DISCONTINUOUS STURM-LIOUVILLE PROBLEMS INVOLVING AN ABSTRACT LINEAR OPERATOR

In this paper we introduce to consideration a new type boundary value problems consisting of an “Sturm-Liouville” equation on two disjoint intervals as −p(x)y + q(x)y +By|x = μy, x ∈ [a, c) ∪ (c, b] together with two end-point conditions whose coefficients depend linearly on the eigenvalue parameter, and two supplementary so-called transmission conditions, involving linearly left-hand and right-hand values of the solution and its derivatives at point of interaction x = c, where B : L2(a, c) ⊕ L2(c, b) → L2(a, c)⊕L2(c, b) is an abstract linear operator, non-selfadjoint in general. For self-adjoint realization of the pure differential part of the main problem we define “alternative” inner products in Sobolev spaces, “incorporating” with the boundary-transmission conditions. Then by suggesting an own approaches we establish such properties as topological isomorphism and coercive solvability of the corresponding nonhomogeneous problem and prove compactness of the resolvent operator in these Sobolev spaces. Finally, we prove that the spectrum of the considered eigenvalue problem is discrete and derive asymptotic formulas for the eigenvalues. Note that the obtained results are new even in the case when the equation is not involved an abstract linear operator B.


Introduction
Boundary value problem for the Sturm-Liouville equation with discontinuous leading coefficients arises in geophysics, electromagnetics, elasticity, and other fields of engineering and physics; for example, modeling toroidal vibrations and free vibrations of the earth, reconstructing the discontinuous material properties of a nonabsorbingmedia, as a rule leads to direct and inverse problems for the Sturm-Liouville equation with discontinuous coefficients. The applications of Sturm-Liouville problems in physics and engineering are numerous. Their use in problems of vibration, heat transfer, quantum mechanics, and a host of other areas have proven successful for many years, and indeed go back to the early 18th century. For example, consider the initial-boundary value problem for the heat equation where q is a given coefficient function. This problem describes the temperature of a heat conducting bar with a nonuniform heat loss term given by −q(x)u. Applying the method of separation of variables we obtain the Sturm-Liouville problem Au := −u ′′ + qu = λu, u(0) = u(1) = 0.
It is important to find a complete set of eigenvectors of A, or, equivalently, to diagonalize A in suitable infinite dimensional Hilbert space. The problem of diagonalizing a linear map on an infinite-dimensional space arises in many other ways, and is part of what is called spectral theory. Spectral theory provides a powerful way to understand linear operators by decomposing the space on which they act into invariant subspaces on which their action is simple. In the finite-dimensional case, the spectrum of a linear operator consists of its eigenvalues. The action of the operator on the subspace of eigenvectors with a given eigenvalue is just multiplication by the eigenvalue. Spectral theory of bounded linear operators on infinite-dimensional spaces is more involved. For example, an operator may have a continuous spectrum, in addition to, or instead of, a point spectrum of eigenvalues. A particularly simple and important case is that of compact, self-adjoint operators since such operators may be approximated by finite-dimensional operators, and their spectral theory is close to that of finite-dimensional symmetric operators. Sturm-liouville problems of spectral analysis consist in recovering operators from their spectral characteristics. Many thousands of papers, by Mathematicians and by others, have been published on this topic since then. Although the history of the subject is long, it remains an active area of research as new applications and concepts as well as computational difficulties continue to arise. The general results on the eigenvalue distribution of the eigenvalues of ordinary differential operators were obtained by Birkhoff [6], and for partial differential operators by Weyl [39]. In 1910 Weyl proved that the essential spectrum, which in this case is just the set of accumulation points of the spectrum, is stable when the boundary condition is modified. Tamarkin [31] introduced a concept of regular boundary conditions and proved that the system of root functions, i.e. eigenfunctions and associated functions of the regular boundary value problem is complete. In 1957 several remarkable papers were published. Rosenblum [27] and Kato [12] proved stability of absolutely continuous spectra for self-adjoint operators under trace class perturbations and Aronszajn [4] showed that the absolutely continuous Darts of spectral measures of Sturm-Liouville problems corresponding to different boundary conditions are equivalent, whereas their singular parts are mutually singular measures. Keldysh [14] elaborated expansions over root functions for weak perturbations of compact self-adjoint operators. Different challenges emerged with the development of 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 [3,13,40]. The most complete and sharp results for compact perturbations and for the so-called β-subordinate perturbations of self-adjoint operators are due to Markus and Matsaev [16] (see more details in [17]). Some recent