On a class of integral systems

We study spectral problems for two--dimensional integral system with two given non-decreasing functions $R_1$, $R_2$ on an interval $[0,b)$ which is a generalization of the Krein string. Associated to this system are the maximal linear relation $T_{\max}$ and the minimal linear relation $T_{\min}$ in the space $L^2(R_2)$ which are connected by $T_{\max}=T_{\min}^*$. It is shown that the limit point condition at $b$ for this system is equivalent to the strong limit point condition for the linear relation $T_{\max}$. In the limit circle case the strong limit point condition fails to hold on $T_{\max}$ but it is still satisfied on a subspace $T_N^*$ of $T_{\max}$ characterized by the Neumann boundary condition at $b$. The notion of the principal Titchmarsh-Weyl coefficient of this integral system is introduced both in the limit point case and in the limit circle case. Boundary triples for the linear relation $T_{\max}$ in the limit point case (and for $T_{N}^*$ in the limit circle case) are constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh-Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of $R_1$ and $R_2$. It is shown that the principal Titchmarsh-Weyl coefficients $q$ and $\widehat q$ of the direct and the dual integral systems are related by the equality $\lambda \widehat q(\lambda) = -1/q(\lambda)$ both in the regular and the singular case.


Introduction
In this paper spectral problems for integral systems, associated dual systems and, in particular, Krein strings are investigated. We consider an integral system of the form where u = [u 1 u 2 ] T , with some spectral parameter λ ∈ C and measures dR 1 and dR 2 associated with non-decreasing functions R 1 (x) and R 2 (x) on an interval [0, b), see [5]. If R 1 (x) ≡ x then u 2 = u ′ Integral systems (1.1) arise in the theory of diffusion processes with two measures [29,26]. In the theory of stochastic processes the equation (1.2) describes generalized diffusion processes which includes both diffusion processes and birth and death processes [13,14,16,23]. In mechanics the equation (1.2) describes small transverse oscillations of the string with the mass distribution function R 2 (x), [19]. Relations for the Titchmarsh-Weyl coefficients of Krein strings and their associated dual strings were studied in [19,22]. Let c(·, λ) and s(·, λ) be the unique solutions of (1.2) satisfying the initial conditions c(0, λ) = 1, c ′ (0, λ) = 0, and s(0, λ) = 0, s ′ (0, λ) = 1. is called the principal Titchmarsh-Weyl coefficient of the string [22] or the dynamic compliance coefficient in the terminology of I. S. Kac and M. G. Krein [19] and describes the spectral properties of the string. The principal Titchmarsh-Weyl coefficient q(λ) is a Stieltjes function and the measure dσ from its integral representation is the spectral measure of the string S 1 [b, R 2 ], which in the limit point case is given by the boundary condition u ′ (0) = 0. Denote the integral system (1.1) by S[R 1 , R 2 ]. In the present paper we define the principal Titchmarsh-Weyl coefficient q of the integral system S[R 1 , R 2 ] by q(λ) := lim x→b s 1 (x, λ) c 1 (x, λ) , (1.6) where c 1 (·, λ), c 2 (·, λ) and s 1 (·, λ), s 2 (·, λ) are pairs of the unique solutions of (1.1) satisfying the initial conditions c 1 (0, λ) = 1, c 2 (0, λ) = 0, and s 1 (0, λ) = 0, s 2 (0, λ) = 1. (1.7) Formula (1.6) requires justification. For this purpose we use the operator approach to the integral system S[R 1 , R 2 ] developed in [31], the boundary triples technique from [24,15] and the theory of associated Weyl functions as introduced in [7,8]. The maximal linear relation T max is defined as the set of pairs u = [u 1 f ] T such that u 1 , f ∈ L 2 (R 2 ) and the equation (2.19) is satisfied for some u 2 ∈ BV loc [0, b), see Definition 2.7. The closure of its restriction to the set of compactly supported functions is called the minimal linear relation T min . In [31] it is shown that T min is symmetric in L 2 (R 2 ), T max = T * min and boundary triples for the linear relation T min were constructed both in the limit point and in the limit circle case.
