HÜSEYIN TUNA Spectral analysis of singular Sturm – Liouville operators on time scales

In this paper, we consider properties of the spectrum of a Sturm– Liouville operator on time scales. We will prove that the regular symmetric Sturm–Liouville operator is semi-bounded from below. We will also give some conditions for the self-adjoint operator associated with the singular Sturm– Liouville expression to have a discrete spectrum. Finally, we will investigate the continuous spectrum of this operator.


Introduction.
A time scale T is an arbitrary nonempty closed set of real numbers.Dynamic equations on time scales has attracted much interest because it unites the theory of differential and difference equations.It has led to several important applications, e.g., in the study of heat transfer, insect population models, epidemic models stock market, and neural networks (see [13], [17], [24], [25]).However, there are very few results known for the Sturm-Liouville operators on time scales.
On the other hand, the spectral analysis of a self-adjoint differential operator is one of the most popular problems in operator theory.The spectrum of such operators depends on the behavior of the coefficients of the corresponding differential expression.This problem has been investigated by many mathematicians (see [4], [8]- [10], [14], [16], [20], [22], [26]).
The aim of this paper is to extend some results for differential operators obtained in [10] to the case of Sturm-Liouville dynamic equation (1) L (y) := − p (t) y ∆ (t) ∇ + q (t) y (t) = λy (t) , t ∈ [a, ∞) T , where p, q are real-valued continuous functions on T and p (t) = 0 for all t ∈ T. We prove that the regular symmetric Sturm-Liouville operator is semi-bounded from below.Using the splitting method [10], we will give some conditions for the self-adjoint operator associated with the singular expression (1) to have a discrete spectrum.We also investigate the continuous spectrum of this operator.

Definition 1.
Let T be a time scale.The forward jump operator σ : T → T is defined by σ (t) = inf {s ∈ T : s > t} , t ∈ T and the backward jump operator ρ : T → T is defined by

It is convenient to consider the graininess operators µ
0 and right dense if µ σ (t) = 0. We introduce the sets T k , T k , T * which are derived from the time scale T as follows.If T has a left scattered maximum t 1 , then Analogously one may define the notion of a ∇-differentiability of some function using the backward jump ρ.One can show (see [5]) for continuously differentiable functions.
If T = Z, then we have Analogously one may define the notion of ∇-antiderivative of some function.
Let L 2 ∇ (T) be the space of all functions defined on T such that Let T be a time scale which is bounded from below and unbounded from above such that inf T = a > −∞ and sup T = ∞.We will denote T also as T is a Hilbert space with the inner product (see [23]) The Wronskian of y (.), z (.) is defined by (see [5]) (2) W t (y, z) := p (t) y (t) z ∆ (t) − y ∆ (t) z (t) , t ∈ T.

Definition 5.
Let D A denote a subset of the complex Hilbert space H.A linear operator A is said to be Hermitian if, for all x, y ∈ D A , (Ax, y) = (x, Ay) holds.A Hermitian operator with a domain of definition dense in H is called a symmetric operator.An operator A * defined on H is called the adjoint of symmetric operator A if for all x, y ∈ D A , (x, Ay) = (A * x, y).An operator with a domain of definition dense in H is said to be self-adjoint if A = A * .An operator A is said to be compact if it maps every bounded set into a compact set (see [21]).