developments of higher order differential operators whose boundary conditions depend on the eigenvalue parameter, including spectral asymptotics and basis properties, have been investigated in [18,30]. General characterizations of self-adjoint boundary conditions have been presented in [38] for singular and regular problems. Gesztesy and Simon [8] found new uniqueness results with partial information on the spectrum for Sturm-Liouville operators with scalar and matrix coefficients, respectively. They showed that more information on the potential can compensate for less information about the spectrum. Martinyuk and Pivovarchik [19] proposed a new method for reconstructing the potential on half the interval. Sakhnovich [28] studied the existence of solutions of half inverse problems. Singular potentials were studied by Hryniv and Mykytyuk [9]. Buterin studied half inverse problem for differential pencils with the spectral parameter in boundary conditions [7]. Trooshin and Yamamoto [35] obtained Hochstadt-Lieberman type theorems for nonsymmetric first order systems. For quadratic pencils of Sturm-Liouville operators without the spectral parameter and transmission conditions Yang and Zettl [41] proved that if p(x) and q(x) are known on half of the domain interval, then one spectrum suffices to determine them uniquely on the other half. These references are certainly not intended to be comprehensive but are given to indicate the wide interest in and variety of half inverse type problems. For the background and applications of the boundary value problems to different areas, we refer the reader to the monographs and some recent contributions as [13,16,20,25,26,31,34,37,40].
In this study we consider a "Sturm-Liouville" equation involving an abstract linear operator B, namely the differential-operator equation and transmission conditions at one interior point x = c , the potential q(x) is real-valued function which continuous in each of the intervals [a, c) and (c, b], and has a finite limits q(c ∓ 0), µ is a complex spectral parameter, α ij , β ± ij , α ′ ij (i = 1, 2 and j = 0, 1) are real numbers. B is an abstract linear operator in Hilbert space This Sturm-Liouville problem is a non-classical eigenvalue problem since it contains an abstract linear operator in the equation, eigenvalue parameter appears also in the boundary conditions and two new conditions added to boundary conditions (so-called transmission conditions). Naturally the spectral theory of this problem is more involved. Boundary value problems with the spectral parameter in boundary conditions and/or with supplementary transmission conditions arise in various problems of mathematics and physical as well as in applications. For example, some boundary value problems with transmission conditions arise in heat and mass transfer problems [15], in vibrating string problems when the string loaded additionally with point masses [32], in diffraction problems [37] and etc. Such properties, as isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel bases of a system of root functions of the similar boundary value problems with transmission conditions and its applications to the corresponding initial boundary value problems for parabolic equations have been investigated in [22] and [23]. Also some problems with transmission conditions which arise in mechanics (thermal conduction problem for a thin laminated plate) were studied in the article [33]. Similar problems for differential equations with discontinuous coefficients were investigated by Rasulov in monograph [26]. Detailed studies on spectral problems for ordinary differential operators depending on the parameter and/or with transmission conditions can be found in various publications, see e.g. [1,2,5,10,11,21,22,24,29,36,42], where further references and links to applications can be found.

Sobolev spaces with "alternative" inner-products "incorporating" with the considered problem
At first we shall introduce an "alternative" inner product in classical Sobolev spaces such a way that the pure differential part of the considered problem can be interpreted as self-adjoint problem in these spaces.

Denote the determinant of the boundary-matrix
and the determinant of the k-th and j-th columns of the transmission-matrix . Throughout in this study we shall assume that the conditions is hold. For self-adjoint realization we shall introduce some "new" Hilbert spaces with alternative inner products. Recall that the direct sum space ) such that f 1 and f 2 has generalized n-th derivatives (in the sense of distributions) in L 2 (a, c) and L 2 (c, b), respectively, with the inner product we shall replace by the "alternative" inner products as respectively and apply operator theory in the new Hilbert space But such realization of this direct sum space allow as to interpret the conditions (1.2)-(1.5) as "self-adjoint boundary-transmission conditions." Let us we define the boundary functionals: and differential operator Φu A suitable inner-product space in which to search for solution of the equation µy − Ψy = f (x), is the linear space , equipped with the inner product and corresponding norm It can be verify easily that, all axioms of inner product are satisfied.
.. be any Cauchy sequence with respect to the norm (2.9). Then by (2.9) the sequence (f n (.)), which consist of the first components of (F n ), will be a Cauchy sequence in the Hilbert

