Compressed Resolvents and Reduction of Spectral Problems on Star Graphs

In this paper a two-step reduction method for spectral problems on a star graph with n + 1 edges e 0 , e 1 , . . . , e n and a self-adjoint matching condition at the central vertex v is established. The ﬁrst step is a reduction to the problem on the single edge e 0 but with an energy depending boundary condition at v . In the second step, by means of an abstract inverse result for Q -functions, a reduction to a problem on a path graph with two edges e 0 , (cid:2) e 1 joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realizations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer e.g. the following question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type?


Introduction
Direct and inverse spectral problems for differential operators on graphs have attracted rapidly increasing interest over the last two decades. Since it seems impossible to list all relevant literature, we mention the three monographs [5,26,33], whose bibliographies reflect the intense research activities in this area very well. The vast majority of these works considers concrete operators on graphs such as Sturm-Liouville and Schrödinger operators [2,3,16,25,37], Laplace and related operators [29], Krein strings [13], Stieltjes strings [30,31], canonical systems and Dirac operators [1,8]. More abstract methods were developed in [32,35].
The results of this paper are of more conceptual character; they have implications to many differential and difference operators, and corresponding Cauchy problems. We establish abstract results for compressed resolvents of self-adjoint extensions of an orthogonal sum of symmetric relations. They allow us to reduce spectral problems on a star graph with self-adjoint matching condition at the central vertex to problems on a single edge and, using inverse results, to problems on a path graph with two edges, see Fig. 1a-c. We also address the question whether for concrete realizations the reduced system on the path graph can be chosen of the same type as the original system.
More precisely, the subsystem on the n edges e 1 , e 2 , . . . , e n (the blue/ non-bold part in Fig. 1a) joined to e 0 (the red/bold part in Fig. 1a) with arbitrary self-adjoint matching condition at the central vertex v is first condensed into an energy depending boundary condition at the endpoint v of the edge e 0 (the blue/non-bold part in Fig. 1b). In the second step the energy depending boundary condition is replaced by a new subsystem on an edge e 1 joined to e 0 by continuity and Kirchhoff conditions (the blue/non-bold part in Fig. 1c). To exemplify our results, we consider a star graph G as in Fig. 1a and we suppose that for each edge e j a symmetric linear relation T j with equal defect numbers 1 in a Hilbert space H j and a boundary triplet (C, j1 , j2 ), j = 0, 1, . . . , n. are given. For concrete realizations, e.g. differential or difference operators as considered below, the boundary triplets depend on the values and derivatives of the functions at the central vertex v. The Hilbert space H on the graph G is the orthogonal sum of the Hilbert spaces H j on the edges e j , j = 0, 1, . . . , n, and in H the symmetric relation T is defined as the orthogonal sum of the relations T j . The self-adjoint extensions of T in H are all described by an interface condition parametrized by two (n + 1) × (n + 1)matrices A, B (see (2.12), (2.13) below); we denote them by T A,B and we fix such an extension.
The key step in our analysis is to determine the compressed resolvent of T A,B corresponding to the distinguished edge e 0 . This means, given an element f 0 ∈ H 0 we describe the first component g 0 ∈ H 0 of the resolvent (T A,B − z) −1 applied to the element f 0 = ( f 0 , 0, . . . , 0) t supported only on H 0 , where P 0 is the projection of H onto the first component H 0 . Note that the compressed resolvent of T A,B (to H 0 ) on the right hand side of (1.1) is also a generalized resolvent of the symmetric relation T 0 in H 0 . Our first main result, Theorem 3.1, shows that g 0 is a solution of the inhomogeneous problem for T * 0 in H 0 with energy depending boundary condition which, if T * 0 is an operator, has the form (T * 0 − z)g 0 = f 0 , 01 g 0 − n 0 (z) 02 g 0 = 0; (1.2) here the function n 0 depends on the matrices A, B and on the Titchmarsh-Weyl functions m j of the other T j , j = 1, 2, . . . , n. In other words, the element g 0 in (1.1) defined by the resolvent of T A,B in the Hilbert space H = H 0 ⊕ H 1 ⊕ · · · ⊕ H n is a solution of a problem in the first component H 0 , or in the graph situation on the edge e 0 , but with a z-depending boundary condition at v, see Fig. 1b. The essential property of the function n 0 in (1.2) is that it is a Nevanlinna function, i.e. holomorphic at least on C + ∪ C − with n 0 (z) = n 0 (z) and Im n 0 (z) ≥ 0 if Im z > 0. According to a general abstract inverse result of Langer and Textorius [28] (see also [11]), each Nevanlinna function is the Q-function, or Titchmarsh-Weyl function, of a symmetric relation with defect numbers 1 (and a self-adjoint extension T 1 of it) in some Hilbert space H 1 . If we apply this result to the function n 0 in (1.2), it follows that the compressed resolvent of T A,B in (1.1) coincides with the compressed resolvent of a self-adjoint relation T A 0 ,B 0 extending T 0 ⊕ T 1 and acting in the Hilbert space here A 0 , B 0 is the matrix pair defining standard matching conditions, see Theorem 4.1.