Definition 6.
A complex number λ is called a regular point of the linear operator A acting in complex Hilbert space H if (R1) the inverse R λ (A) = (A − λI) −1 (where I is the identity operator in H) exists, and (R2) R λ (A) is a bounded operator defined on the whole space H. Assume that (R3) R λ (A) is defined on a set which is dense in H.
The operator R λ (A) is then called the resolvent of the operator A. All non-regular points λ are called points of the spectrum of the operator A.
The point spectrum or discrete spectrum σ p (A) is the set such that R λ (A) does not exist.A number λ ∈ σ p (A) is called an eigenvalue of A. The spectrum of the operator A is said to be purely discrete if it consists of a denumerable set of eigenvalues with no finite point of accumulation.
The continuous spectrum σ c (A) is the set such that R λ (A) exists and satisfies (R3) but not (R2).
The residual spectrum σ r (A) is the set such that R λ (A) exists but does not satisfy (R3) (see [18]).Theorem 7 ([18]).The residual spectrum σ r (A) of a self-adjoint linear operator acting on a complex Hilbert space H is empty.Theorem 8 ([21]).All self-adjoint extensions of a closed, symmetric operator which has equal and finite deficiency indices have one and the same continuous spectrum.

and
It is easily seen that if A 1 and A 2 are each self-adjoint operators, then their direct sum A 1 ⊕ A 2 is also a self-adjoint operator.

Definition 10 ([21]
).A symmetric operator A is said to be semi-bounded from below if there is a number m such that, for all x ∈ D A , the inequality holds.Similarly, if there is a number M such that for all x ∈ D A , the inequality holds, then A is said to be semi-bounded from above.Theorem 11 ([21]).If a symmetric operator A with finite deficiency indices (n, n) satisfies the condition then the part of the spectrum of every self-adjoint extension of A which lies to the left of m or to the right of M can consist of only a finite number of eigenvalues and the sum of their multiplicities does not exceed n.For every y, z ∈ D max we have Green's formula [11]).Let D min be the linear set of all vectors y ∈ D max satisfying the conditions The operator L min , that is the restriction of the operator L max to D min is called the minimal operator and the equalities L max = L * min holds.Further (it follows from (3)), L min is a closed symmetric operator with deficiency indices (2, 2) ( [7], [21]).
, then the regular operator L min is semi-bounded from below.Further, the negative part of the spectrum of every self-adjoint extension of L min consists of not more that a finite number of negative eigenvalues of finite multiplicity.By integration by parts, we get We set For y ∈ D min we have Hence we get (4) Let L 2 ∇,p (a, b) be the Hilbert space of all complex-valued functions defined on [a, b] with the inner product ∇t.
On the other hand, the dimension of the manifold D min modulo D is N , and consequently, the operator L min is semi-bounded from below on the whole manifold D min .By Theorem 11, we get the desired result.
Let H denote the set of all functions Further, let L min denote the restriction of the operator L min to D min .Then L min is the closure of the operator L min , i.e., L min = L min ( [21]).Now we restrict D min by imposing the additional conditions where c is a fixed point of the interval (a, ∞) T .By this restriction, we obtain the manifold D min .
The restriction L min of the operator L min to D min is called the splitting of the operator L min at the point c of the interval (a, ∞) T .It is clear that ( 6) i.e., the operator L min is the direct sum of two operators L 1 and L 2 in the spaces L 2 ∇ (0, c) T and L 2 ∇ (c, ∞) T , where L 1 and L 2 are generated in these spaces from the Sturm-Liouville expression L in the same way as L min was.
If L 1 = L 1 and L 2 = L 2 are the closures of the operators L 1 and L 2 , then (6) implies that If we extend the symmetric operators L 1 and L 2 into self-adjoint operators L 1,s and L 2,s in the spaces L 2 ∇ (0, c) T and L 2 ∇ (c, ∞) T respectively, then the direct sum will be a self-adjoint extension of the symmetric operator L min .The spectrum of the operator A is the set-theoretic sum of the spectra of L 1,s and L 2,s .Since the deficiency indices of the operator L min are finite, by Theorem 8, all its self-adjoint extensions have one and the same continuous spectrum.Both the operator A and also each self-adjoint extension L s of the operator L min are such extensions.Hence, the continuous parts of the spectrum of the two operators A and L s coincide.
Therefore, we have the following theorem: Theorem 13.The continuous parts of the spectrum of every self-adjoint extension of the operator L min is the set-theoretic sum of the continuous parts of the spectra of L Proof.Let N > 0 be an arbitrary number.From (7), one can choose a number c such that ( 9) By the condition (8), via integration by parts, we obtain (y ∈ D L 2 ) Hence the operator L 2 is bounded from below and its closure L 2 is also bounded from below by the number N .Therefore, by Theorem 11, the half-axis −∞ < λ < N , contains no point of the continuous spectrum of the self-adjoint extension L 2,s of L 2 .
On the other hand, since the operator L 1 is regular, the spectrum of any self-adjoint extension L 1,s of L 1 is purely discrete.Hence the half-axis −∞ < λ < N , contains no point of the continuous spectrum of A = L 1,s ⊕ L 2,s .
By Theorem 13, every self-adjoint extension L s of the operator L min has this property.Since the number N is arbitrary, the spectrum of the operator L s has no continuous part at all.Proof.If we decompose the operator at a point c such that then we obtain L 2 y, y > (M − ε) (y, y) .
Hence, the part of the spectrum of L 2 lying in the interval (−∞, M − ε) can consist only of a finite number of eigenvalues of finite multiplicity.On the other hand, by Theorem 12, the operator L 1 is regular and bounded below.Hence its spectrum is purely discrete and any point of accumulation of the spectrum L 1,s can only be at λ = +∞.Thus, from Theorem 13, we get the desired result.Now, we need the following lemma.Then any interval, of length greater than 2M , of the positive half-axis contains points of the continuous spectrum of any self-adjoint extension L s of the singular operator L min .
Proof.Suppose, contrary to our claim, that an interval [λ 0 − δ, λ 0 + δ] of the half-axis λ > 0 contains no point of the continuous spectrum of L s , δ > M .Then, though the operator may be decomposed, this interval would contain no point of the continuous spectrum of any self-adjoint extension of L 2 .If we choose the point c such that |q (t)| ≤ M + ε < δ for t > c, then, by Lemma 16, λ 0 can not belong to the continuous spectrum of the self-adjoint extension of the minimal operator generated by the expression − p (t) y ∆ (t) ∇ and the same boundary conditions.But this is contradiction because the continuous spectrum of the last operator covers the whole of the positive half-axis.
In particular, for M = 0 we have the following corollary.Then the whole positive half-axis is covered by the continuous spectrum of any self-adjoint extension L s of the singular operator L min .