Topological isomorphism and coercive solvability. The resolvent operator.
Let us construct the operator £ : H → H with the domain with the a finite limits f (c ∓ 0) and f ′ (c ∓ 0); ΨF ∈ L 2 (a, c) ⊕ L 2 (c, b); and action low Obviously, the operator £ is well-defined in the Hilbert space H. Then the problem (1.1)-(1.5) acquires the operator equation form in the Hilbert space H.
between eigenfunctions y k (x) of the problem (1.1) − (1.5) and eigenelements Y k of the operator £.
are dense in L 2 (a, c) and L 2 (c, b), respectively, we have that the function g 0 (x) vanishes on [a, c) ∪ (c, b]. It is easy to see that, there is an element Putting F = F 0 in (3.1) we have g 1 = g 2 = 0. Consequently (dom( £)) ⊥ = (0, 0, 0). The proof is complete. Now, consider the nonhomogeneous boundary value-transmission problem in the Hilbert space H. Throughout in below we shall use the notations and U ∞ (r) = {µ ∈ C :| µ |> r}, r > 0. c, b). Then, for any δ > 0 there exists r δ > 0 such that for all complex numbers µ ∈ G δ ∩ U ∞ (r δ ) the operator µI − £ is an topological isomorphism from H onto H and for these µ the coercive estimate

Theorem 3.1. Suppose that the linear operator B is compact from
holds for the solution Y = Y µ (.) of the equation (3.4) where C δ is a constant, which depend only of δ.
Proof. From the definitions of £, H and H it follows immediately that the linear operator µI − £ acts from H into H continuously for all µ ∈ C. Following the same procedure as in [23], we obtain that for any δ > 0 there exists r δ > 0 such that for all complex numbers µ ∈ G δ ∩ U ∞ (r δ ) the operator ℑ(µ) : y → (µy − Ψ(µ)y, Ψ 1 (µ)y, Ψ 2 (µ)y) is an isomorphism from W 2 2 (a, c)⊕W 2 2 (c, b) onto (L 2 (a, c)⊕ L 2 (c, b)) ⊕ C 2 and for these µ the coercive estimate holds for a solution y(x) of the problem (3.2)-(3.3). Consequently, the operator £−µI is an isomorphism from H onto H. The estimate (3.5) follows from (3.6). c, b) then, for any δ > 0 there exists r δ > 0 such that for all complex numbers µ ∈ G δ ∩ U ∞ (r δ ) are regular point of the operator £ and for the resolvent operator of £ the estimate

Corollary 3.1. If the linear operator B is compact from
holds, where C δ > 0 is a constant which depend only of δ.
, n = 1, 2, ... be any bounded sequence in H. Then the sequence (f n (.)) consisting of the first components of (F n ) will be bounded in the direct sum space are compact, the sequence(f n k (.)) has an convergent subsequence (f n ks (.)) in spaces C[a, c] and C[c, b] respectively. Consequently the numerical sequences (B ′ a (f n ks )) and (B ′ b (f n ks )) are convergent. Let f 1 , f 2 ∈ C are limits of this numerical sequences respectively. Now defining F 0 = (f 0 (.), f 1 , f 2 ), we see that F 0 ∈ H and the subsequence (F n ks ) converges to F 0 in the Hilbert space H, so the embedding H ⊂ H is compact. Further, from Corollary 3.2, follows that the resolvent operator R(µ, £) is bounded from H into H. Consequently, the resolvent operator R(µ, £) is compact from H into itself.

A pure differential operator associated with the problem
Consider the pure differential part (i.e. without operator B) of the considered problem (1.1)-(1.5). Let £ be linear differential operator in Hilbert space H with domain D(£) = D( £) and action low where, as usual, W (f, g; x) denotes the Wronskians of the functions f and g. From the definitions of boundary functionals we get that  Proof. Since £ is symmetric, it is enough to prove that D(£ * ) = D(£). Let F ∈ D(£ * ). Then for all G ∈ D(£) Let µ 0 be any regular value of £ such that Imµ 0 ̸ = 0. Then Since µ 0 is a regular point of £ the operator µ 0 I − £ has the inverse (µ 0 I − £) −1 which is defined on whole Hilbert space H. Then defining F 0 ∈ D(£) as By using the last equalities and applying the previous Theorem we have for an arbitrary G ∈ D(£). Consequently, for an arbitrary element G ∈ D(£) the equality holds. Since µ 0 is a regular point of £, we can put G = R(µ 0 , £)(F − F 0 ) in the last equality. Then we have ∥F − F 0 ∥ H = 0, namely, F = F 0 . Thus we find that F ∈ D(£) i.e. D(£ * ) = D(£). The proof is complete. Let µ = s 2 . Following the same procedure as in [24] we have the next Theorem.

