Let A denote the operator generated in L2(ℛ+) by the Sturm-Liouville problem: -y′′+q(x)y=λ2y, x∈ℛ+=[0,∞), (y′/y)(0)=(β1λ+β0)/(α1λ+α0), where q is a complex valued function and α0,α1,β0,β1∈𝒞, with α0β1-α1β0≠0. In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of A. In particular, we obtain the conditions on q under which the operator A has a finite number of the eigenvalues and the spectral singularities.

1. Introduction

Let L denote the non-self-adjoint Sturm-Liouville operator generated in L2(ℛ+) by the differential expression

l(y)=-y′′+q(x)y,x∈ℛ+
and the boundary condition y(0)=0, where q is a complex valued function. The spectral analysis of L with continuous and discrete spectrum was studied by Naĭmark [1]. In this article, the spectrum of L was investigated and shown that it is composed of the eigenvalues, the continuous spectrum and the spectral singularities.The spectral singularities of L are poles of the resolvent which are imbedded in the continuous spectrum and are not the eigenvalues. If the function q satisfies the Naĭmark condition, that is,

∫0∞eɛx|q(x)|dx<∞
for some ɛ>0, then L has a finite number of the eigenvalues and spectral singularities with finite multiplicities.

The results of Naĭmark were extended to the Sturm-Liouville operators on the entire real axis by Kemp [2] and to the differential operators with a singularity at the zero point by Gasymov [3]. The spectral analysis of dissipative Sturm-Liouville operators with spectral singularities was considered by Pavlov [4]. A very important development in the spectral analysis of L was made by Lyance [5, 6]. He showed that the spectral singularities play an important role in the spectral theory of L. He also investigated the effect of the spectral singularities in the spectral expansion. The spectral singularities of the non-self-adjoint Sturm-Liouville operator generated in L2(ℛ+) by (1.1) and the boundary condition

∫0∞K(x)y(x)dx+αy′(0)-βy(0)=0,
in which K∈L2(ℛ+) is a complex valued function and α,β∈𝒞, was studied in detail by Krall [7–9].

Some problems of spectral theory of differential and difference operators with spectral singularities were also investigated in [10–16]. Note that, the boundary conditions used in [1–17] are independent of spectral parameter. In recent years, various problems of the spectral theory of regular Sturm-Liouville problem whose boundary conditions depend on spectral parameter have been examined in [18–22].

Let us consider the boundary value problem

-y′′+q(x)y=λ2y,x∈ℛ+,y′y(0)=β1λ+β0α1λ+α0,
where q is a complex valued function and α0,α1,β0,β1 are complex numbers such that α0β1-α1β0≠0. By A we will denote the operator generated in L2(ℛ+) by (1.4) and (1.5). In this paper we discuss the discrete spectrum of A and prove that the operator A has a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity if

limx→∞q(x)=0,∫0∞eɛxδ|q′(x)|dx<∞
for some ɛ>0 and 1/2≤δ<1. We also show that the analogue of the Naĭmark condition for A is the form

limx→∞q(x)=0,∫0∞eɛx|q′(x)|dx<∞
for some ɛ>0.

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

We will denote the solution of (1.4) satisfying the condition

limx→∞y(x,λ)e-iλx=1,λ∈𝒞¯+:={λ:λ∈𝒞,Imλ≥0},
by e(x,λ). The solution e(x,λ) is called the Jost solution of (1.4). Under the condition

∫0∞x|q(x)|dx<∞,
the Jost solution has a representation

e(x,λ)=eiλx+∫x∞K(x,t)eiλtdt
for λ∈𝒞¯+, where the kernel K(x,t) satisfies

K(x,t)=12∫(x+t)/2∞q(ξ)dξ+12∫x(x+t)/2∫t+x-ξt+ξ-xK(ξ,η)q(ξ)dηdξ+12∫(x+t)/2∞∫ξt+ξ-xK(ξ,η)q(ξ)dηdξ.
Moreover, K(x,t) is continuously differentiable with respect to its arguments and

|K(x,t)|≤c∫(x+t)/2∞|q(ξ)|dξ,|Kx(x,t)|,|Kt(x,t)|≤14|q(x+t2)|+c∫(x+t)/2∞|q(ξ)|dξ,
where c>0 is a constant [23, Chapter 3].

The solution e(x,λ) is analytic with respect to λ in 𝒞+:={λ:λ∈𝒞,Imλ>0} and continuous on the real axis.

Let 𝒜𝒞(ℛ+) denote the class of complex valued absolutely continuous functions in ℛ+. In the sequel we will need the following.

Lemma 2.1.