In the graph situation this means that, for given f 0 ∈ H 0 , on the common edge e 0 the 'values' of (T A,B − z) −1 ( f 0 0 · · · 0) t for the original star graph G and of ( T A 0 ,B 0 − z) −1 ( f 0 0) t for the simpler path graph G coincide, see Theorem 4.4. This result also has implications for first and second order Cauchy problems on star graphs (see e.g. [4,23]). Consider, for example, the Cauchy probleṁ  [38,Thm. 1]) yields that every (scalar) Nevanlinna function is the Titchmarsh-Weyl function of a canonical system. Hence T 1 can even be chosen to be generated by a trace normed canonical system on the interval [0, ∞) = e 1 . Thus, in the graph situation, the compressed resolvent of T A,B on the star graph, see (1.1), coincides with the compressed resolvent of an operator T A 0 ,B 0 on a path graph with two edges e 0 , e 1 , given by T 0 on e 0 and a canonical system on e 1 joined by continuity and Kirchhoff conditions at v, see Theorem 4.6. If, in particular, T 0 is given by a canonical system while all other relations T j , j = 1, 2, . . . , n, are arbitrary, then T A 0 ,B 0 is given by two canonical systems on the path graph G, see Corollary 5.1.
According to an inverse result of Krein [18] (see also [12,Sect. 5]), a Nevanlinna function that belongs to the subclass S of Stieltjes functions, i.e. is also holomorphic on (− ∞, 0) and non-negative there, is the Titchmarsh-Weyl function of a Krein string. Hence, if the function n 0 in (1.2) is a Stieltjes function, the relation T 1 can be chosen to be generated by a Krein string. In the case where the relations T j , j = 1, 2, . . . , n, are generated by Krein strings joined to T 0 by continuity and Robin type interface conditions at v, a sufficient condition for n 0 ∈ S is that the Robin parameter τ satisfies the inequality While the mass distribution function of a Krein string is an arbitrary non-decreasing (left-continuous) function, for a Stieltjes string it is a step function with steps accumulating at most at the right end-point. For a star graph of Stieltjes strings as in Fig. 1a this means that the steps on the edges e j accumulate at most at the outer vertices v j , j = 0, 1, . . . , n. Employing asymptotic expansions of Titchmarsh-Weyl functions, it follows that if, in the above situation, inequality (1.5) holds and all T j , j = 0, 1, . . . , n, are Stieltjes strings, the reduced system T A 0 ,B 0 consists of two Stieltjes strings, see Corollary 5.4. If all T j , j = 0, 1, . . . , n, are Sturm-Liouville operators, Theorem 3.1 yields a reduction to the single Sturm-Liouville operator T 0 on e 0 with an energy depending boundary condition at v. Moreover, Theorem 4.6 based on de Branges' inverse result applies and yields a reduction to a problem on a path graph of two edges for the Sturm-Liouville operator T 0 and a canonical system T 1 with standard interface conditions. However, due to a lack of corresponding inverse results for Sturm-Liouville operators, the latter can in general not be replaced by a Sturm-Liouville operator.
The paper is organized as follows. In Sect. 2 we present some general results on Krein's formula for an orthogonal sum of n + 1 symmetric linear relations T j , j = 0, 1, . . . , n, each with equal defect numbers 1. In Sect. 3 we establish the relation between the compressed resolvent (1.1) and the extension of T 0 determined by an energy depending boundary condition, see (1.2). Section 4 contains the main result where we use the abstract inverse result on Q-functions to 'replace' the z-depending boundary condition by a new symmetric linear relation T 1 or, correspondingly, by attaching a realization on a new edge e 1 , see Fig. 1c. Implications of our reduction result for Cauchy problems of first and second order on star graphs are given in Sect. 4.2. In Sect. 5 we consider concrete operators on star graphs: canonical systems and Krein strings, which cover Dirac operators and Stieltjes strings as special cases, respectively, and Sturm-Liouville operators. Combining our results with the inverse results of de Branges and Krein, we obtain a more complete picture of the structure of the reduced problems.

