Abstract
We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.
1. Introduction
Various modifications of classical Sturm-Liouville problems have attracted a lot of attention in the recent past because of the appearance of new important applications in mathematics, mechanics, physics, electronics, geophysics, meteorology, and other branches of natural sciences (see [1–9] and references cited therein). For example, they describe the vibrational modes of various systems, such as the vibrations of a string or the energy eigenfunctions of a quantum mechanical oscillator, in which case the eigenvalues correspond to the resonant frequencies of vibration or energy levels. It was, in part, the idea that the discrete energy levels observed in atomic systems could be obtained as the eigenvalues of a differential operator which led Schrödinger to propose his wave equation. The simplest example of a Sturm-Liouville operator is the constant-coefficient second-derivative operator, whose eigenfunctions are trigonometric functions. Many other important special functions, such as Airy functions and Bessel functions, are associated with variable-coefficient Sturm-Liouville operators. One feature that occurs for Sturm-Liouville operators, which does not occur for matrices, is the possibility of an absolutely continuous spectrum. Instead of eigenfunction expansions, we then get integral transforms, of which the Fourier transform is an example. Other, more complicated spectral phenomena can also occur. For example, eigenvalues embedded in a continuous spectrum, singular continuous spectrum, and pure point spectrum consisting of eigenvalues that are dense in an interval (see [2]).
In this paper we will examine a new type of Sturm-Liouville equation involving an abstract linear operator , namely, the equation on , together with eigen-dependent boundary conditions: and transmission conditions at one interior point : where , , , , and are real numbers, is a real-valued function and continuous in each and which have finite limits , is a complex spectral parameter, and is an abstract linear operator (unbounded and non-self-adjoint in general) in Hilbert space . Naturally, everywhere we will assume that , , , and . By standard arguments (see [10]) we will also assume that . Note that the “Sturm-Liouville” problem studied here is new and nonstandard, since it contains a nondifferential term, namely, an abstract linear operator in the equation. Moreover, spectral parameter appears not only in the equation, but also in the boundary conditions and two supplementary transmission conditions are given at one interior point.
Some special cases of this problem arise after an application of the method of separation of variables to the varied assortment of physical problems. For example, some boundary value problems with transmission conditions arise in heat and mass transfer problems [7], in vibrating string problems when the string is loaded additionally with point masses [8], and in diffraction problems [9]. Different challenges emerged with the development of the Sturm theory and a corresponding awareness of the importance of distinguishing the absolutely continuous component from other parts of the essential spectrum, in connection with existence and completeness of the wave operators [3, 11, 12].
2. Operator Treatment in Associated Hilbert Spaces
For operator-theoretic interpretation we will introduce some modified Hilbert spaces according to boundary-transmission conditions. For this throughout in the below we will assume that , and define modified Hilbert spaces and as follows. Firstly, we will replace the standard inner product in direct sum space , which is given by for , , by modified inner product Then we will replace the standard inner product in the direct sum space , which is given by for , , by the modified inner product accordance with boundary-transmission conditions as and apply operator theory in the Hilbert space
Remark 1. It is readily seen that the modified inner product (8) is equivalent to the standard of inner product , so is also Hilbert space and can be seen as different realization of the Hilbert space . But such realization of direct sum space allows us to interpret the conditions (2)–(4) as self-adjoint boundary-transmission conditions.
Denoting and , define a linear operator in direct sum space by action low: on the domain of definition consisting of all which satisfy the following conditions:(i) and are absolutely continuous on both and for arbitrary and have finite limits and ,(ii),(iii),(iv).
Consequently we can reformulate the boundary value-transmission problem (BVTP) (1)–(4) in the operator-equation form as in the Hilbert space .
Lemma 2. The linear operator is densely defined in .
Proof. It is enough to prove that, if is orthogonal to all , then . Suppose that Denote by the set of infinitely differentiable functions in , each of which vanishes on some neighborhoods of the points , , and . Since for each , we have from (12) that for all which in turn implies that for all . Taking into account that and are dense in and , respectively, we have that the function vanishes on . Furthermore, by choosing an element such that and putting in (12) we have . Hence . The proof is complete.
Now let be linear differential operator in Hilbert space with domain and action low:
Lemma 3. The operator is symmetric in .