If
q∈𝒜𝒞(ℛ+),limx→∞q(x)=0,∫0∞x2|q′(x)|dx<∞,
then Kxt(x,t):=(∂2/∂t∂x)K(x,t) exists and
Kxt(x,t)=-18q′(x+t2)-14K(x+t2,x+t2)q(x+t2)-12∫x(x+t)/2[Kt(ξ,t+x-ξ)+Kt(ξ,t-x+ξ)]q(ξ)dξ-12∫(x+t)/2∞Kt(ξ,t-x+ξ)q(ξ)dξ.

The proof of the lemma is the direct consequence of (2.4).

From (2.5)–(2.8) we find that

|Kxt(0,t)|≤c[|q(t2)|+|q′(t2)|+∫t/2∞|q(ξ)|dξ],
where c>0 is a constant.

3. The Green Function and the Continuous Spectrum

Let φ(x,λ) denote the solution of (1.4) subject to the initial conditions φ(0,λ)=α0+α1λ,φ′(0,λ)=β0+β1λ. Therefore φ(x,λ) is an entire function of λ.

Let us define the following functions:

D±(λ)=(α0+α1λ)ex(0,±λ)-(β0+β1λ)e(0,±λ)λ∈𝒞¯±,
where 𝒞¯±={λ:λ∈𝒞,±Imλ≥0}. It is obvious that the functions D+ and D- are analytic in 𝒞+ and 𝒞-:={λ:λ∈𝒞,Imλ<0}, respectively and continuous on the real axis.

Let

G(x,t;λ)={G+(x,t;λ),λ∈𝒞+,G-(x,t;λ),λ∈𝒞-
be the Green function of A (obtained by the standard techniques), where

G±(x,t;λ)={-e(x,±λ)φ(t,λ)D±(λ),0≤t≤x,-e(t,±λ)φ(x,λ)D±(λ),x≤t<∞.
We will denote the continuous spectrum of A by σc. Using (3.1)–(3.3) in a way similar to Theorem 2 [17, page 303], we get the following:

σc=ℛ.

4. The Discrete Spectrum of the Operator <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M83"><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:math></inline-formula>

Let us denote the eigenvalues and the spectral singularities of the operator A by σd and σss respectively. From (2.3) and (3.1)–(3.4) it follows that

σd={λ:λ∈𝒞+,D+(λ)=0}∪{λ:λ∈𝒞-,D-(λ)=0},σss={λ:λ∈ℛ*,D+(λ)=0}∪{λ:λ∈ℛ*,D-(λ)=0},
where ℛ*=ℛ-{0}.

Definition 4.1.

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

In order to investigate the quantitative properties of the eigenvalues and the spectral singularities of A we need to discuss the quantative properties of the zeros of D+ and D- in 𝒞¯+ and 𝒞¯-, respectively. For the sake of simplicity we will consider only the zeros of D+ in 𝒞¯+. A similar procedure may also be employed for zeros of D- in 𝒞¯-.

Let us define

M1±={λ:λ∈𝒞±,D±(λ)=0},M2±={λ:λ∈ℛ,D±(λ)=0}.
So we have, by (4.1), that

σd=M1+∪M1-,σss=M2+∪M2--{0}.

Theorem 4.2.

Under the conditions in (2.7):

the discrete spectrum σd is a bounded, at most countable set and its limit points lie on the bounded subinterval of the real axis;

the set σss is a bounded and its linear Lebesgue measure is zero.

Proof.

From (2.3) and (3.1) we obtain that D+ is analytic in 𝒞+, continuous on the real axis and has the form
D+(λ)=iα1λ2+aλ+b+∫0∞f(t)eiλtdt,
where
a=iα0-α1K(0,0)-β1,b=-(α0+iβ1)K(0,0)-β0+iα1Kx(0,0),f(t)=-β0K(0,t)-iβ1Kt(0,t)+α0Kx(0,t)+iα1Kxt(0,t).
Using (2.5), (2.6), and (2.9) we get that f∈L1(ℛ+). So
D+(λ)=iα1λ2+aλ+b+o(1),λ∈𝒞¯+,|λ|→∞.
From (4.3), (4.6) and uniqueness theorem for analytic functions [24], we get (i) and (ii).

Theorem 4.3.

If
q∈𝒜𝒞(ℛ+),limx→∞q(x)=0,∫0∞x3|q′(x)|dx<∞,
then
∑ν|lν|ln1|lν|<∞,
where |lν| is the lengths of the boundary complementary intervals of σss.

Proof.

From (2.5), (2.6), (2.9), (4.4) and (4.7) we see that D+ is continuously differentiable on ℛ. Since the function D+ is not identically equal to zero, by Beurling's theorem we obtain (4.8) [25].

Theorem 4.4.

Under the conditions
q∈𝒜𝒞(ℛ+),limx→∞q(x)=0,∫0∞eɛx|q′(x)|dx<∞,ɛ>0,
the operator A has a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity.

Proof.