In Theorem 4.3 we show that the system S[R 1 , R 2 ] is in the limit point case at b if and only if it satisfies the strong limit point condition at b, see [11], which in our case is of the form lim x→b u 1 (x)u 2 (x) = 0 for all u ∈ T max . (1.8) As a consequence of (1.8) we conclude that in the limit point case the linear relation T min and its von Neumann extension A N , characterized by the boundary condition u 2 (0) = 0, are nonnegative, the corresponding Weyl function is a Stieltjes function and coincides with the principal Titchmarsh-Weyl coefficient of the system S[R 1 , R 2 ]. The strong limit point condition for second order differential operators was introduced by W. Everitt [11].
In the limit circle case the linear relation T min has defect numbers (2,2), in this case an intermediate symmetric extension T N with defect numbers (1, 1) of T min is considered as the restriction of T max to the set of elements u ∈ T max such that u 1 (0) = u 2 (0) = u 2 (b) = 0. In this case we show in Lemma 3.3 that the strong limit point condition (1.8) fails to hold, but still the limit in (1.8) is vanishing on the subspace T * N of T max , i.e. the following Evans-Everitt condition holds, cf. [12]: lim x→b u 1 (x)u 2 (x) = 0 for all u ∈ T * N . (1.9) This result implies the nonnegativity of the linear relation T N and its selfadjoint extension In [25] another analytical object -the Neumann m-function of the system S[R 1 , R 2 ] was introduced by the equality m N (λ) := lim x→b s 2 (x, λ) c 2 (x, λ) , (1.10) which is a special case of a more general definition of the Neumann m-function presented in [5]. In Proposition 3.6 it is shown that the Neumann m-function m N (λ) is a Stieltjes function and it coincides with the principal Titchmarsh-Weyl coefficient of In the regular case we construct the canonical singular extension S[ R 1 , R 2 ] of the system S[R 1 , R 2 ] with R 1 , R 2 extended to non-decreasing functions R 1 , R 2 on the interval (0, ∞), so that the principal Titchmarsh-Weyl coefficients of both systems coincide.
The dual system S[R 1 , R 2 ] of the integral system S[R 1 , R 2 ] in the singular case is obtained by changing the roles of R 1 and R 2 . In the regular case the dual system of the integral system S[R 1 , R 2 ] is defined as the dual of the canonical singular extension S[ R 1 , R 2 ] of the system S[R 1 , R 2 ]. The main result of the paper is Theorem 5.2 where it is shown that the principal Titchmarsh-Weyl coefficient q of the dual system is related to the principal Titchmarsh-Weyl coefficient q of the system S[R 1 , R 2 ] by the equality . (1.11) both in the regular and the singular case.
In the case of a string (R 1 (x) = x) the notion of the dual string and the formula (1.11) connecting the principal Titchmarsh-Weyl coefficients of the direct and the dual string in the singular case was presented in [17]. In [22] some further relations between strings, dual strings and canonical systems of differential equations were studied. Analogues of these relations between integral systems and canonical systems can also be established and will be presented in a forthcoming paper.

Preliminaries
2.1. Linear relations. Let H be a Hilbert space. A linear relation T in H is a linear subspace of H × H. Let us recall some basic definitions and properties associated with linear relations in [1,4].
The domain, the range, the kernel, and the multivalued part of a linear relation T are defined as follows: (2. 2) The adjoint linear relation T * is defined by The set of all closed linear operators (relations) is denoted by C(H) ( C(H)). Identifying a linear operator T ∈ C(H) with its graph one can consider C(H) as a part of C(H).
Let T be a closed linear relation, λ ∈ C, then A linear relation T is called symmetric if T ⊆ T * . A point λ ∈ C is called a point of regular type (and is written as λ ∈ ρ(T )) for a closed symmetric linear relation T , if λ / ∈ σ p (T ) and the subspace ran(T − λI) is closed in H. For λ ∈ ρ(T ) let us set N λ (T * ) := ker(T * − λI) and The deficiency indices of a symmetric linear relation T are defined as n ± (T ) := dim ker(T * ∓ iI). (2.7) 2.2. Boundary triples and Weyl functions. Let T be a symmetric linear relation with deficiency indices (1, 1). In the case of a densely defined operator the notion of the boundary triple was introduced in [24,15]. Following the papers [28,8] we shall give a definition of a boundary triple for the linear relation T * .
