MULTI-INTERVAL STURM-LIOUVILLE PROBLEMS WITH DISTRIBUTIONAL COEFFICIENTS

The paper investigates spectral properties of multi-interval SturmLiouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions and also all generalized resolvents in terms of boundary conditions are given.


Introduction
Differential operators generated by the Sturm-Liouville expression l(y) = −(py ) + qy, arise in numerous problems of analysis and its applications. The classic assumptions on its coefficients are the following: Principal statements of the theory of such operators remain true under more general assumptions that are q, 1/p ∈ L 1 ([a, b], C) .
However, many problems of mathematical physics require study of differential operators with complex coefficients which are Radon measures or even more singular distributions. In papers [1,2,3,4] a new approach to investigation of such operators was proposed based on definition of these operators as quasi -differential, which allows also to consider differential operators of higher order [5,3]. The purpose of this paper is to develop a spectral theory of Sturm-Liouville operators given on a finite system of bounded intervals under minimal conditions for the regularity of the coefficients.

Preliminary results
Let [a, b] be a compact line segment, m ∈ N, and let a = a 0 < a 1 < · · · < a m = b be a partition of [a, b] into m parts. Let us consider the space L 2 ([a, b], C) as a direct sum ⊕ m k=1 L 2 ([a k−1 , a k ], C) which consists of vector functions f = ⊕ m k=1 f k such that f k ∈ L 2 ([a k−1 , a k ], C).
Let on every interval (a k−1 , a k ), k ∈ {1, . . . , m} the formal Sturm-Liouville differential expression (1) l k (y) = −(p k (t)y ) + q k (t)y + i((r k (t)y) + r k (t)y ), be given with the coefficients p k , q k , and r k satisfying the conditions where the derivatives Q k are understood in the sense of distributions. Similarly to [3] (see also [1,4]), using the coefficients of the expression (1) we introduce the quasi-derivatives on every segment [a k−1 , a k ] in the following way: k y := y; D [1] k y := p k y − (Q k + ir k )y; D [2] k y := (D [1] k y(t), D [1] k y(t) ∈ C 2 for all t ∈ [a k−1 , a k ]. Under assumptions (2) these expressions are Shin-Zettl quasiderivatives (see [12,11]). One can easily verify that for the smooth coefficients p k , q k and r k the equality l k (y) = −D [2] k y holds. Therefore one may correctly define expressions (1) under assumptions (2) as Shin-Zettl quasi-differential expressions, l k [y] := −D [2] k y. The corresponding Shin-Zettl matrices (see [11,12]) have the form Then on the Hilbert spaces L 2 ((a k−1 , a k ), C) minimal and maximal differential operators are defined, which are generated by the quasi-differential expressions l k [y] (see [12,11]), L k,1 : y → l k [y], Dom(L k,1 ) := y ∈ L 2 y, D [1] k y ∈ AC([a k−1 , a k ], C), D [2] k y ∈ L 2 , Results of [11,12] for general Shin-Zettl quasi-differential operators together with formula (3) imply that the operators L k,1 , L k,0 are closed and densely defined on the space In the case where p k , q k and r k are real-valued, the operator L k,0 is symmetric with the deficiency index (2, 2) and L * k,0 = L k,1 , L * k,1 = L k,0 .

