SPECTRAL PROPERTIES OF THE KLEIN-GORDON s-WAVE EQUATION WITH SPECTRAL PARAMETER-DEPENDENT BOUNDARY CONDITION

We investigate the spectrum of the differential operator Lλ defined by the Klein-Gordon s-wave equation y′′ +(λ−q(x))2y = 0, x ∈R+ = [0,∞), subject to the spectral parameterdependent boundary condition y′(0)−(aλ+b)y(0)= 0 in the space L(R+), where a≠±i, b are complex constants, q is a complex-valued function. Discussing the spectrum, we prove that Lλ has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions limx→∞q(x) = 0, supx∈R+{exp(ε √ x)|q′(x)|} <∞, ε > 0, hold. Finally we show the properties of the principal functions corresponding to the spectral singularities.


Introduction.
Let L denote the operator generated in L 2 (R + ) by differential expression l(y) = −y + q(x)y, x ∈ R + , (1.1) and the boundary condition y (0) − hy(0) = 0; here q is a complex-valued function, and h ∈ C. The spectral analysis of L was first investigated by Naȋmark [6].In this paper, he has proved that some of the poles of the resolvent's kernel of L are not the eigenvalues of the operator.Moreover, he has proved that these poles are on the continuous spectrum.(Schwartz named these poles as spectral singularities [11].)Furthermore, Naȋmark has shown that if the condition ∞ 0 e εx q(x) dx < ∞, ε >0, ( holds, then L has a finite number of eigenvalues and spectral singularities with finite multiplicities. The effect of spectral singularities in the spectral expansion of the operator L in terms of principal functions has been investigated in [4,5].The dependence of the structure of spectral singularities of L on the behavior of q at infinity has been studied in [9].In [10], some problems of spectral theory of the operator L with real potential and complex boundary condition were studied under the condition on the potential sup x∈R+ exp ε √ x q(x) < ∞, ε >0, (1.3) which is weaker than that considered in [6].
We consider the operator L λ generated in L 2 (R + ) by the Klein-Gordon s-wave equation for a particle of zero mass with static potential and the spectral parameter-dependent boundary condition where a, b ∈ C, a ≠ ±i, a ≠ 0, q is a complex-valued function and is absolutely continuous on each finite subinterval of R + .Some problems of the spectral theory of the Klein-Gordon equation have been investigated in [2,3] with real potential, and in [1] with complex potential subject to the boundary condition y(0) = 0.
In this paper, using the similar technique used in [1,12], we discuss the spectrum of L λ and prove that this operator has a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditions hold.Therefore we find the principal functions corresponding to the eigenvalues and the spectral singularities of L λ .In the rest of the paper, we use the following notations: (1.7)