Definition 4 .
Let f : T → R be a function, and a, b ∈ T. If there exists a function F : T → R, such that F ∆ (t) = f (t) for all t ∈ T k , then F is a ∆-antiderivative of f .In this case the integral is given by the formula b a f (t) ∆t = F (b) − F (a) for a, b ∈ T.

3 .
Main Results.Let us consider the linear set D max consisting of all vectors y ∈ L 2 ∇ [a, ∞) T such that y and py ∇ are locally ∆ absolutely continuous functions on [a, ∞) T and Ly ∈ L 2 ∇ [a, ∞) T .We define the maximal operator L max on D max by the equality L max y = Ly.

Theorem 15 .
Let lim t→∞ q (t) = M and p (t) > 0 (t ∈ [0, ∞) T ).Then the interval (−∞, M ) contains no point of the continuous spectrum of any self-adjoint extension L s of the singular operator L min ; on the contrary, any L s can only have at most point-eigenvalues on this interval and these can have a point of accumulation only at λ = M .

Lemma 16 .Theorem 17 .
If the interval [λ 0 − δ, λ 0 + δ] contains no point of the spectrum of a self-adjoint operator A except perhaps for a finite number of eigenvalues, each of finite multiplicity, and if Q is a bounded Hermitian operator satisfying the condition Q < δ, then the point λ 0 does not lie in the continuous part of the spectrum of the operator A + Q. Proof.See [21].Let p(t) ≡ 1 and lim t→∞ |q (t)| = M.
1,s and L 2,s , where L 1,s and L 2,s have been obtained by the splitting of the operator L min .