Boundary-value problems. Symmetric case
We consider the space L 2 ([a, b], C) as a direct sum ⊕ m k=1 L 2 ([a k−1 , a k ], C) which consists of vector functions f = ⊕ m k=1 f k such that f k ∈ L 2 ([a k−1 , a k ], C). In this space L 2 ([a, b], C) we consider the maximal and the minimal operators L max = ⊕ m k=1 L k,1 and L min = ⊕ m k=1 L k,0 . It is easy to see that the operators L max , L min are closed and densely defined on the space L 2 ([a, b], C).
Throughout the rest of the paper we assume the functions p k , q k , and r k to be realvalued for all k and therefore the operators L k,0 are symmetric with the deficiency indices (2,2). Then the operator L min is symmetric with the deficiency index (2m, 2m) and L * min = L max , L * max = L min . Then naturally arises the problem of describing all its self-adjoint, maximal dissipative, and maximal accumulative extensions in terms of homogeneous boundary conditions of the canonical form. For this purpose it is convenient to apply the approach based on the concept of boundary triplets. It was developed in the papers by Kochubei [13], see also the monograph [14] and references therein.
Note that the minimal operator L min can not be semi-bounded even in the case of a single-interval boundary-value problem since the function p can change the sign.
Recall that a boundary triplet of a closed densely defined symmetric operator T with equal (finite or infinite) deficiency indices is a triplet (H, Γ 1 , Γ 2 ) where H is an auxiliary Hilbert space and Γ 1 , Γ 2 are linear maps from Dom(T * ) into H such that (1) for any f, g ∈ Dom (T * ) there holds with dim H = n exists for any symmetric operator T with equal non-zero deficiency indices (n, n) (n ≤ ∞), but it is not unique.
For the minimal operator L min in the space L 2 ([a, b], C) the boundary triplet is explicitly given by the following theorem.
Proof. We need to verify that both conditions of the definition of the boundary triplet are fulfilled. But since for every k = 1, . . . , m (C 2 , Γ 1,k , Γ 2,k ) is a boundary triplet for the operator L k,0 , then we have the following.
1) Let y = ⊕ m k=1 y m , z = ⊕ m k=1 z m be two arbitrary functions which belong to Dom(L max ). Then 2) Let g 1 , g 2 ∈ C 2m , and therefore g 1 = (g 1,1 , . . . , g 1,m ), g 2 = (g 2,1 , . . . , g 2,m ), where g ik ∈ C 2 . Then for all k = 1, . . . , m there are functions y k ∈ Dom(L k,1 such that Γ 1,k y k = g 1,k , Γ 2,k y k = g 2,k , which implies that the definition of boundary triplet is fulfilled. Denote by L K the restriction of the operator L max onto the set of functions y ∈ Dom(L max ) satisfying the homogeneous boundary condition where K is an arbitrary bounded operator on the space C 2m . Similarly, denote by L K the restriction of L max onto the set of functions y ∈ Dom(L max ) satisfying the homogeneous boundary condition where K is an arbitrary bounded operator on the space C 2m . Theorem 1 leads to the following description of all self-adjoint, maximal dissipative, and maximal accumulative extensions of the operator L max .
Theorem 3. Every L K with K being a contracting operator in the space C 2m , is a maximal dissipative extension of the operator L min . Similarly every L K with K being a contracting operator in C 2m , is a maximal accumulative extension of the operator L min .
Conversely, for any maximal dissipative (respectively, maximal accumulative) extension L of the operator L min there exists a unique contracting operator K such that L = L K (respectively, L = L K ).
The extensions L K and L K are self-adjoint if and only if K is a unitary operator on C 2m .
The mappings K → L K and K → L K are injective.
Proof. The assertion of the theorem follows from Theorem 1 together with [14, Th. 1.6] for boundary triplets of a general symmetric operator.

Real extensions and local boundary conditions
Recall that a linear operator T acting on the Hilbert space L 2 ([a, b], C) is called real if (1) for every function f from Dom(T ) the complex conjugate function f also lies in Dom(T ); (2) the operator T maps complex conjugate functions into complex conjugate functions, that is, T (f ) = T (f ).

