Boundary value problems with eigenvalue-dependent boundary and transmission conditions
© Aydemir; licensee Springer. 2014
Received: 4 March 2014
Accepted: 8 May 2014
Published: 22 May 2014
In this paper the operator-theoretical method to investigate a new type boundary value problems consisting of a two-interval Sturm-Liouville equation together with boundary and transmission conditions dependent on eigenparameter is developed. By suggesting our own approach, we construct modified Hilbert spaces and a linear operator in them in such a way that the considered problem can be interpreted as a spectral problem for this operator. Then we introduce so-called left- and right-definite solutions and give a representation of solution of the corresponding nonhomogeneous problem in terms of these one-hand solutions. Finally, we construct Green’s vector-function and investigate some important properties of the resolvent operator by using this Green’s vector-function.
KeywordsSturm-Liouville problems eigenparameter-dependent boundary and transmission conditions Green’s function resolvent operator
This example makes it clear that the Sturm-Liouville problems are of broad interest. There is a well-developed theory for classical Sturm-Liouville problems (see, e.g., [1–5] and the references therein). Details of the derivation of the theory and of related background results can be found in the cited references. Although the subject of Sturm-Liouville problems is over 160 years old, these problems are an intensely active field of research today. The main tool for solvability analysis of such problems is the concept of Green’s function. Green’s functions have played an important role as a theoretical tool in the field of physics, since the possibility of a transition from the problems in mathematical physics to integral equations is based on the fundamental concept of Green’s function. Therefore, the powerful and unifying formalism of Green’s functions finds applications not only in standard physics subjects such as perturbation and scattering theory, bound-state formation, etc., but also at the forefront of current and, most likely, future developments (see ). Green’s function transforms the differential equation into the integral equation, which, at times, is more informative. In terms of Green’s function, the BVP with arbitrary data can be solved in a form that shows clearly the dependence of the solution on the data. Namely, Green’s function approach would allow us to have an integral representation of the solution instead of an infinite series. Determination of Green’s functions is also possible using Sturm-Liouville theory. This leads to series representation of Green’s functions (see, e.g., the monograph  as well as the recent results in  and the references therein).
where is a real-valued piecewise constant function, for , for , the potential is a real-valued function continuous in each of the intervals and , and has finite limits , μ is a complex spectral parameter, the coefficients , , , ( and ), are real numbers. This Sturm-Liouville problem is a non-classical eigenvalue problem since the eigenvalue parameter μ appears not only in the differential equation, but also in the boundary and transmission conditions. Moreover, in the differential equation there appears a singularity at one interior point. Because of these reasons the spectral theory of this problem is more complicate. Naturally, eigenfunctions of this problem may have discontinuity at the singular interior point. Some special cases of this problem arise after an application of the Fourier method to a varied assortment of physical problems. For instance, some boundary value problems with transmission conditions arise in heat and mass transfer problems , in vibrating string problems when the string is loaded additionally with point masses , in diffraction problems , in quantum mechanics , in thermal conduction problems for a thin laminated plate etc. Such properties as isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel bases property of a system of root functions of some boundary value problems with transmission conditions and its applications to the corresponding initial boundary value problems for parabolic equations have been investigated in [16–19]. For the background and applications of boundary value transmission problems to different areas, we refer the reader to the monographs and some recent contribution [8–11, 17, 18, 20–25].
2 Hilbert space formulation of the problem
In certain cases the boundary value problem can be characterized by means of a uniquely determined unbounded self-adjoint operator. In these cases the eigenvalues and eigenfunctions of the boundary value problem are determined by the eigenvalues and eigenvectors of the corresponding operator; these will be called a self-adjoint case of the boundary value problem. In some cases such a characterization is not possible and these will be referred to as ‘symmetric’ cases in general. In classical point of view, our problem cannot be characterized as ‘self-adjoint case’. For ‘self-adjoint characterization’ of the considered problem (1)-(5), we shall define a new Hilbert space as follows.
for . It is easy to see that the relation (6) really defines a new inner product in the direct sum space .
Lemma 1 ℋ is a Hilbert space.
Proof Let , , be any Cauchy sequence in ℋ. Then by (6) the sequences and will be Cauchy sequences in the Hilbert spaces and , respectively. Therefore they are convergent. Let and be limits of these sequences, respectively. Defining we have that and in ℋ. The proof is complete. □
Then problem (1)-(5) can be written in the operator equation form as , in the Hilbert space ℋ.
Theorem 1 The linear operator ℜ is symmetric in the Hilbert space ℋ.
The proof is complete. □
Theorem 2 The linear operator ℜ is self-adjoint in ℋ.
From this equality, by applying the technique of Theorem 2.5 in our previous work , it can be derived easily that , , , and , , . The proof is complete. □
Theorem 3 The operator ℜ has only point spectrum, i.e., .
Proof It suffices to prove that if is not an eigenvalue of ℜ, then is a regular point of ℜ, i.e., . Let not be an eigenvalue of ℜ. The resolvent operator exists and is defined on all of ℋ. By Theorem 2 and the closed graph theorem, we get that is bounded. Thus, . Hence . □
3 Left-definite and right-definite solutions
respectively. The existence of these solutions follows from the well-known Cauchy-Picard theorem of ordinary differential equation theory. Moreover, by applying the method of , we can prove that each of these solutions are entire functions of the parameter for each fixed x.
4 Construction of Green’s function
5 Representations of the resolvent operator in terms of Green’s vector-function
where . Consequently, we have the following theorem.
where , holds.
holds for all regular value such that .
The proof is complete. □
Theorem 6 The resolvent operator is compact in the Hilbert space ℋ.
Similarly to  we can easily show that . Thus for . Consequently, the series (35) is strongly convergent. It is obvious that the orthogonal projections , , are compact operators since each of them are of finite rank. Consequently, the sum of series (35) is also compact in ℋ. The proof is complete. □
The author is grateful to anonymous referees for their constructive comments and suggestions, which led to the improvement of the original manuscript.
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