2.
Preliminaries.We suppose that the functions q and q satisfy ∞ 0 x q(x) + q (x) dx < ∞. (2.1) Obviously we have from (2.1) that lim x→∞ q(x) = 0, x|q(x)| is bounded and (see [1]).Under condition (2.1), equation (1.4) has the following solutions: for λ ∈ C + , and x q(t)dt and K ± (x, t) are solutions of integral equations of Volterra type and are continuously differentiable with respect to their arguments.Moreover, |K ± (x, t)|, |K ± x (x, t)|, |K ± t (x, t)| satisfy the following inequalities: where and c > 0 is a constant [3].Therefore f + (x, λ) and f − (x, λ) are analytic with respect to λ in C + and C − , respectively, and continuous in λ up to the real axis.f ± (x, λ) also satisfy the following asymptotic equalities: ( Moreover, from (2.3) and (2.4), we have From (2.7), the Wronskian of the solutions of f + (x, λ) and f − (x, λ) is for λ ∈ R. Hence f + (x, λ) and f − (x, λ) are the fundamental solutions of (1.4) for λ ∈ R * .Let ϕ(x, λ) denote the solution of (1.4) satisfying the initial conditions which is an entire function of λ.
3. Eigenvalues and spectral singularities.Let ψ ± (x, λ) denote the solutions of (1.4) satisfying the following conditions: (see [3]).We obtain from (2.9) and (3.2) that if then we can write the solution ϕ(x, λ) of the boundary value problem (1.4) and (1.5) as follows: be Green's function of L λ .Then, using classical methods, we easily obtain the kernel of the resolvent as follows [6]: Now we denote the eigenvalues and the spectral singularities of L λ by σ d (L λ ) and σ ss (L λ ), respectively.From (2.8), (3.1), (3.4), (3.5), and (3.7), we obtain that ) The multiplicity of a zero of g ± in C ± is defined as the multiplicity of the corresponding member of the spectrum.
It is clear from (3.8) and (3.9) that, in order to investigate the quantitative properties of the members of the spectrum of L λ , we must investigate the zeros of g + (λ) in C + and g − (λ) in C − .We will consider the zeros of g + (λ) in C + .The zeros of g − (λ) in C − will be similar then.
We define the following sets:  (3.12)where It is clear from (2.5) and (3.12) that A + ∈ L 1 (R + ) and Hence the proof is obtained from (3.14) by the assumption i ≠ a.
Now we suppose that the following conditions hold: We then have Using (2.5) and (2.6) we find (3.17) From (3.17) we get that the functions f + x (0,λ) and f + (0,λ) can be continued analytically from C + into the half-plane Im λ > −ε/2.Hence g + (λ) can be continued analytically from C + into the half-plane Im λ > −ε/2.Therefore the sets N + 1 and N + 2 cannot have limit points on the real axis.From Lemma 3.2, we find that N + 1 and N + 2 have a finite number of points.Moreover, multiplicities of the zeros of g + (λ) in C + are finite.(Similarly, we can prove the finiteness of the zeros, and their multiplicities, of g − (λ) in C − .) From (3.8) and (3.9), we get the following theorem.
Theorem 3.3.Under conditions (3.15), the operator L λ has a finite number of eigenvalues and spectral singularities with finite multiplicities.Now we suppose that the following conditions hold: Obviously (3.18) is weaker than (3.15) which was considered in [10] for Sturm-Liouville case with real potential.Using (3.18), (2.6), and (2.5), we obtain Here and c > 0 is a constant.Let N + 3 and N + 4 denote the sets of limit points of N + 1 and N + 2 , respectively, and let N + 5 denote the set of all zeros of g + with infinite multiplicity in C + .Clearly, Since all the derivatives of g + (λ) are continuous on R, we then have

.23)
In order to show the finiteness of the zeros of g + (λ) and their multiplicities, we need to prove that N + 5 = φ.So we will use the following uniqueness theorem given by Pavlov.
Pavlov's theorem.Assume that the function g is analytic in C + , all of its derivatives are continuous on C + , and there exists T > 0 such that If the set Q, with linear Lebesgue measure zero, is the set of all zeros of the function g with infinite multiplicity and if where E(s) = inf n (A n s n /n!), n = 0, 1,..., µ(Q s ) is the linear Lebesgue measure of s-neighborhood of Q, and h is an arbitrary positive constant, then g(z) ≡ 0 [9].

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Theorem 3 . 5 .
30) holds for an arbitrary s if and only if µ(N + 5,s ) = 0 or N + 5 = φ, which proves the lemma.Under conditions (3.18), the operator L λ has a finite number of eigenvalues and spectral singularities with finite multiplicities.Proof.From Lemma 3.4 and (3.23), we get N + 3 = N + 4 = φ.Hence the sets N + 1 and N + 2 have no limit points.So g + has only a finite number of zeros in C + .From Lemma 3.4, the multiplicities of these zeros are finite.(Similarly, we can show that g − has a finite number of zeros with finite multiplicities in C − .)The proof follows from (3.8) and (3.9).
From (2.3) we get that g + is analytic in C + , continuous in C + , and it has the form g .11) Lemma 3.2.Under condition (2.1),(a) the set N +1 is bounded and has at most a countable number of elements, and its limit points can lie only in a bounded subinterval of the real axis, (b) the set N + 2 is compact.Proof.
.19)Here c > 0 is a constant.(3.19) cannot let g + (λ) continue analytically into a domain containing the real axis.So the technique of analytic continuation fails to apply here.From (2.3), (2.5), condition(3.18)givesthat the function g + is analytic in C + , and all of its derivatives are continuous in C + .So

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