Let L denote the operator generated in L2(R+) by Sturm-Liouville equation −y′′+q(x)y=λ2y, x∈R+=[0,∞), y′(0)/y(0)=α0+α1λ+α2λ2, where q is a complex-valued function and αi∈ℂ, i=0,1,2 with α2≠0. In this article, we investigate the eigenvalues and the spectral singularities of L and obtain analogs of
Naimark and Pavlov conditions for L.

1. Introduction

Let L0 denote Sturm-Liouville operator generated in L2(ℝ+) by the differential expression
l0(y):=-y′′+q(x)y,x∈ℝ+,
and the boundary condition y(0)=0, where q:ℝ+→ℂ. Sinceqis a complex-valued function, the operator L0 is a non-selfadjoint. The spectral analysis of L0 has been investigated byNaĭmark[1]. He proved that some of the poles of the kernel of resolvent of L0 are not the eigenvalues of the operator. He also showed that those poles (which are called spectral singularities by Schwartz [2]) are on the continuous spectrum. Moreover, he has shown the spectral singularities play an important role in the spectral analysis of L0 , and if
∫0∞eεx|q(x)|dx<∞,ε>0,
then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity.

One very important step in the spectral analysis of L0 was taken by Pavlov [3]. He studied the dependence of the structure of the eigenvalues and the spectral singularities of L0 on the behavior of potential function at infinity. He also proved that if
supx∈R+[eεx|q(x)|]<∞,ε>0,
then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity.

Conditions (N) and (P) are called Naimark and Pavlov conditions for L0, respectively.

Lyance showed that the spectral singularities play an important role in the spectral analysis of L0 [4, 5]. He also investigated the effect of the spectral singularities in the spectral expansion.

The spectral singularities of non-selfadjoint operator generated in L2(ℝ+) by (1.1) and the boundary condition
∫0∞K(x)y(x)dx+αy′(0)-βy(0)=0
was investigated in detail by Krall [6, 7].

Some problems of spectral theory of differential operator and some other types of operators with spectral singularities were studied by some authors [8–14]. Note that in all papers the boundary conditions are not depending on the spectral parameter.

In a recent series of papers, Bindinget al.and Browne[15–18] have studied the spectral theory of regular Sturm-Liouville operators with boundary conditions depending on the spectral parameter.

Let L denote the operator generated in L2(ℝ+) by
-y′′+q(x)y=λ2y,x∈R+,y′(0)y(0)=α0+α1λ+α2λ2,
whereqis a complex-valued function, αi∈ℂ,i=0,1,2, with α2≠0. In this paper, we investigate the eigenvalues and the spectral singularities of L. In particular, we show that the analogs of Naimark and Pavlov conditions for L are
q∈AC(ℝ+),limx→∞q(x)=0,∫0∞eεx|q′(x)|dx<∞,ε>0,q∈AC(ℝ+),limx→∞q(x)=0,supx∈R+[eεx|q′(x)|]<∞,ε>0,
respectively, where AC(ℝ+)denotes the class of complex-valued absolutely continuous functions on ℝ+.

2. Jost Functions of (<xref ref-type="disp-formula" rid="EEq1.2">1.3</xref>)-(<xref ref-type="disp-formula" rid="EEq1.3">1.4</xref>)

Under the condition
∫0∞x|q(x)|dx<∞,
(1.3) has a solution e(x,λ)satisfying
limx→∞e(x,λ)e-iλx=1,λ∈ℂ¯+,
where ℂ¯+={λ:λ∈ℂ,Imλ≥0}. The solution e(x,λ) is called Jost solution of (1.3). Note that Jost solution has a representation [19]
e(x,λ)=eiλx+∫x∞K(x,t)eiλtdt,λ∈ℂ¯+,
where K(x,t) is the solution of the integral equation
K(x,t)=12∫(x+t)/2∞q(s)ds+12∫x(x+t)/2∫t+x-st+s-xq(s)K(s,u)duds+12∫(x+t)/2∞∫st+s-xq(s)K(s,u)duds,
and K(x,t) are continuously differentiable with respect to their arguments. We also have
|K(x,t)|⩽cw(x+t2),|Kx(x,t)|,|Kt(x,t)|≤14|q(x+t2)|+cw(x+t2),
where w(x)=∫x∞|q(s)|ds and c>0 is a constant.

Let
E+(λ):=e′(0,λ)-(α0+α1λ+α2λ2)e(0,λ),λ∈ℂ¯+,E-(λ):=e′(0,-λ)-(α0+α1λ+α2λ2)e(0,-λ),λ∈ℂ¯-,
where ℂ¯-={λ:λ∈ℂ,Imλ≤0}. Therefore, E+and E- are analytic in ℂ+={λ:λ∈ℂ,Imλ>0} and ℂ-={λ:λ∈ℂ,Imλ<0}, respectively, and continuous up to real axis. The functions E+ and E- are called Jost functions of L.

Let us denote the eigenvalues and the spectral singularities of L by σd(L) and σss(L), respectively. It is evident that
σd(L)={λ:λ∈ℂ+,E+(λ)=0}∪{λ:λ∈ℂ-,E-(λ)=0},σss(L)={λ:λ∈ℝ*,E+(λ)=0}∪{λ:λ∈ℝ*,E-(λ)=0},{λ:λ∈ℝ*,E+(λ)=0}∩{λ:λ∈ℝ*,E-(λ)=0}=∅,
where ℝ*=ℝ∖{0}.

Definition 2.1.

The multiplicity of a zero E+(or E-) in ℂ¯+ (or ℂ¯-) is defined as the multiplicity of the corresponding eigenvalue and spectral singularity of L.