The Abstract Schema
In this section we set up a framework for pasting a finite number of symmetric linear relations with general self-adjoint interface conditions, comp. [35]. For this we need some notation and basic properties of symmetric linear relations, boundary triplets and corresponding Titchmarsh-Weyl functions, see e.g. [34,Chapter 14], [9,10].
Then all self-adjoint extensions of T j in H j are described by 'boundary' conditions with a, b ∈ R. By T j;a,b we denote the corresponding self-adjoint relation in H j , i.e. the restriction of T * j by (2.2). Choose an H j -valued function h j on C\R such that h j (z) is a defect element of T j at z, i.e. h j (z) ∈ H j \{0} and and such that h j is analytic on C\R. In fact, if we fix z 0 ∈ C\R and a defect element it is easy to check that h j satisfies (2.3) and is analytic on C\R. Further, In the following we choose h j (z 0 ) such that j2 h j (z 0 ) = 1, and hence j2 h j (z) = 1, z ∈ C\R. (2.5) Then the Titchmarsh-Weyl function m j , corresponding to this choice of the boundary triplet, is defined by the formula 6) or equivalently, due to (2.5), The function m j is a Nevanlinna function since by (2.5), (2.6), for z, ζ ∈ C\R, z = ζ , and hence in particular, Im m j (z) > 0 if Im z > 0.

The 'Interface' Condition
Now we consider the orthogonal sum T := n j=0 T j of linear relations T j in the Hilbert space H := n j=0 H j with elements Then T has equal defect numbers n + 1 and a boundary triplet for T * = n j=0 T * j is given by

If we define the H-valued function h by
then {h(z), zh(z)} ∈ T * and the corresponding Titchmarsh-Weyl function is the (n + 1) × (n + 1)-matrix function M defined by Clearly, due to (2.6), and hence, using (2.5) we conclude that A description of all self-adjoint extensions of symmetric linear relations with equal deficiency numbers in terms of a boundary triplet has long been known (see [6,22], [10]). Correspondingly, all self-adjoint extensions of T in H are given by 'interface' conditions where A, B is a pair of real (n + 1) × (n + 1)-matrices with the properties By T A,B we denote the corresponding self-adjoint relation in H, i.e. the restriction of T * by (2.12).
It is well-known that the inverse (AM(z) − B) −1 exists for all z ∈ C\R. In fact, the Nevanlinna property of M and (2.13) imply that, for whence A * x 0 = 0 and thus B * x 0 = 0, a contradiction to the first condition in (2.13).

A Version of M. G. Krein's Resolvent Formula
The resolvent of any self-adjoint relation T A,B is related to the resolvent of the particular self-adjoint relation T 0,I with the special 'interface' condition 2 y = 0 as follows.