where Γ 0 and Γ 1 are linear mappings from T * to C, is called a boundary triple for the linear relation T * , if: Notice, that in contrast to [28] the linear relation T is not supposed to be single-valued. The following linear relations A 0 := ker Γ 0 , A 1 := ker Γ 1 (2.9) are selfadjoint extensions of the symmetric linear relation T . 7,8]). Let Π = (C, Γ 0 , Γ 1 ) be a boundary triple for the linear relation T * . The scalar function m(·) and the vector valued function γ(·) defined by are called the Weyl function and the γ-field of the symmetric linear relation T corresponding to the boundary triple Π.
The Weyl function and the γ-field are connected via the next identity (see [8]) It follows from (2.11) that the Weyl function m(·) is a Herglotz-Nevanlinna function. A Herglotz-Nevanlinna function m which admits a holomorphic continuation to R − and takes nonnegative values for all λ ∈ R − is called a Stieltjes function. Every Stieltjes function m admits an integral representation (1.5) with a non-decreasing function σ(t) such that R+ (1 + t) −1 dσ(t) < ∞.

2.3.
Minimal and maximal relations associated with the integral system S[R 1 , R 2 ]. Let I = [0, b) be an interval with b ≤ ∞, let R(x) be a non-decreasing left-continuous function on I such that R(0) = 0, let dR be the corresponding Lebesgue-Stieltjes measure, and let L 2 (R, I) be an inner product space which consists of complex valued functions f such that with inner product defined by From now on the following convention is used for the integration limits for any measure dσ on an interval: (2.14) Thus, an integral as a function of its upper limit is always left-continuous. With every function of bounded variation f we associate the left-continuous and the right-continuous functions f − and f + defined by Let u and v be left-continuous functions of bounded variation, du and dv be the corresponding Lebesgue-Stieltjes measures. The following integration-by-parts formula for the Lebesgue-Stieltjes integral (see e.g. [32]) is used throughout the paper (2.16) If u and u + have no zeros then it follows with v = 1/u from (2.16) This leads to the quotient-rule formula (2.17) The following existence and uniqueness theorem for integral systems was proven in [5, Theorem 1.1]. Theorem 2.4. Let dS be a complex n×n matrix-valued measure. For every left continuous (either n × n or n × 1 matrix valued) function A(x) in BV loc [0, b) there is a unique function U such that the equality holds for every point x ∈ [0, b).
Remark 2.5. Due to the properties of the Lebesgue-Stieltjes integral and the used convention, any solution U to (2.18) is left continuous and belongs to BV loc [0, b), componentwise.
Now we focus on integral systems S[R 1 , R 2 ] of the form (1.1), where R 1 (x) and R 2 (x) are nondecreasing and left-continuous real-valued functions on the interval I = [0, b) such that R 1 (0) = R 2 (0) = 0. We define the corresponding inhomogeneous system. Definition 2.6. Let f ∈ L 2 (R 2 ) and [u 1 u 2 ] T be a vector-valued function such that the following equation Due to Remark 2.5 for every (u 1 , u 2 , f ) ∈ T both functions u 1 and u 2 belong to BV loc [0, b). Theorem 2.4 implies that for every f ∈ L 2 (R 2 ) the vector-valued function [u 1 u 2 ] T satisfying (2.19) is uniquely determined by its initial values at zero, however u 1 ∈ L 2 (R 2 ) is not guaranteed for an arbitrary f . Definition 2.7. We define the maximal and the pre-minimal relations T max , T ′ ⊂ L 2 (R 2 )×L 2 (R 2 ) by (2.20) Everywhere in the paper, except Remark 3.10, we suppose that the following two natural assumptions hold.  where π : is the corresponding quotient map.
Assumption 2.8 has the important consequence that the first component of a solution has no discontinuity in common with the second component of any solution (u 1 , u 2 , f ) ∈ T . Assumption 2.9 makes it possible to assign correctly the values u 1 (x) and u 2 (x) for every u ∈ T max . In case of absolutely continuous functions R 1 and R 2 the equivalent to S[R 1 , R 2 ] differential system is definite in the sense of [27,Definition 2.14] if and only if Assumption 2.9 holds. Definition 2.10. Let (u 1 , u 2 , f ) ∈ T and u ∈ T max be its image under the mapping The following Proposition provides an analog of [27, Proposition 2.15] for integral system Proposition 2.11. If Assumptions 2.8 and 2.9 hold then the mappings φ 1,2 [x] are well-defined.