(2.5), (2.7), (2.9), (4.4) and (4.9) imply that the function D+ has an analytic continuation to the half-plane Imλ>-ɛ/2. Hence the limit points of its zeros on 𝒞¯+ cannot lie in ℛ. Therefore using Theorem 4.2, we have the finiteness of zeros of D+ in 𝒞¯+. Similarly we find that the function D- has a finite number of zeros with finite multiplicity in 𝒞¯-. Then the proof of the theorem is the direct consequence of (4.3).

Note that the conditions in (4.9) are analogous to the Naĭmark condition (1.2) for the operator A.

It is clear that the condition (4.9) guarantees the analytic continuation of D+ and D- from the real axis to the lower and the upper half-planes respectively. So the finiteness of the eigenvalues and the spectral singularities of A are obtained as a result of these analytic continuations.

Now let suppose that
q∈𝒜𝒞(ℛ+),limx→∞q(x)=0,∫0∞eɛxδ|q′(x)|dx<∞,
for some ɛ>0 and 1/2≤δ<1, which is weaker than (4.9). It is obvious that under the condition (4.10) the function D+ is analytic in 𝒞+ and infinitely differentiable on the real axis. But D+ does not have analytic continuation from the real axis to the lower half-plane. Similarly, D- does not have analytic continuation from the real axis to the upper half-plane either. Consequently, under the conditions in (4.10) the finiteness of the eigenvalues and the spectral singularities of A cannot be shown in a way similar to Theorem 4.4.

Let us denote the sets of limit points of M1+ and M2+ by M3+ and M4+ respectively and the set of all zeros of D+ with infinite multiplicity in 𝒞¯+ by M∞+. Analogously define the sets M3-,M4- and M∞-.

It is clear from the boundary uniqueness theorem of analytic functions that [24]
M1±∩M∞±=∅,M3±⊂M2±,M4±⊂M2±,M∞±⊂M2±,M3±⊂M∞±,M4±⊂M∞±,
and μ(M3±)=μ(M4±)=μ(M∞±)=0, where μ denote the Lebesgue measure on the real axis.

Theorem 4.5.

If (4.10) holds, then M∞+=M∞-=∅.

Proof.

We will prove that M∞+=∅. The case M∞-=∅ is similar. Under the condition (4.10) D+ is analytic in 𝒞+ all of its derivatives are continuous on the real axis and there exists N>0 such that
|dndλnD+(λ)|≤Bn,n=0,1,2,…,λ∈𝒞¯+,|λ|<2N,B0=4|α1|N2+2|a|N+|b|+∫0∞|f(t)|dt,B1=4|α1|N+|a|+∫0∞t|f(t)|dt,B2=2|α1|+∫0∞t2|f(t)|dt,Bn=∫0∞tn|f(t)|dt,n≥3.
From Theorem 4.2, we get that
|∫-∞-Nln|D+(λ)|1+λ2dλ|<∞,|∫N∞ln|D+(λ)|1+λ2dλ|<∞.
Let us define the function
T(s)=infnBnsnn!.
Since the function D+ is not equal to zero identically, by Pavlov's theorem [4],
∫0hlnT(s)dμ(M∞,s+)>-∞
holds, where h>0 is a constant and μ(M∞,s+) is the Lebesgue measure of s-neighborhood of M∞+. Using (2.5), (2.6), (2.9) and (4.4) we obtain that
Bn≤Bdnn!nn(1/δ-1),
where B and d are constants depending on ɛ and δ. Substituting (4.16) in the definition of T(s) we get
T(s)≤Bexp{-(1δ-1)e-1d-δ/(1-δ)s-δ/(1-δ)}.
Now (4.15) and (4.17) imply that
∫0hs-δ/(1-δ)dμ(M∞,s+)<∞.
Since δ/(1-δ)≥1, consequently (4.18) holds for arbitrary s if and only if μ(M∞,s+)=0 or M∞+=∅.

Theorem 4.6.

Under the condition (4.10) the operator A has a finite number of the eigenvalues and the 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 D+ and D- have finite number of zeros with finite multiplicities in 𝒞¯+ and 𝒞¯-, respectively. We will prove it only for D+. The case of D- is similar.

It follows from (4.11) that M3+=M4+=∅. So the bounded sets M1+ and M2+ have no limit points, that is, the D+ has only a finite number of zeros in 𝒞¯+. Since M∞+=∅ these zeros are of a finite multiplicity.

Theorem 4.7.

If the condition (2.7) is satisfied then the set σss is of the first category.

Proof.

From the continuity of D+ it is clear that the set M2+ is closed and is a set of Lebesgue measure zero which is of type Fσ. According to Martin's theorem [26] there is measurable set whose metric density exists and is different from 0 and 1 at every point of M2+. So, M2+ is of the first category from the theorem due to Goffman [27]. We also have obviously same things for M2-. Consequently σss is of the first category by (4.3).

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