Proposition 2.1 For z ∈ C\R and f ∈ H, we have
where h j (z) and M(z) are as in (2.4) and (2.10), respectively.
In the following, for ease of presentation, we restrict ourselves to the case that all T j , j = 0, 1, . . . , n, are operators, and hence so are T A,B and T 0,I . We consider the difference of the resolvents of T A,B and T 0,I . Recall that, due to (2.2) and (2.12), Then, for f ∈ H, we have and hence, by (2.3), 0 ∈ H, taking the scalar product with g in (2.16), we conclude that and, analogously, Since T A,B and T 0,I are self-adjoint, it follows that the two right-hand sides above are equal. It is not difficult to see that this implies that there exists an In order to find an expression for F(z), we apply the functionals 1 and 2 to (2.17) and observe (2.11), (2.5), and (2.15) to obtain By means of the relation (2.12) for y = (T A,B − z) −1 f, we arrive at 3), the expression on the left hand side equals ( f j , h j (z)). The first term on the right hand side is zero because the first factor vanishes, and the second factor of the last term is equal to 1 due to (2.5). It follows that Using this in (2.18), we find Inserting this expression into (2.17) we arrive at (2.14).

The Compressed Resolvent
In this section we characterize the compression of the resolvent of T A,B to the first component H 0 of H, in terms of T * 0 and a z-depending boundary condition. Here P 0 is the projection onto the first component, where m j are the Titchmarsh-Weyl functions defined by (2.6), j = 0, 1, . . . , n, and let e 0 ∈ C n+1 be the first unit vector. Then, for z ∈ C\R, the problem is equivalent to the inhomogeneous problem in H 0 with z-depending boundary con- where n 0 is a Nevanlinna function not depending on m 0 and given by

Remark 3.2
The statement of Theorem 3.1 may also be formulated as follows.
the solution of the boundary value problem for T * 0 − z on H 0 with the z-depending boundary condition in (3.2). Note that, if T 0 is a linear operator, then (3.2) becomes Proof of Theorem 3.1 For ease of presentation, we assume that T 0 is a linear operator. Let z ∈ C\R. First we prove that if f 0 , g 0 ∈ H 0 satisfy (3.1), then they satisfy (3.2). By (3.1) and since T = n j=0 T j ⊂ T A,B , we conclude that, for arbitrary u 0 ∈ dom T 0 , This implies g 0 ∈ dom T * 0 and (T * 0 − z)g 0 = f 0 , which proves the first claim in (3.2). Further, applying P 0 to (2.14) for f = P * 0 f 0 with f 0 ∈ H 0 , we find If we use (2.19) and (2.7), it follows that Similarly, observing (2.15) and (2.5), we conclude that Altogether, we obtain and hence g 1 satisfies the boundary condition in (3.2) if we show that n 0 = n 0 . Next we prove that if f 0 , g 0 ∈ H 0 satisfy (3.2), then they satisfy (3.1). This means that we have to show that g 0 : The definition of g 0 implies that f 0 and g 0 satisfy (3.1) and hence, by what has been already proved in the first part, also (3.2), i.e.
Thus the difference h := g 0 − g 0 is a solution of the homogeneous problem This together with (2.1) yields that If h = 0, this would imply that Im n 0 (z)/ Im z < 0, which is a contradiction if we prove that n 0 is a Nevanlinna function.
It remains to be shown that n 0 = n 0 and that n 0 is a Nevanlinna function. Since the latter clearly holds for n 0 ≡ 0, we may assume that we have It is not difficult to show that N (z) −1 exists for z ∈ C\R and is given by By (3.3), this implies that, for z ∈ C\R, Hence n 0 = n 0 and n 0 is a Nevanlinna function if − N (z) −1 A is a matrix Nevanlinna function. Since the m j are Nevanlinna functions and BA * is self-adjoint, (3.4) shows that N (z)A * is a matrix Nevanlinna function and hence so is That n 0 is independent of m 0 is immediate from (3.3).
Theorem 3.1 may be used to relate the spectra of T A,B in H and of the z-depending spectral problem (3.2) in H 0 . For simplicity, we consider the case that the self-adjoint extensions of the T j have discrete spectra, hence all the functions m j and also n 0 are meromorphic. Note that, if z = λ is a pole of n 0 , then the second relation in (3.2) becomes 02 g = 0; if z = λ is a zero of n 0 or if n 0 ≡ 0, then the second relation in (3.2) becomes 01 g = 0. A,B ), then for each f 0 ∈ H 0 the problem (3.2) has the solution g 0 given by (3.1). (ii) If λ ∈ σ p (T A,B ) with eigenvector g and n 0 is holomorphic in a neighbourhood of λ, then

Corollary 3.3 (i) If z ∈ ρ(T
is satisfied for z = λ, and λ is an eigenvalue of (3.5) with eigenvector P 0 g if P 0 g = 0.
Proof The first claim is immediate from Theorem 3.1, the second claim follows because for u ∈ dom(T 0 ).

