A) =0 (n € N).

Substituting (19) in (16), we obtain the equality (14). ⊲

Theorem 8. Let Im A = a > 0. The circles C_q- 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 j |x(x,A)+ nv(x,A)|2- .x<

ω 0

q-n j |x(x,A)+ nv(x,A)|2 dx<

ω 0

Im n

σ

This implies that the point η must be inside the circle C _q - k . ⊲

Corollary 1. The circles C _q - n may converge either to a circle or a point as n → ∞ .

Definition 3. If C _∞ is a point, then the equation (4) is said to be in the limit-point case .

Similarly, if C _∞ 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 C ^ , and Ф(х, A) : = x(x, A) + nv(x, A), Im A = a > 0, be the solution of (4). Then Ф(> A) € L ^,_q (w q , to), i. e.,

∞ j |x(x,A)+ nv(x,A)|2 dx< to.

ω 0

We note that n is called a Titchmarsh—Weyl function, and Ф(х, A) is called a Weyl solution of the equation (4).

⊳ Let η be a point lying on or inside the limiting circle C∞ . Then we have q-n j |x(x,A)+ nv(x,A)|2dш,qx< Iml^ (n € N).

ω 0

Since the right-hand side of (20) is independent of the point q-n, we may pass to the limit as n → ∞. Thus, we get

∞

Im n

, σ

j | ^(x,A) | 2 d _ш,q x <

ω 0

which completes the proof. ⊲