Non-real Eigenvalues of Symmetric Sturm–liouville Problems with Indefinite Weight Functions

The present paper deals with non-real eigenvalues of regular Sturm–Liouville problems with odd symmetry indefinite weight functions applying the two-parameter method. Sufficient conditions for the existence and non-existence of non-real eigenval-ues are obtained. Furthermore, an explicit expression of the bound of non-real eigen-values will be given in the paper.


Introduction
Consider the Sturm-Liouville problem − y (x) − µy(x) = λw(x)y(x), x ∈ [−1, 1], ( with the Dirichlet boundary condition where µ is real, λ is the spectral parameter and the weight function w is a real-valued integrable function satisfying the following conditions. For a.e.x ∈ [0, 1], w(x) is a monotone nonincreasing function.

D(T) =
Here AC[−1, 1] is the set of absolutely continuous functions on [−1, 1] and W is the operator of multiplication by w.Then T is self-adjoint, bounded below with compact resolvents and W is self-adjoint in Hilbert space L 2 [−1, 1].
Under the condition that the first two eigenvalues of have contrary signs, papers [15,25] tell us that (1.1), (1.2) has exactly two non-real eigenvalues.
In more general conditions, Volkmer [23, pp. 233-234] studies the existence of non-real eigenvalues for the Richardson equation (1.5) associated to the Dirichlet conditions y(±1) = 0 (see Corollary 3.12).For the general Sturm-Liouville problems, Mingarelli [14] made a summary of regular indefinite Sturm-Liouville problems and posed many questions about the bounds and the existence of the non-real eigenvalues.Recently, Behrndt, Philipp and Trunk [3] and Behrndt, Schmitz and Trunk [4] studied the existence and obtained a bound on non-real eigenvalues in a special singular case.For the regular case, Behrndt, Chen, Philipp and Qi [1], Kikonko, Mingarelli [11] and other papers [10,15,24] got bounds on non-real eigenvalues.
In this paper, we will prove the existence of non-real eigenvalues of problem (1.1), (1.2), for µ ∈ (µ 2m−1 (0), µ 2m (0)), m = 1, 2, . . ., (see Theorem 3.11).And a sufficient condition for the non-existence of non-real eigenvalues of (1.1), (1.2) is obtained in Theorem 4.3.The arrangement of the present paper is as follows.The next section gives some preliminary knowledge and some properties of real eigencurves.The main result of this paper, the existence of nonreal eigenvalues, Theorems 3.11 and its proof are stated in Section 3. Furthermore, an explicit expression of non-real eigenvalues' bound will be given in Lemma 3.2 of Section 3. The last section, Section 4, gives the non-existence of non-real eigenvalues, Theorems 4.3 and its proof.

Properties of real eigencurves and preliminary knowledge
This section gives some preliminary knowledge and some properties of real eigencurves, µ = µ n (λ), n = 1, 2, 3, . . .In paper [6], Binding and Volkmer have made a comprehensive summary and further research on real eigencurves about two-parameter Sturm-Liouville problems.The following first five lemmas are from paper [6].
Lemma 2.4 (see [6,Corollary 2.6]).The intersection of any straight line in the (Re λ, µ)-plane with the union of the first n eigencurves consists of at most 2n points for every positive integer n.
Lemma 2.5 (see [6,Theorem 2.9]).For λ ∈ R, the order of µ n (λ) is at most 2n for every positive integer n, i.e., µ We call the point λ 0 is a critical point of u, if u (λ 0 ) = 0.If there are 2n critical points about u n (λ), then it can lead that there exists a point λ 0 such that µ (2n) n (λ 0 ) = 0, by the mean value theorem.Applying (2.1) and Lemma 2.5 to real eigencurves, we can obtain the next result.Lemma 2.6.
(ii) For every positive integer n, there are at most 2n − 1 critical points for u n (λ).
With the same method, we can get the sign of µ 2m−1 (0) is as same as the sign of