Example 3.4
As a special case of interface conditions (2.12) we consider 01 f 0 = 11 f 1 = · · · = n1 f n , where τ ∈ R ∪ {∞}. Here τ = 0 corresponds to continuity and Kirchhoff type conditions, while τ = ∞ corresponds to Dirichlet type conditions where the relations T j are decoupled. If τ = 0, the matrices A and B in (2.12) can be chosen as and hence we conclude that the function n 0 in (3.3) takes the form . (3.8) It is not difficult to see that (3.8) continues to hold if τ = 0; note that here the matrices A 0 , B 0 have to be chosen differently from (3.7), see (3.10) below for the case n = 1.
In the special case of Robin type interface conditions, the properties of n 0 proved in Theorem 3.1 for general interface conditions (2.12) are immediate from the explicit formula where A, B in (2.12) can be chosen as Here formula (3.8) reduces to that is, the z-depending coefficient of the boundary condition in the first component H 0 is the Titchmarsh-Weyl function of T 1 in the second component H 1 . This is well-known when performing interval truncation e.g. in numerics of Sturm-Liouville equations.

Remark 3.6 A natural generalization of the Robin type interface condition in (3.6) is
with coefficients τ j ∈ R ∪ {∞}, j = 0, 1, . . . , n. If all τ j are positive or all τ j are negative, the function n 0 which can be introduced in a similar way as above as is a Nevanlinna function and induces a self-adjoint operator in some Hilbert space.

The Reduced System
In the following theorem we show that there exists a single linear relation T 1 with Titchmarsh-Weyl function m 1 = n 0 such that the compressed resolvents on H 0 of the corresponding self-adjoint extensions of T = T 0 ⊕ T 1 ⊕ · · · ⊕ T n with interface condition (2.12) and of T = T 0 ⊕ T 1 with continuity and Kirchhoff type conditions (3.9) coincide. The main tool here is an abstract inverse result [28].

Corollary 4.3 With the notation and assumptions of Theorem 4.1, we have:
(i) For z ∈ C\R and f 0 ∈ H 0 , the first components of the solutions g and g of the inhomogeneous equations coincide, that is, P 0 g = P 0 g. (ii) If λ ∈ σ p (T A,B ) with eigenvector g and P 0 g = 0, then λ ∈ σ p ( T A 0 ,B 0 ) with eigenvector g such that P 0 g = P 0 g, and vice versa if P 0 g = 0.
Proof The claim in (i) is immediate from Theorem 4.1. The claims in (ii) follow from the fact that for a self-adjoint operator A in a Hilbert space with eigenvalue λ 0 ∈ R the orthogonal projection P λ 0 onto the corresponding eigenspace may be obtained as the strong limit P λ 0 = s − lim z→λ 0 (z 0 − z)(A − z) −1 if z tends to λ 0 perpendicularly to R.

Implications for Cauchy Problems
If T A,B and T A 0 ,B 0 are self-adjoint operators, they generate unitary groups (U A,B (t)) t∈R and ( U A 0 ,B 0 (t)) t∈R , respectively. The elements of these groups can be expressed as contour integrals with respect to the resolvents of the generators (see e.g. [21, IX.1]). Therefore the relation (4.1) in Theorem 4.1 readily implies that is, the compressions to H 0 of the two groups coincide. A consequence of this is the following result on Cauchy problems for the operators T A,B and T A 0 ,B 0 .
Then the solutions of the Cauchy problem for T A,B in H = H 0 ⊕ H 1 ⊕ · · · ⊕ H n given bẏ and of the Cauchy problem for given bẏ have the same first components, i.e.