Proof. Let , be any two elements. Integrating twice by parts, we have where . From the fact, that and satisfied the first boundary condition (2) we have in turn; since and satisfies both transmission conditions (4) it follows that Further, the direct calculations give Now, putting these equalities in (16) we have needed equality
Corollary 4. The eigenvalues of are real.
Theorem 5. is self-adjoint linear operator in the Hilbert space .
Proof. For shortening denote . Since are symmetric and densely defined in it is enough to show that if for all then and where and ; that is, (i) , , , , and ; (ii) ; (iii) ; (iv) ; and (v) . For all , we find that . Hence, by standard Sturm-Liouville theory, (i) and (iv) hold. By (iv), the equation for all becomes On the other hand, by two partial integrations we get Therefore, By Naimark's patching lemma [13] there is such that , and . Putting in (22) we conclude that . Thus, (ii) holds. Similarly, we can prove (v). It remains to show that (iii) holds. Choose so that , and . Then . Putting in (22), we obtain . Let satisfy , and . Then, . By (22) we get . Lastly, choose so that , and . In this case . Putting in (22) we get . Thus . The proof is complete.
3. Maximal Decreasing of the Resolvent Operator and Discreteness of the Spectrum
To establish the topological isomorphism and coerciveness we need to introduce a new inner product space as linear space: equipped with the inner product It can be verified easily that all axioms of inner product are satisfied.
Lemma 6. is a Hilbert space.
Proof. Let , , be any Cauchy sequence in . Then by (24) the sequence , which consists of the first components of , will be a Cauchy sequence in the Hilbert space and therefore is convergent in this space. Let be limit of this sequence. By virtue of the fact that the embeddings and are continuous, the sequences , , and are converges to , , and , accordingly. Hence since for all by (23). Now, defining we see that and the sequence converges to in ; so, the arbitrary Cauchy sequence in is convergent. The proof is complete.
Theorem 7. If the operator T is compact from into then, for any , there exists such that for all complex numbers satisfying , , the operator is an isomorphism from onto and for these the coercive estimate holds for the solution of the equation , , where is a constant, which depends only on .
Proof. It is obvious that the linear operator acts from into continuously for all . Furthermore, proceeding in a similar manner as in [14], we obtain that for any there exists such that for all complex numbers satisfying , , the operator from onto is an isomorphism and for these the coercive estimate holds for a solution of the problem , , , and , where . Consequently, the operator is an isomorphism from onto . The estimate (25) follows from (26).
Definition 8 (see [9]). Let be densely defined closed operator in complex Hilbert space . The point of the complex plane is called a regular point of an operator in , if the operator is invertible (i.e., has a bounded inverse operator which is defined on whole ). In this case the operator is called the resolvent of the operator . The complement of the set of regular points to the entire complex plane is called the spectrum of the operator (obviously, all eigenvalues belong to the spectrum).
Corollary 9. From the coercive estimate (25), in particular, it follows that the maximal decreasing of the resolvent operator , namely, the estimate holds for all complex as in the formulation of the last Theorem.
Definition 10 (see [12]). Let be eigenvalue of . The linear manifold is called a root lineal corresponding to eigenvalue . The dimension of the lineal is called an algebraic multiplicity of the eigenvalue . The spectrum of the operator is called discrete if consist of isolated eigenvalues with finite algebraic multiplicities and infinity is the only possible limit point of .
Theorem 11. If the operator acts compactly from into , then the spectrum of the problem (1)–(4) is discrete.
Proof. At first show that the embedding is compact. For this, let , , be any bounded sequence in . Then the sequence consisting of the first components of will be bounded in the direct sum space . Since the embeddings and are compact, the sequence has a convergent subsequence in the space . Let be limit of this subsequence. Further, since the embedding is compact, the sequence has a convergent subsequence in space . Consequently the numerical sequence is convergent. Let be limit of this numerical sequence. Now defining , we see that and the sequence converges to in the Hilbert space , so the embedding is compact. Further, from the coercive estimate (25) in particular, it follows that the resolvent operator acts boundedly from into . Consequently, the resolvent operator acts compactly from into itself. Then by virtue of well-known theorem of functional analysis (see, e.g., ([11], Chapter III, Section 6)) the spectrum of is discrete.
