Non-Self-Adjoint Singular Sturm-Liouville Problems with Boundary Conditions Dependent on the Eigenparameter
Let denote the operator generated in by the Sturm-Liouville problem: , , , where is a complex valued function and , with In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of . In particular, we obtain the conditions on under which the operator has a finite number of the eigenvalues and the spectral singularities.
Let denote the non-self-adjoint Sturm-Liouville operator generated in by the differential expression
and the boundary condition , where is a complex valued function. The spectral analysis of with continuous and discrete spectrum was studied by Naĭmark . In this article, the spectrum of was investigated and shown that it is composed of the eigenvalues, the continuous spectrum and the spectral singularities.The spectral singularities of are poles of the resolvent which are imbedded in the continuous spectrum and are not the eigenvalues. If the function satisfies the Naĭmark condition, that is,
for some , then 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  and to the differential operators with a singularity at the zero point by Gasymov . The spectral analysis of dissipative Sturm-Liouville operators with spectral singularities was considered by Pavlov . A very important development in the spectral analysis of was made by Lyance [5, 6]. He showed that the spectral singularities play an important role in the spectral theory of . 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 by (1.1) and the boundary condition
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
where is a complex valued function and are complex numbers such that . By we will denote the operator generated in by (1.4) and (1.5). In this paper we discuss the discrete spectrum of and prove that the operator has a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity if
for some and . We also show that the analogue of the Naĭmark condition for is the form
2. Jost Solution of (1.4)
We will denote the solution of (1.4) satisfying the condition
by . The solution is called the Jost solution of (1.4). Under the condition
the Jost solution has a representation
for where the kernel satisfies
Moreover, is continuously differentiable with respect to its arguments and
where is a constant [23, Chapter 3].
The solution is analytic with respect to in and continuous on the real axis.
Let denote the class of complex valued absolutely continuous functions in In the sequel we will need the following.
Lemma 2.1. If then exists and
The proof of the lemma is the direct consequence of (2.4).
where is a constant.
3. The Green Function and the Continuous Spectrum
Let denote the solution of (1.4) subject to the initial conditions . Therefore is an entire function of
Let us define the following functions:
where It is obvious that the functions and are analytic in and respectively and continuous on the real axis.
be the Green function of (obtained by the standard techniques), where
4. The Discrete Spectrum of the Operator
Definition 4.1. The multiplicity of a zero of (or ) in (or ) is defined as the multiplicity of the corresponding eigenvalue or spectral singularity of .
In order to investigate the quantitative properties of the eigenvalues and the spectral singularities of we need to discuss the quantative properties of the zeros of and in and , respectively. For the sake of simplicity we will consider only the zeros of in A similar procedure may also be employed for zeros of in
Let us define
So we have, by (4.1), that
Theorem 4.2. Under the conditions in (2.7):(i)the discrete spectrum is a bounded, at most countable set and its limit points lie on the bounded subinterval of the real axis;(ii)the set is a bounded and its linear Lebesgue measure is zero.
Proof. From (2.3) and (3.1) we obtain that is analytic in , continuous on the real axis and has the form where Using (2.5), (2.6), and (2.9) we get that So From (4.3), (4.6) and uniqueness theorem for analytic functions , we get (i) and (ii).
Theorem 4.3. If then where is the lengths of the boundary complementary intervals of
Theorem 4.4. Under the conditions the operator 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 has an analytic continuation to the half-plane Im. Hence the limit points of its zeros on cannot lie in . Therefore using Theorem 4.2, we have the finiteness of zeros of in . Similarly we find that the function has a finite number of zeros with finite multiplicity in . 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
It is clear that the condition (4.9) guarantees the analytic continuation of and from the real axis to the lower and the upper half-planes respectively. So the finiteness of the eigenvalues and the spectral singularities of are obtained as a result of these analytic continuations.
Now let suppose that for some and which is weaker than (4.9). It is obvious that under the condition (4.10) the function is analytic in and infinitely differentiable on the real axis. But does not have analytic continuation from the real axis to the lower half-plane. Similarly, 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 cannot be shown in a way similar to Theorem 4.4.
Let us denote the sets of limit points of and by and respectively and the set of all zeros of with infinite multiplicity in by Analogously define the sets and
It is clear from the boundary uniqueness theorem of analytic functions that  and where denote the Lebesgue measure on the real axis.
Theorem 4.5. If (4.10) holds, then
Proof. We will prove that The case is similar. Under the condition (4.10) is analytic in all of its derivatives are continuous on the real axis and there exists such that From Theorem 4.2, we get that Let us define the function Since the function is not equal to zero identically, by Pavlov's theorem , holds, where is a constant and is the Lebesgue measure of -neighborhood of Using (2.5), (2.6), (2.9) and (4.4) we obtain that where and are constants depending on and Substituting (4.16) in the definition of we get Now (4.15) and (4.17) imply that Since , consequently (4.18) holds for arbitrary if and only if or
Theorem 4.6. Under the condition (4.10) the operator 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 and have finite number of zeros with finite multiplicities in and respectively. We will prove it only for The case of is similar.
It follows from (4.11) that So the bounded sets and have no limit points, that is, the has only a finite number of zeros in Since these zeros are of a finite multiplicity.
Theorem 4.7. If the condition (2.7) is satisfied then the set is of the first category.
Proof. From the continuity of it is clear that the set is closed and is a set of Lebesgue measure zero which is of type . According to Martin's theorem  there is measurable set whose metric density exists and is different from and at every point of So, is of the first category from the theorem due to Goffman . We also have obviously same things for Consequently is of the first category by (4.3).
M. G. Gasymov, “On the decomposition in a series of eigenfunctions for a non-selfadjoint boundary value problem of the solution of a differential equation with a singularity at zero point,” Soviet Mathematics. Doklady, vol. 6, pp. 1426–1429, 1965.View at: Google Scholar
B. S. Pavlov, “On seperation conditions for spectral components of a dissipative operator,” Mathematics of the USSR-Izvestiya, vol. 9, pp. 113–137, 1975.View at: Google Scholar
V. E. Lyance, “A differential operator with spectral singularities—I,” American Mathematical Society Translations, vol. 2, no. 60, pp. 185–225, 1967.View at: Google Scholar
V. E. Lyance, “A differential operator with spectral singularities—II,” American Mathematical Society Translations, vol. 2, no. 60, pp. 227–283, 1967.View at: Google Scholar
E. Bairamov, Ö. Çakar, and A. O. Çelebi, “Quadratic pencil of Schrödinger operators with spectral singularities: discrete spectrum and principal functions,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 303–320, 1997.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. M. Krall, E. Bairamov, and Ö. Çakar, “Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition,” Journal of Differential Equations, vol. 151, no. 2, pp. 252–267, 1999.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. A. Naimark, Linear Differential Operators II, Ungar, New York, NY, USA, 1968.
P. A. Binding, P. J. Browne, W. J. Code, and B. A. Watson, “Transformation of Sturm-Liouville problems with decreasing affine boundary conditions,” Proceedings of the Edinburgh Mathematical Society, vol. 47, no. 3, pp. 533–552, 2004.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. I,” Proceedings of the Edinburgh Mathematical Society, vol. 45, no. 3, pp. 631–645, 2002.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II,” Journal of Computational and Applied Mathematics, vol. 148, no. 1, pp. 147–168, 2002.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. A. Binding, P. J. Browne, and B. A. Watson, “Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter,” Journal of Mathematical Analysis and Applications, vol. 291, no. 1, pp. 246–261, 2004.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. A. Marchenko, Sturm-Liouville Operators and Applications, vol. 22 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1986.View at: MathSciNet