Realization by a Canonical System
In this subsection we employ a fundamental inverse theorem by de Branges [7,Thm. 7], [38,Thm. 1] to show that the linear relation T 1 in the reduced system in Theorem 4.1 can always be realized as a canonical system. To this end, we first need to recall some basic results for canonical systems on a single edge, where we omit the subscript j for simplicity.
Let H be the Hilbert spaceL 2 H (e, C 2 ) of measurable 2-vector functions on e with inner product arising from L 2 H (e, C 2 ) by forming the factor space with respect to H -indivisable intervals (see [17,38]). InL 2 H (e, C 2 ) we consider the symmetric linear relation T defined as follows: if and only if f is absolutely continuous, f satisfies the canonical differential equation with Hermitian H given by together with the boundary condition and, if the edge is of finite length, i.e. < ∞, a boundary condition with some α ∈ [0, π); if = ∞, no boundary condition at is needed. Note that also in the regular case < ∞ the right endpoint can be replaced by ∞, see [38], if H is extended to [ , ∞) by Under these conditions T is a symmetric relation inL 2 H (e, C 2 ) with equal defect numbers 1 and T * is the extension of T obtained by omitting the boundary condition (4.8) at 0.
We introduce the boundary functionals (4.10) and, for z ∈ C, the fundamental matrix Y (·; z) = (y i j (·; z)) 2 i, j=1 , given as the 2 × 2 matrix function satisfying Then, if < ∞, the Titchmarsh-Weyl function m is defined by the condition that the function where the limit exists and is independent of α ∈ [0, π). Note that for the canonical system (4.7) the relation (2.8) for ζ = z takes the form with h(·, z) given by (4.11).
In the following theorem we make use of a fundamental inverse result of de Branges, see [7,Thm. 7], [38,Thm. 1]. Given any Nevanlinna function m, there exists a Hermitian H on [0, ∞) with the properties mentioned at the beginning of this subsection such that m is the Titchmarsh-Weyl function of the canonical system corresponding to H . This means that the symmetric linear relation in Theorem 4.1 can always be chosen to be a canonical system.

Differential Operators on Star Graphs
In this section we investigate the structure of the reduced problem if the symmetric relations T j , j = 0, 1, . . . , n, (or some of them) are differential or difference operators on the edges of a star graph G. In particular, we ask e.g. the following question: If all differential operators are of one type, when can the reduced system constructed in the previous section be chosen to consist of two differential operators of the same type?
We denote the central vertex of G by v and the outer vertices by v 0 , v 1 , . . . , v n (see Fig. 2 below); the coordinate x j on the edge e j increases from 0 at v to j at the outer vertex v j , j = 0, 1, . . . , n.
The symmetric relations T j , j = 0, 1, . . . , n, in the Hilbert spaces H j may be induced by three different types of differential or difference expressions on the edges e j , which are all supposed to be regular at v: canonical systems, Sturm-Liouville problems, and Krein strings; they include, as special cases, Dirac systems and Stieltjes strings. Here the 'interface' condition (2.12) becomes a matching condition at the common vertex v.

Canonical Systems and Sturm-Liouville Operators
By Theorem 4.6, independently of the type of differential operators inducing the relations T j , j = 1, 2, . . . , n on the subgraph of edges different from e 0 (the blue/nonbold part in Fig. 2), the new part T 1 of the reduced system T A 0 ,B 0 extending T 0 ⊕ T 1 can always be chosen to be a canonical system on an edge e 1 .
Thus the answer to the question when the whole reduced system consists of two canonical systems is immediate from Theorem 4.6: it suffices that only T 0 is induced by a canonical system.