Proof. In general, the mapping defined by (2.23) is not invertible. Suppose that (u 1 , u 2 , f ) and ( u 1 , u 2 , f ) are two pre-images of u = πu 1 πf ∈ T max as it is shown on the following diagram.
Let us show that due to Assumption 2.9 The mapping π to applied the first line of (2.26) gives Further in the text we will simply write u 1,2 (x) instead of φ 1,2 [x]u unless this can lead to confusion. For a pair of vector-valued functions u = u 1 u 2 belong to T then the following generalized first and second Green's identities hold Proof. We recall that due to Assumption 2.8 the functions R 1 and R 2 do not have common points of discontinuity, so neither do the functions v 1 and u 2 . By virtue of (2.19) we get and hence, using the integration-by-parts formula (2.16), Integrating (2.32) over [0, x) provides (2.29). Swapping the tuples (u 1 , u 2 , f ) and (v 1 , v 2 , g) in (2.32) and subtracting the obtained expression from (2.32) proves (2.30).
Theorem 2.4 provides that system S[R 1 , R 2 ] has a unique solution for every choice of initial values. Let c(·, λ) = [c 1 (·, λ) c 2 (·, λ)] T and s(·, λ) = [s 1 (·, λ) s 2 (·, λ)] T be its unique solutions satisfying the initial conditions (1.7). Corollary 2.13. For every λ ∈ C and x ∈ [0, b) the following formulas hold: Proof. Equality (2.33) follows immediately from either (3.2) or (2.30). Further we subtract the left-hand side of (2.33) from the left-hand side of (2.34): One can immediately see that the expression (2.36) is equal to zero at every point of continuity of R 1 . Let x 0 be a point of discontinuity of R 1 . From (2.19) one can see that The proof of (2.35) is similar.
It follows from (2.30) that the pre-minimal relation T ′ is symmetric in L 2 (R 2 ).
Definition 2.14. The minimal relation T min is defined as the closure of the pre-minimal linear relation T ′ : T min = clos T ′ .
As was shown in [31] the linear relation T min is also symmetric, T * min = T max and The coefficient m is well-defined and can be calculated as and its radius can be calculated as Proof. (i) From (2.41) and the condition ψ 1 (l, λ) + hψ 2 (l, λ) = 0 we get which results as (2.42).
(ii) It is clear from formula (2.42) that the function m(λ, l, ·) maps R + ∪ {∞} into a circle. Let h ∈ clos C + ∪ {∞} and ω ∈ D l (λ). Applying the second Green's identity (2.30) to the tuples and therefore D l2 ⊆ D l1 . Assume now ω ∈ ∩ l<b D l (λ). Passing to the limit as l → b in (2.43), one gets Assume that the point b is singular for the system (1.1). Then the following alternative holds: (i) either the discs D l (λ) shrink to a limit point as l → b and then dim N λ (T max ) = 1; (ii) or the discs D l (λ) converge to a limit disc as l → b and then dim N λ (T max ) = 2.
Remark 2.17. A matrix version of integral equation equivalent to the integral system S[R 1 , R 2 ] with R 1 (x) ≡ x and R 2 (x) continuous was considered in [2]. Such equation can be reduced to a canonical differential system, see [2,Section 2.2]. Condition of definiteness of general matrix canonical differential system was found in [27]. In the scalar case this condition coincides with Assumption 2.9.

3.
To show (iii), expanding c 1 (x, λ) and c 2 (x, λ) in series in λ one obtains from (1.1) that This implies that ϕ n (x) ≥ 0, ψ n (x) ≥ 0 for n ∈ N and hence Moreover, it follows from (3.10) that Therefore, the relations (3.4) hold since The proof of (3.5) is similar. 4. The identity (2.33) yields and hence and hence the function c2(x,λ) s2(x,λ) is increasing on [0, b). This proves (iv). Notice, that the function 3.2. The Evans-Everitt condition in the limit circle case.