Existence of non-real eigenvalues
In this section, we will obtain sufficient conditions of the existence about non-real eigenvalues of problem (1.1), (1.2).In Lemma 3.2, we will give an a priori bound on the modulus of the largest non-real eigenvalue which might appear.For this purpose, the lower bound about µ for any non-real eigenpair (λ, µ) must be given first.
It is well known that if the indefinite problem (1.1), (1.2), is a left-definite problem, then the problem only has real eigenvalues (see [12,13,26]).Since T ≥ π 2 4 , hence as the problem is left-definite and thus has real spectrum.
4 .An explicit bound for the non-real eigenvalues will be obtained.
Let φ(x; λ, µ) be the solution of (1.1) satisfying the initial conditions Here λ and µ can be arbitrary complex numbers.By analytic parameter dependence, the function is an entire function and the zeros (λ, µ) of D are the eigenpairs of (1.1), (1.2).Hence by the continuity of zeros of analytic functions (see [8, p. 248] or the next proposition), we can obtain the corresponding conclusion about the analytic function D, in Lemma 3.4.

Proposition 3.3 (The continuity of zeros of analytic functions).
Let A be an open set in the complex plane C, X a metric space, f a continuous complex valued function on A × X such that for each α ∈ X, the map z → f (z, α) is an analytic function on A. Let B be an open set of A whose closure B in C is compact and contained in A, and let α 0 ∈ X be such that no zero of f (z, α 0 ) is on the boundary of B. Then there exists a neighborhood W of α 0 in X such that (1) for any α ∈ W, f (z, α) has no zero on the boundary of B; (2) for any α ∈ W, the sum of the order of the zeros of f (z, α) contained in B is independent of α.
Using Proposition 3.3, let the metric space X be R, then (2) for any α ∈ W, the sum of the order of the zeros of D(z, α) contained in B is independent of α.
From Lemma 3.2, Λ is bounded, by the boundedness of the interval J. Then it follows from D(λ(µ (n) ), µ (n) ) = 0 and the continuity of the function D that D(ξ, η) = 0 for any ξ ∈ Λ.We only need to prove that Λ has only one point.
Suppose on the contrary, if Λ has more than one point ξ 1 , ξ 2 .Then by the continuity of λ(µ), µ ∈ J, we know that for any fixed r, 0 < r < |ξ 1 − ξ 2 |, and any δ > 0 such that (η − δ, η] ⊂ J, the set {(λ(µ), µ) : µ ∈ (η − δ, η]} ∩ S(ξ 1 , r) must contain infinite points, where S(ξ 1 , r) denotes the sphere in C with the center ξ 1 and the radius r, respectively.This means that the number of the λ-solutions about the η-equation D(λ, η) = 0 on the compact set S(ξ 1 , r) is infinite.Hence for any 0 < r < |ξ 1 − ξ 2 | there exists at least one accumulation point λ r for these λ-solutions and D(λ r , η) = 0.That is to say that the zeros of D(λ, η) are uncountable and hence D(λ, η) = 0 for any λ ∈ R, since for the fixed η, D(λ, η) is analytic about λ.Clearly, this is a contradiction since for the fixed η the eigenvalue problem has only countable eigenvalues.Therefore, Λ has only one point, say ξ 0 , and ξ 0 is a finite point of C. The proof about the left end-point of J is the same as the one above and Lemma 3.9 is proved.
where μ is a maximum of a real eigencurve and μ a minimum of a real eigencurve.Another description can be given that for any non-real eigencurve, it must start from a minimum of a real eigencurve and downwards end at a maximum of a real eigencurve.In such case, any non-real eigencurve downwards at most arrives at µ 1 (0) = π 2 4 .

Nonexistence of non-real eigenvalues
In this section, we will obtain sufficient conditions for the non-existence of non-real eigenvalues.The next two lemmas give some properties about the maximum and minimum of the real eigencurves.

Lemma 3 . 4 .
Let B be an open set of C whose closure B is compact, and let α 0 ∈ R be such that no zero of D(z, α 0 ) is on the boundary of B. Then there exists a neighborhood W of α 0 in R such that(1) for any α ∈ W, D(z, α) has no zero on the boundary of B;
Remark 3.13.Any non-real eigencurve λ(µ) must start from a maximum of a real eigencurve and go upwards, ending at a minimum of a real eigencurve or to +∞, i.e., sup{µ : λ(t) is non real, for any t ∈ ( μ, µ)}