Theorem 4.
Suppose that all the coefficients r k ≡ 0, k ∈ {1, . . . , m}. Then the maximal and the minimal quasi-differential operators L max and L min are real. All real maximal dissipative and maximal accumulative extensions of the real symmetric quasi-differential operator L min are self-adjoint.
The self-adjoint extensions L K or L K are real if and only if the unitary matrix K is symmetric.
Proof. As we assumed at the beginning of this section, all functions p k , Q k ∈ L 2 ([a k−1 , a k ], R). As for any k the coefficients of the quasi-derivatives are real-valued functions, one has D [1] k y = D [1] k y, k = 1, . . . , m, which implies that l k (y) = l k (y). Therefore for any y ∈ Dom(L k,1 ) we have y ∈ Dom(L k,1 ) and L k,1 (y) = L k,1 (y) and for any y ∈ Dom(L k,0 ) we have y ∈ Dom(L k,0 ). Thus the assertion of the theorem yields that L k,1 and L k,0 are real. Therefore the operators L max and L min are real being the direct sums.
Let now L K be a real maximal dissipative extension of L min . Again, one easily sees that Γ 1 y = Γ 1 y and Γ 2 y = Γ 2 y. Then for an arbitrary function y ∈ Dom(L K ) we have (K − I) Γ 1 y + i (K + I) Γ 2 y = 0, therefore K − I Γ 1 y − i K + I Γ 2 y = 0, so, in other words, L K ⊂ L K . Thus the maximal dissipative extension L K is also accumulative, therefore, the operator L K is self-adjoint and we obtain L K = L K .
Due to [2,Remark 4.3] we have that Similarly one can show that the maximal accumulative extension L K is real if and only if it is self-adjoint and K = K T .
All functions from Dom(L max ), together with their first quasi-derivatives, belong to ⊕ m k=1 AC ([a k−1 , a k ], C). This implies that the following definition is correct.
Denote by [f ] t− the left germ and by [f ] t+ the right germ of the continuous function f at a point t. Similarly to the paper [2] we say that boundary conditions that define the operator L ⊂ L max are called local if, for any functions y ∈ Dom(L) and for any y 1 , . . . , y m ∈ Dom(L max ), the identities imply that y j ∈ Dom(L). For j = 0 and j = m we consider only the right and left germs, respectively.
The following statement gives a description of the extensions L K and L K which are given by local boundary conditions. Theorem 5. The boundary conditions (5) and (6) defining the extensions L K and L K , respectively, are local if and only if the matrix K has the block form where K a1 and K an ∈ C and other K aj ∈ C 2×2 .
Proof. Let us consider the extension L K defined by the boundary conditions (5).
Let us return to boundary conditions (5) and rewrite them in a parametric form. For any vector F = (F 1 , . . . , F 2m ) ∈ C 2m , we consider the vectors −i(K + I)F and (K − I)F . Due to the definition of a boundary triplet the is a function y F ∈ Dom(L max ) such that −i (K + I) F = Γ 1 y F , One easily sees that y F satisfies boundary conditions (5) and therefore y F ∈ Dom(L K ). We rewrite the last system in the form [1] p y F (a p −), r = 2p + 1, p = 0, . . . , n, p+1 y F (a p +), r = 2p, p = 1, . . . , n, which results in −D [1] p y F (a p −) + iy F (a p −) = −2iF 2p and D [1] p+1 y F (a p +) + iy F (a p +) = −2iF 2p+1 for arbitrary numbers F j ∈ C.

Generalized resolvents
Let us recall that a generalized resolvent of a closed symmetric operator L on a Hilbert space H is an operator-valued function λ → R λ , defined on C\R which can be represented as R λ f = P + L + − λI + −1 f, f ∈ H, where L + is a self-adjoint extension of the operator L that acts on a certain Hilbert space H + ⊃ H, I + is the identity operator on H + , and P + is the orthogonal projection operator from H + onto H. It is known that an operator-valued function R λ is a generalized resolvent of a symmetric operator L if and only if it can be represented as where F µ is a generalized spectral function of the operator L. This implies that the operator-valued function F µ has the following properties.
(3) For any x ∈ H the following identities hold: The following theorem provides a parametric description of all generalized resolvents of the symmetric operator L min (see also [15]). Theorem 6. 1) Every generalized resolvent R λ of the operator L min in the half-plane Imλ < 0 acts by the rule R λ h = y, where y is a solution of the boundary-value problem l(y) = λy + h, (K(λ) − I) Γ 1 f + i (K(λ) + I) Γ 2 f = 0.
2) In the half-plane Imλ > 0 every generalized resolvent of the operator L min acts by R λ h = y, where y is a solution of the boundary-value problem l(y) = λy + h, Here h(x) ∈ L 2 ([a, b], C) and K(λ) is a 2m × 2m matrix-valued function which is holomorphic in the upper half-plane and satisfies ||K(λ)|| ≤ 1.
This parametrization of the generalized resolvents by the matrix-valued functions K(λ) is bijective.
Proof. The Theorem is just an application of Theorem 2 and [15, Theorem 1 and Remark 1] which proves the description of generalized resolvents in terms of boundary triplets. Namely, one requires to take as an auxiliary Hilbert space C m and as the operator γy := {Γ 1 y, Γ 2 y}.