Proposition 3.2. The system S[R 1 , R 2 ] is limit circle at b if and only if 1, R 1 ∈ L 2 (R 2 ).
If the system S[R 1 , R 2 ] is regular at b, then the following limits exist: Assume now that the system S[R 1 , R 2 ] is in the limit circle at b. One can check (see [19,Section 10.7], [30,Theorem 3.8]) that for every element u = πu 1 πf ∈ T max the limit exists and is well defined. Therefore, the limits (3.17) exist.
Lemma 3.3. Let the system S[R 1 , R 2 ] be limit circle at b. Then for every u = πu 1 πf ∈ T * N one has u 2 ∈ L 2 (R 1 ) and the following two equalities hold: Proof. Let u = πu 1 πf ∈ T * N . Applying the integration-by-parts formula (2.16) to the first line of (2.19) one gets (3.21) We recall that in the limit circle case 1, R 1 ∈ L 2 (R 2 ) and f ∈ L 2 (R 2 ) by the assumption of the lemma. The condition u 2 (b) = 0 provides u 2 (x) = b x f dR 2 and hence (3.21) can be rewritten as Note the following estimation: The claim u 2 ∈ L 2 (R 1 ) for u = πu 1 πf ∈ T * N follows from (3.19) and the first Green's iden- Remark 3.4. The condition (3.20) for Sturm-Liouville operators in the limit circle case was introduced and studied by Evans and Everitt in [12]. We will call it the Evans-Everitt condition.
3.3. Boundary triples for integral systems in the limit circle case.
Definition 3.5 (see [5,25]). The function m(λ, b, ∞) from (2.41) for which the solution Proposition 3.6. Let the system S[R 1 , R 2 ] be singular and limit circle at b, let T N be defined by (3.28), and let m(λ, b, ∞) be the Neumann m-function of the system S[R 1 , R 2 ] given by (3.27). Then: (i) T N is a symmetric nonnegative linear relation in L 2 (R 2 ) with deficiency indices (1, 1).
where m is a function from S such that lim y→0 y m(iy) = 0.
Proof. 1. To show (i) and (ii), let the tuples (u 1 , u 2 , f ) and (v 1 , v 2 , g) satisfy the system (2.19) and assume that u 2 (b) = v 2 (b) = 0, i.e. u, v ∈ T * N . Let µ ∈ R. By formula (2.33) at least one of the values c 2 (b, µ) and s 2 (b, µ) is not equal to 0. Assume that c 1 (b, µ) = 0 and let us set c(x) := c(x, µ). Due to the identity    ∩ (a, b)). Then by Assumption 2.9 a 1 < b and due to (1.1) and Lemma 3.1 (iii) Now we must consider two cases. In case if c 1 (·, λ) has a jump at point a 1 , which is only possible if a 1 > a, we get 1 and hence by the Lebesgue bounded convergence theorem one obtains from (3.14) The last equality in (3.38) follows from a 1 > a and the definition of the points a, a 1 . In case if c 1 (·, λ) has no jump at point the a 1 , which is possible either if a 1 = a or a 1 > a and R 1 has no jump at a 1 , we get 1 and similarly to (3.38) Since R 1 (b) + R 2 (b) = +∞ it follows from (3.12) that lim x→b c 1 (x, λ)c 2 (x, λ) = +∞ for all λ ∈ R − and hence it follows from (3.13) that Since q and m N are holomorphic on C \ R + this proves that q(λ) ≡ m N (λ), and (iv) is shown. 4. Now we prove (v). It follows from (1.1) that and by (3.27) that , λ ∈ C \ R.

3.4.
Integral systems in the regular case. Assume that the system S[ Then for every tuple (u 1 , u 2 , f ) ∈ T it follows from (3.18) that the function u 2 is bounded and hence the limit exists and well defined. Therefore, the limits (3.16) exist.
As follows from (2.30) the adjoint linear relation T * D is of the form Proposition 3.8 (cf. [30]). Let the system S[R 1 , R 2 ] be regular at b, and let T D be defined by (3.50). Then: (i) T D is a symmetric nonnegative linear relation in L 2 (R 2 ) with deficiency indices (1, 1) and is a boundary triple for T * D .