Corollary 5.1
Suppose that in Theorem 4.6 the symmetric relation T 0 is given by a canonical system on an edge e 0 with Hermitian H 0 and that the other symmetric relations T j , j = 1, 2, . . . , n, therein are arbitrary. Then the reduced system T A 0 ,B 0 can be chosen to consist of two canonical systems on the path graph G with the two edges e 0 , e 1 with Hermitians H 0 , H 1 , respectively, and by the interface condition (4.14) which takes the form Fig. 3

below).
We mention that two canonical systems coupled at a common end-point, as obtained in Corollary 5.1, were studied in [17,Sect. 6].
For Sturm-Liouville operators the situation is very different. Since there are no suitable inverse results in this case, one cannot hope to be able to reduce a system of Sturm-Liouville operators on a star graph with n + 1 edges to a system of two Sturm-Liouville operators on a path graph. Nevertheless the results of the two previous sections can still be applied.
To explain this, suppose that the symmetric relations T j , j = 0, 1, . . . , n, are given by Sturm-Liouville operators in the Hilbert spaces H j = L 2 (e j , C), where q j is real-valued and locally summable on [0, j ), with boundary conditions at v and, if T j is regular at v j , cos α j y j ( j ) + sin α j y ( j ) = 0 ( 5 . 2 ) ... with some α j ∈ [0, π). If T j is singular at v j , the boundary condition at v j is either omitted (in the limit point case), or has to be modified to a limit circle type boundary condition. In any case, under these conditions T j is a symmetric operator in L 2 (e j , C) with equal defect numbers 1 and the operator T * j is the extension of T j obtained by omitting the boundary condition (5.1) at v.
We introduce the boundary functionals j1 y j := y j (0), j2 y J := y j (0), y ∈ dom T * j , (5.3) and, for z ∈ C, the fundamental system ϕ j (·; z), ψ j (·; z) of the homogeneous equation Then the Titchmarsh-Weyl coefficient m j is determined by the condition that the solution χ j (·; z) = −ψ j (·; z) + m j (z)ϕ j (·; z) satisfies the boundary condition (5.2) at v j , that is, We mention that here the relation (2.8) takes the form In this situation Theorem 3.1 as well as Theorem 4.6 apply and yield the following reductions: Theorem 3.1 reduces the system of Sturm-Liouville operators T j , j = 0, 1, . . . , n, on the star graph G with arbitrary interface conditions (2.12) to the Sturm-Liouville operator T 0 on the single edge e 0 with an energy depending boundary condition at v given by (3.3) with m j as in (5.6), j = 1, 2, . . . , n.
According to Theorem 4.6, the energy depending boundary condition at v can be replaced by attaching a canonical system on a new edge e 1 . In order to decide whether the latter can be replaced by a Sturm-Liouville operator would require suitable inverse results for Sturm-Liouville problems, which to the best of our knowledge do not exist.