The structure of the spectrum and asymptotic behaviour of the eigenvalues in complex plane
Let us present here some needed basic definitions about spectrum of the general linear operators in Hilbert space( see, for example, [24]). Let A be densely defined closed operator in complex Hilbert space E. The point µ of the complex plane is called a regular point of an operator A in E, if the operator A − µI is invertible(i.e has a bounded inverse operator (A − µI) −1 which defined on whole E). In this case the operator R(µ, A) = (A − µI) −1 is called the resolvent of the operator A.The complement of the set of regular points ρ(A) to the entire complex plane is called the spectrum σ(A) of the operator(obviously, all eigenvalues belongs to the spectrum).
Let µ 0 be eigenvalue of A. The linear manifold is called a root lineal corresponding to eigenvalue µ 0 . The dimension of the lineal N µ0 is called an algebraic multiplicity of the eigenvalue µ 0 . The spectrum σ(A) of the operator A is called discrete if σ(A) consist of isolated eigenvalues with finite algebraic multiplicities and infinity is the only possible limit point of σ(A).
Let A be an linear operator with discrete spectrum and let S be a subset of complex plane C. In below we shall denote by N (r, S, A) the sum of the algebraic multiplicities of all the eigenvalues of A contained in S ∩ {µ ∈ C : |µ| < r}. Definition 5.1. Let A 1 be any closed linear operator having at least one regular point. A linear operator A 2 is said to be A 1 -compact (or compact with respect to A 1 ) if D(A 2 ) ⊇ D(A 1 ) and if for some regular point µ 0 ∈ ρ(A 1 ) the operator A 2 R(µ 0 , A 1 ) = A 2 (A 1 − µ 0 I) −1 is compact (see, for example, [16]).
Putting (5.11) in these formulas yields the needed equalities (5.9)-(5.10). Now, we are ready to prove the main result of this section. c, b). Then, the spectrum of the problem (1.1)-(1.5) is discrete and consist of precisely denumerable many eigenvalues µ n , n = 1, 2, ... which, when listed according to decreasing real parts and repeated according to algebraic multiplicity, has the following asymptotic representation:

Discussion of the results
Note that such generalization of Sturm-Liouville problems involving abstract linear operator in the equation has been investigated by us for the first time in literature. It is easy to verify that the pure differential part of the considered problem (1.1)-(1.5) is not self-adjoint in the usual direct sum space L 2 [a, c) ⊕ L 2 (c, b] (i.e. the generated differential operator is not self-adjoint in L 2 ([a, c) ⊕ L 2 (c, b]) ⊕ C 2 ). For self-adjoint realization of this problem we develop an own approach. Namely, we define "alternative" inner product in this space "incorporating" with the considered problem. We must emphasize that in our approach the sign of the boundarytransmission determinants θ i (i = 1, 2), ∆ 12   for which the condition (2.1) is not hold. By direct calculation we can show that this problem has only one eigenvalue µ = 1 in contract with standard Sturm-Liouville problems which has infinitely many eigenvalues. We find "minimal" conditions (2.1) on coefficients of boundary and transmission conditions under which the pure differential part of the problem (1.1)-(1.5) can be interpreted as selfadjoint eigenvalue problem in some "alternative realization" of usual direct sum space L 2 [a, c) ⊕ L 2 (c, b] ⊕ C 2 . Even under condition (2.1) the spectral properties of the considered problem is essentially different from the corresponding spectral properties of standard Sturm-Liouville problem. For instance, the eigenvalues of the problem (1.1)-(1.5) are not real numbers in general. But the leading term in asymptotic expansion of eigenvalues is real sequence. Note also that, the second term in asymptotic expansion of eigenvalues appears in the more "weak" form as o(n 2 ) because of the abstract linear operator B in the equation.