Remark 3.9. The functions R 1 and R 2 are not uniquely defined by the principal Titchmarsh-Weyl coefficient of the system S[R 1 , R 2 ]. As was shown in [25,Lemma 2.12] if functions R 1 (ξ) and R 2 (ξ) are connected by where x(ξ) is an increasing function on the interval [0, β], such that x(0) = 0 and x(β) = b, then the principal Titchmarsh-Weyl coefficient q of the system coincides with the principal Titchmarsh-Weyl coefficient q of the system S[R 1 , R 2 ]. Therefore we can always assume that for regular systems S[R 1 , R 2 ] the parameter x ranges over a finite interval [0, b], b < ∞. Remark 3.10. As is known, see [19, Section A13], a truncated moment problem can be reduced to a regular integral system S[R 1 , R 2 ] with where H(x) is the Heaviside function. The corresponding monodromy matrix U (x n , λ) is of the form The system S[R 1 , R 2 ] satisfies Assumption 2.9 if n > 1. If n = 1 then and does not satisfy the Assumption 2.9. However, in this case one can still introduce a boundary triple (C, Γ 0 , Γ 1 ) for T max = C × C by and the corresponding Weyl function is m(λ) = m 0 λ. 2] is equivalent to the system S[R 1 , R 2 ] in the sense that its Weyl function corresponding to the boundary triple (3.58) coincides with m(λ) = m 0 λ and the monodromy matrix U (2, λ) of this system coincides with U (l 1 , λ). The advantage of system S[ R 1 , R 2 ] is that the elementary factors of U (2, λ) from its factorization can be also treated as monodromy matrices of systems S[0, R 2 ] on the interval [0, 1] and S[ R 1 , 0] on [1,2], respectively. 4. Integral systems in the limit point case 4.1. The strong limit point condition. The next lemma is an analog of one result in [11,Lemma] in the case of integral systems.  Proof. We will prove the lemma in the case f > 0, f → 0. The proof in the other cases is similar. Let D f be the set of the points of discontinuity of f on [b 0 , b). One can write Notice that both the integrals on the right hand side of (4.2) are negative, therefore if one of them diverges (as x → b) then the assertion of the lemma holds.
Consider the following inequality and the associated series If series (4.4) diverges then the following integral diverges as well, so the assertion of the lemma holds immediately. Assume now that series (4.4) converges and denote a n := 1 − f + (x n )/f − (x n ). Notice that the measure d log(f ) is absolutely continuous with respect to df and therefore there exists the Radon-Nikodym derivative d log(f )/df ∈ L 1 (df ) which has a representative (see [6, 5.3 Now we get by the Radon-Nikodym theorem and hence One can see from the following inequality converges provided the series ∞ 1 a n converges. Hence we obtain that the integral on the left hand side of (4.8) diverges which completes the proof. where d ∈ R if the Dirichlet property holds and d = +∞ otherwise. Let us start with the implication (LP) ⇒ (D). For this purpose we assume the contrary i.e. the system S[R 1 , R 2 ] is in the limit point case but d = +∞. Notice, that the functions R 1 and R 2 do not have common points of discontinuity, therefore neither do the functions u 1 and u 2 . It implies that both u 1 and u 2 preserve their signs on some interval [b 0 , b) (otherwise they would have to share a jump from a positive to a negative value or vice versa), so the function u 1 is either positive and increasing or negative and decreasing. If 1 / ∈ L 2 (R 2 ) then it immediately results as u 1 / ∈ L 2 (R 2 ). In the case if 1 ∈ L 2 (R 2 ) (and hence R 1 / ∈ L 2 (R 2 )) the implication f ∈ L 2 (R 2 ) ⇒ f ∈ L 1 (R 2 ) is valid and hence (see (3.18)) there exists a finite limit u 2 (b) := lim x→b u 2 (x). The limit u 2 (b) must be zero, otherwise from one gets u 1 / ∈ L 2 (R 2 ). One can see that 1/u 2 / ∈ L 2 (R 2 ). Indeed, if 1/u 2 ∈ L 2 (R 2 ) then the integral converges as x → b, which contradicts to Lemma 4.1. Since d = +∞, the estimate |u 1 | > 1/|u 2 | hold on some interval [b 0 , b) and provides again u 1 / ∈ L 2 (R 2 ). This completes the proof of the implication (LP) ⇒ (D). Now let us prove the implication (D) ⇒ (SLP*). We first will show that d = 0. In the case 1 ∈ L 2 (R 2 ) the reasoning of the previous paragraph can be used to show that u 1 / ∈ L 2 (R 2 ) for every non-zero d. In the case 1 / ∈ L 2 (R 2 ) the reasoning above shows again that u 1 / ∈ L 2 (R 2 ) for every d > 0. Therefore we assume d < 0 and get that u 1 is either positive and decreasing or negative and increasing on some interval The left hand side converges by our assumption but the right hand side diverges due to Lemma 4.1. This contradiction proves that d = 0. Thus, implication (D) ⇒ (SLP) is valid.