Krein Strings and Stieltjes Strings
For a star graph of n + 1 Krein strings, the reduced system need not consist of two Krein strings. For the case of Robin type interface conditions, we establish a sufficient condition involving the Titchmarsh-Weyl functions of the n Krein strings of the subsystem and the Robin parameter. In this subsection we strongly rely on [18] (see also [20]).
A Krein string is given by a bounded left-continuous and non-decreasing function M on an (edge) interval e = [0, ), 0 < ≤ ∞; here, for x ∈ (0, ), the value M(x) is considered to be the mass of the string on the interval [0, x). We assume that the string has no concentrated mass at 0 and at so that, in particular, M(0) = M(0+) = 0, and we set M( ) := lim x M(x) ≤ ∞.
By H = L 2 M (e, C) we denote the Hilbert space of measurable functions f on e with inner product M (e, C).
In order to realize the symmetric relation T in L 2 M (e, C), we have to distinguish the two cases that is a regular boundary point, i.e. + M( ) < ∞, or that is a singular boundary point, i.e. + M( ) = ∞. ( with some α ∈ [0, π). Due to (5.7) every f ∈ dom T is absolutely continuous with left and right derivatives and f (0) = f (0) = 0. Note that, formally, the first equation on the right hand side of (5.7) reads It is not difficult to check that T is symmetric but not necessarily densely defined and that, with f , g ∈ L 2 M (e, C), the adjoint operator or relation T * is given by cos α f ( ) + sin α f ( ) = 0, (5.8) with c, d ∈ C. Hence T has equal defect numbers 1. For f ∈ dom T * as in (5.8) we introduce the boundary functionals A fundamental role is played by the solutions ϕ(·; z), ψ(·; z) of the integral equations For every z ∈ C, they form a fundamental system of the (formal) homogeneous equation Then, in the regular case, the Titchmarsh-Weyl function m corresponding to the boundary triplet with 1 , 2 given by (5.9), i.e. for the boundary condition 2 f = f (0) = 0 is defined by the property that the function − ψ(·; z)+m(z)ϕ(·; z) satisfies the boundary condition at , and hence Recall that a function F belongs to the Stieltjes class S or is a Stieltjes function if F is a Nevanlinna function that is holomorphic also on (− ∞, 0) with F(x) ≥ 0 for x ∈ (− ∞, 0), see [19]. According to [19,Thm. 5.2], F ∈ S if and only if F has an integral representation with some γ ≥ 0 and a measure σ on [0, ∞) with 1+x < ∞. Since F ∈ S is non-decreasing on (− ∞, 0), the limits at the boundary points exist, It is not difficult to see that, with the representation (5.11), As a consequence of this inverse result, if the function n 0 from (3.3) in Theorem 3.1 belongs to the class S, then the operator T 1 can be chosen to be a Krein string on some interval e 1 = [0, 1 ). In the following theorem we establish a sufficient condition for n 0 ∈ S for the case of Robin type interface conditions. Theorem 5.2 Suppose that in Theorem 4.1 the linear relations T j , j = 1, 2, . . . , n, are given by Krein strings with mass distributions M j on the edges e j of the star graph G so that M j (0+) = 0. If the vertex v j is regular, T j is defined as in (5.7) with α j ∈ [0, π/2] in the boundary condition at v j ; if the vertex v j is singular, T j is defined as in (5.14). In both cases, we denote by m j , j = 1, 2, . . . , n, the Titchmarsh-Weyl functions corresponding to the boundary condition f j (0) = 0. Further, let the self-adjoint extension T A τ ,B τ of T = T 0 ⊕ T 1 ⊕ · · · ⊕ T n be given by a Robin type interface condition (3.6) at v with τ ∈ R. If Because m 1 , m 2 , . . . , m n are increasing on (− ∞, 0), (5.19) is equivalent to (5.17). Therefore (5.17) implies n 0 ∈ S, and according to the inverse result of Krein mentioned above, T 1 with Titchmarsh-Weyl function m 1 = n 0 can be chosen a (regular or singular) Krein string. Next we prove the claim about the mass-free interval of T 1 . Since m 1 = n 0 and m j , j = 1, 2, . . . , n, belong to S, by (5.11) they have representations (5.20) with γ 1 , γ j ≥ 0, j = 1, 2, . . . , n. Then the first relation in (5.12) and (5.18) show that Because τ = ∞, this implies that γ 1 > 0 if and only if all γ j > 0, j = 1, 2, . . . , n, which proves the claim about the mass-free interval of T 1 .
To prove that the concentrated mass M 1 (0+) of T 1 at 0 equals 0, we first note that (see [18, 11.
Clearly, the first term in the denominator tends to 0; for the other terms we have Altogether, this proves M 1 (0+) = 0. Finally, if T 0 is also supposed to be given by a Krein string, then the last claim follows from the fact that Kirchhoff conditions on the vertex v joining e 0 and e 1 amount to the required continuity conditions at v. Remark 5.3 A Krein string with mass distribution function M describes a gap diffusion with state space supp M and speed measure M (comp. [27]). Hence, for a given system of gap diffusions on the graph G with state space X := n j=0 supp M j , Theorem 5.2 provides a system of two gap diffusions on the path graph G with state space X := supp M 0 ∪ supp M 1 . According to relation (4.2), the corresponding transition probabilities P(t; x, ), t > 0, x ∈ X , ⊂ X , measurable, and P(t; x, ), t > 0, x ∈ X , ⊂ X , measurable, satisfy P(t; x, ) = P(t; x, ), t > 0, x ∈ supp M 0 , ⊂ supp M 0 , measurable.
A Krein string is called a Stieltjes string (see [18,Sect. 13]) if the mass distribution function M is a step function with jumps μ k > 0 at points l 0 + l 1 + · · · + l k−1 , k = 1, 2, . . . , where l 0 ≥ 0 is the length of the possibly mass-free interval at 0 and