In order to investigate the quantitative properties of the eigenvalues and the spectral singularities of L, we need to discuss the quantitative properties of the zeros of E+ and E- in ℂ¯+and ℂ¯-, respectively.

Define
M1±={λ:λ∈ℂ±,E±(λ)=0},M2±={λ:λ∈ℝ*,E±(λ)=0},
then by (2.7), we have
σd(L)=M1+∪M1-,σss(L)=M2+∪M2-.

Now, let us assume that
q∈AC(R+),limx→∞q(x)=0,∫0∞x3|q′(x)|dx<∞.

Theorem 2.2.

Under condition (2.11), the functions E+ and E- have the representations
E+(λ)=-α2λ2+β+λ+δ++∫0∞f+(t)eiλtdt,λ∈ℂ¯+,E-(λ)=-α2λ2+β-λ+δ-+∫0∞f-(t)e-iλtdt,λ∈ℂ¯-,
where β±,δ±∈ℂ, and f±∈L1(ℝ+).

Proof.

Using (2.3),(2.4), and (2.6), we get (2.12), where
β+=i-α1-iα2K(0,0),δ+=-K(0,0)-α0-iα1K(0,0)+α2Kt(0,0),f+(t)=Kx(0,t)-α0K(0,t)-iα1Kt(0,t)+α2Ktt(0,t).
From (2.4), we see that
|Ktt(0,t)|≤c[t|q(t2)|+|q′(t2)|+tw(t2)+w1(t2)]
holds, where w1(t)=∫t∞w(s)ds and c>0 is a constant. It follows from (2.5), (2.14), and (2.15) that f+∈L1(ℝ+). In a similar way, we obtain (2.13).

Theorem 2.3.

Under condition (2.11), we have the following.

The set of σd(L) 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.

The set of σss(L) is bounded and its linear Lebesgue measure is zero.

Proof.

From (2.14) and (2.15), we see that
E+(λ)=-α2λ2+β+λ+δ++o(1),λ∈ℂ¯+,|λ|→∞,E-(λ)=-α2λ2+β-λ+δ-+o(1),λ∈ℂ¯-,|λ|→∞.
Using (2.10), (2.16), and the uniqueness theorem of analytic functions [20], we get (i) and (ii).

3. Naĭmark and Pavlov Conditions for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M96"><mml:mrow><mml:mi mathvariant="bold-italic">L</mml:mi></mml:mrow></mml:math></inline-formula>

We will denote the set of all limit points of M1+ and M1- by M3+ and M3-, respectively, and the set of all zeros of E+ and E- with infinity multiplicity in ℂ¯+ and ℂ¯-, by M4+ and M4-, respectively. It is obvious that
M3±⊂M2±,M4±⊂M2±,M3±⊂M4±,
and the linear Lebesgue measures of M3± and M4± are zero.

Theorem 3.1.

If
q∈AC(ℝ+),limx→∞q(x)=0,∫0∞eεx|q′(x)|dx<∞,ε>0,
then the operator L has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.

Proof.

From (2.5), (2.14), (2.15), and (3.2), we find that
|f+(t)|≤ce-(ε/2)t,
where c>0 is a constant. By (2.12) and (3.3), we observe that the function E+ has an anlytic continuation to the half-plane Imλ>-ε/4. So we get that M4+=∅. It follows from (3.1) that M3+=∅. Therefore the sets M1+ and M2+ have a finite number of elements with a finite multiplicity. We obtain similar results for the sets M1- and M2-. By (2.10) we have the proof of the theorem.

Now let us assume that
q∈AC(ℝ+),limx→∞q(x)=0,supx∈R+[eεx|q′(x)|]<∞,ε>0.

Hence, we have the following lemma.

Lemma 3.2.

It holds that M4+=M4-=∅.

Proof.

From (2.12) and (3.4), we find that the function E+ is analytic in ℂ+, and all of its derivatives are continuous in ℂ¯+. For a sufficiently large T>0, we have
|dkdλkE+(λ)|≤Ak,λ∈ℂ¯+,|λ|≤T,k=0,1,2,…,
where
Ak=2kc∫0∞tke-(ε/2)tdt,k=0,1,2,…,
and c>0 is a constant. Since the function E+ is not equal to zero identically, then by Pavlov's theorem, M4+ satisfies
∫0hlnA(s)dμ(M4+,s)>-∞,
where A(s)=infk(Aksk/k!),μ(M4+,s) is the linear Lebesgue measure of s-neighborhood of M4+,[3]. Now, we obtain the following estimates for Ak:Ak≤Bbkkkk!,
where B and b are constants depending on c and ε. From (3.8), we get that
A(s)≤Binfk(bkskkk)≤Bexp(-b-1e-1s-1).
Now, (3.7) yields that
∫0h1sdμ(M4+,s)<∞.
However, (3.10) holds for an arbitrary s, if and only if μ(M4+,s)=0 or M4+=∅. In a similar way we can prove that M4-=∅.

Theorem 3.3.

Under condition (3.4), the operator L has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity.

Proof.

To be able to prove the theorem, we have to show that the functions E+ and E- have a finite number of zeros with finite multiplicities in ℂ¯+and ℂ¯-, respectively. We give the proof for E+.

From Lemma 3.2 and (3.1), we find that M3+=∅. So the bounded sets M1+ and M2+ have no limit points, that is, the function E+ has only a finite number of zeros in ℂ¯+. Since M4+=∅, these zeros are of finite multiplicity.

Acknowledgment

This work was supported by TUBITAK.

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