4. Distribution of the Eigenvalues in the Complex Plane
Define a linear operator in the Hilbert space with domain and action low: for . Then, the considered problem (1)–(4) can be written in the operator-equation form as
Remark 12. The eigenvalues of the problems (1)–(4) and (30) coincide, and the corresponding eigenfunctions of (1)–(4) coincide with the first components of the corresponding eigenelements of .
Let be densely defined closed operator in complex Hilbert space and let be any subset of complex plane and any real number. By we will denote the number of eigenvalues of belonging to , which are smaller than and are counted according to their algebraic multiplicity.
Definition 13 (see [15]). Let be any closed linear operator having at least one regular point. A linear (in general, unbounded) operator is said to be -compact if and if for some regular point the operator is compact.
The following theorem can be deduced from Theorem 3.2 in [15].
Theorem 14. Let be self-adjoint operator in Hilbert space the spectrum of which is discrete, be -compact operator, and . Then if has precisely denumerable many positive eigenvalues and then for any where , , and as is the abbreviation for .
Lemma 15. The operator has precisely denumerable many eigenvalues , , with the asymptotic representation
Proof. Let and let be solution of the equation for which , , , and . Obviously, this solution satisfies the first boundary condition (2) and both transmission conditions (4). Consequently, eigenvalues of coincide with the zeros of the entire function Consider the case . By proceeding with the same procedure as in [16] we have Putting in previous equality we get Now by applying the well-known Rouche’s theorem (see, e.g., [13]) we can prove that the function has precisely denumerable many zeros , , with the asymptotic representation The proof is complete.
Lemma 16. Let the operator be compact with respect to in the Hilbert space . Then (i)the spectrum of is discrete and consists of precisely denumerable many eigenvalues;(ii)for any arbitrary small , all eigenvalues of with the possible exception of a finite number lie in the sector of angular ;(iii)for the sequence of eigenvalues , belongs to the sector , which, when listed according to nondecreasing modulus and repeated according to algebraic multiplicity, the asymptotic formula holds.
Proof. From asymptotic formula (33) it follows that for some real numbers and . In turn, from this we can easily derive that Consequently Taking into account the above, we have that, for any small , where, as usual, the expression , , is the abbreviation for . Putting in the last equality we have From that it immediately follows that The proof is complete.
Theorem 17. Under conditions of previous lemma the spectrum of the operator is discrete and consists of denumerable many eigenvalues , , which, when arranged in nondecreasing modulus and counted to their algebraic multiplicity, the asymptotic formulas hold.
Proof. Taking in view that for all small there are at most finite number eigenvalues of outside the angle , from Lemma 16 it follows that Again, by Theorem 7 for all , small enough, there are such that for all the inequalities and hold. Letting we have that Combining with (46) yields the needed formulas
The main result of this section is the following theorem.
Theorem 18. Let the operator be acted compactly from into . Then, the spectrum of BVTP (1)–(4) is discrete and consists of precisely denumerable many eigenvalues , , which, when listed according to decreasing real part and repeated according to algebraic multiplicity, has the following asymptotic representation:
Proof. By virtue of Theorem 11 the resolvent acted boundedly from to . On the other hand, the operator , defined by (29), acted compactly from to , by assumption on and definition of . Consequently the operator is compact in the Hilbert space ; that is, is -compact. Consequently it is enough to apply the previous theorem to complete the proof.
5. Examples
Let us give some examples of abstract linear operator as follows: where the functions and satisfy the same conditions as ; , are interior points; where the Kernels and are defined and continuous in and , respectively.
Consequently, the results of this study can be applied to the wide variety class of boundary value problems.
6. Concluding Remarks
All results in this study are derived under condition and . Let us show that this simple condition cannot be omitted. For this, consider the following simple special case of the problem (1)–(4) for which the condition does not hold: We can show that this problem has only the trivial solution for any . Thus, if then the spectrum of the problem (52) may be empty. Moreover, it is well known that for the standard Sturm-Liouville problems the eigenvalues are real and the first asymptotic term has the form . But for our problem, the eigenvalues may be also nonreal complex numbers and the asymptotic term appears in the “weak” form as because of the abstract linear operator in the equation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.