As is known (see [31,Theorem 4.3]), the system S[R 1 , R 2 ] is in the limit point case if and only if for every (u 1 , u 2 , f ) and In order to prove the implication (SLP*) ⇒ (SLP) we notice first that by Lemma 3.3 the system S[R 1 , R 2 ] cannot be in the limit circle case since (4.13) holds for every (u 1 , u 2 , f ) ∈ T . The condition (4.11) follows from (4.13), (4.18) and the following equality (cf. [11]) Assume that the statement (SLP) holds, i.e. condition (4.11) is satisfied for every (u 1 , u 2 , f ) and (v 1 , v 2 , g) from T . Then, clearly, (4.18) holds for every (u 1 , u 2 , f ) and (v 1 , v 2 , g) from T and hence the system S[R 1 , R 2 ] is in the limit point case. This proves the implication (SLP) ⇒ (LP).
Remark 4.4. In the case of absolutely continuous R 1 and R 2 the implication (LP ) ⇒ (SLP ) for the system S[R 1 , R 2 ] was proved in [20], see also [11].

4.2.
Boundary triples for integral systems in the limit point case.
Definition 4.5. Let the system S[R 1 , R 2 ] be in the limit point case at b. Then for each λ ∈ C \ R there is a unique coefficient m N (λ), such that (4.20) The function m N is called the Neumann m-function of the system (1.1) on I and the function ψ(t, λ) is called the Weyl solution of the system S[R 1 , R 2 ] on I.
Let us collect some statements concerning boundary triples for S * , which were partially formulated in [30,31]. is a boundary triple for T * .
(iii) The defect subspace N λ (T ) is spanned by the Weyl solution ψ 1 (t, λ), and the Weyl function m(λ) of T corresponding to the boundary triple Π coincides with the Neumann m-function of the system S[R 1 , R 2 ] on I: (iv) The Weyl function m(λ) of T corresponding to the boundary triple Π coincides with the principal Titchmarsh-Weyl coefficient q(λ) of the system S[R 1 , R 2 ] on I and belongs to the Stieltjes class S.
where m is a function from S such that lim y↓0 y m(iy) = 0.
Proof. 1. At first we show (i) − (ii). Since (1.1) is in the limit point case at b, and hence the generalized Green's identity (2.30) is of the Therefore, the triple Π in (4.21) is a boundary triple for T * . It follows from the first Green's identity (2.29) and Lemma 3.3 that for every u ∈ T the identity (3.35) holds and thus the linear relation T is nonnegative.

4.3.
The canonical singular continuation of a regular integral system. If the integral system S[R 1 , R 2 ] is regular at b then due to Remark 3.9 we can assume without loss of generality that b < ∞.
Definition 4.7. For a regular system S[R 1 , R 2 ] with b < ∞ we define the extended functions (4.29) The integral system S[ R 1 , R 2 ] corresponding to will be called the canonical singular continuation of a regular integral system S[R 1 , R 2 ].
Proof. Let the pair u 1 , u 2 satisfy the integral system S[R 1 , R 2 ] for some λ ∈ C \ R and let u 1 , u 2 be the continuations of u 1 , u 2 to the interval [0, +∞) given by

λmN (λ)
is the Neumann m-function of the system S[R 1 , R 2 ].
Remark 5.4. Formula (5.4) was proven in [21] for Krein strings and in [25] for integral systems. However, in [25] it was overlooked that in the regular case the formula (5.4) fails to hold and should be replaced by (5.5).