The Classical Moment Problem and Generalized Indefinite Strings

We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, we introduce the class of Krein-Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of M. G. Krein on the connection between the Stieltjes moment problem and Krein-Stieltjes strings and I. S. Kac on the connection between the Hamburger moment problem and 2x2 canonical systems with Hamburger Hamiltonians.


Introduction
Let {s k } k≥0 be a sequence of real numbers. The classical Hamburger moment problem is to find a positive Borel measure ρ on R such that the numbers s k are its moments of order k, that is, such that Every positive Borel measure ρ on R that satisfies (1.1), is called a solution of the Hamburger moment problem with data {s k } k≥0 . Similarly, the Stieltjes moment problem is to find a positive Borel measure ρ on R ≥0 such that the numbers s k are its moments of order k, that is, such that There are two principal questions: (i) For which sequences {s k } k≥0 are the moment problems solvable? (ii) Are solutions unique? If not, how to describe the set of all solutions?
We are neither going to provide comprehensive historical details nor a complete discussion of solutions to both of these problems here. Instead, let us only refer to the book by Akhiezer [1] (see also [29]). It is widely known that the Hamburger moment problem is closely connected with the spectral theory of symmetric Jacobi (tri-diagonal) matrices. On the other side, it was discovered by Krein [23] that the Stieltjes moment problem is closely connected with the spectral theory of strings (now known as Krein strings), that is, spectral problems of the form on an interval [0, L), where L ∈ (0, ∞] and ω is a positive Borel measure on [0, L). The quantities L and ω are usually referred to as the length and the mass density of the string, respectively. Both objects, Jacobi matrices and Krein strings, serve as certain canonical models for operators with simple spectra (for a nice account on canonical representations of self-adjoint operators we refer to a lecture by Krein [24]). Another such model for operators with simple spectra is a 2 × 2 canonical system [17,32] and it was shown by Kac [18,19] that the Hamburger moment problem can be included in the spectral theory of canonical systems with a special class of Hamiltonian functions termed Hamburger Hamiltonians. Motivated by the study of the indefinite moment problem, in [25,26] (see also [27,28] [11] that there is a one-to-one correspondence between spectral problems (1.4) and canonical systems. In particular, this entails that every Herglotz-Nevanlinna function can be identified with the Weyl-Titchmarsh function of a unique spectral problem (1.4), however, for this, the assumptions on the coefficients have to be relaxed to allow ω to be a real-valued distribution in H −1 loc ([0, L)) and υ to be a positive Borel measure on [0, L). Similarly to Krein strings, we shall call such a triple (L, ω, υ) a generalized indefinite string; see [11]. In this respect, let us also mention briefly that a lot of the interest in spectral problems of the form (1.4) stems from the fact that they arise as isospectral problems for the conservative Camassa-Holm flow [3,4,[7][8][9]12,16]. In particular, our motivation to include the classical moment problem in the spectral theory of generalized indefinite strings is dictated by the study of infinite multi-peakon solutions of the Camassa-Holm equation (see [2,9,12] for the case of finitely many peakons).
Our main aim here is to establish a connection between the Hamburger moment problem and the spectral theory of generalized indefinite strings. More precisely, we will show that there is a one-to-one correspondence between moment sequences and a special class of generalized indefinite strings (which we decided to call Krein-Langer strings). This can be done in various ways. For example, one can use the results of Kac [18,19] in conjunction with the correspondence between canonical systems and generalized indefinite strings (see Sect. 6). On the other hand, one can also prove this result by identifying moment sequences with (formal) continued fractions of the form , (1.5) which is the approach that we will follow in Sect. 5. Notice that this kind of continued fraction is a slight modification of the one studied by Stieltjes [30] and subsequently applied to solve the Stieltjes moment problem. In fact, this generalization allows one to deal with the full (Hamburger) moment problem (as an alternative to employing the continued fractions used by Hamburger [15]). Continued fractions of the form (1.5) naturally lead to spectral problems of the form (1.4) with coefficients ω and υ supported on a discrete set; Krein-Langer strings. A significant part of this article is of preliminary character. In Sects. 2 and 3, we first collect basic notions and facts on Hamburger as well as Stieltjes moment problems and describe their respective relations to Jacobi matrices and Krein strings. Section 4 then contains necessary information on canonical systems, Hamburger Hamiltonians and their connection with the Hamburger moment problem. After these preparations, we proceed to introduce the class of Krein-Langer strings in Sect. 5 and subsequently prove our main result,  Suppose now that {s k } k≥0 is a strictly positive sequence and that ρ is a solution to the Hamburger moment problem (1.1). Without loss of generality, we can assume that s 0 = 1, which means that ρ is a probability measure. First, let us define the polynomials of the first kind associated with the measure ρ: for all n ≥ 1. Clearly, we have the asymptotics

The Hamburger Moment Problem and Jacobi Matrices
Here we set Δ 0,−1 := 1, Δ 0,−1 := 0, Δ 0,0 := s 1 , and Δ 0,n := The family {P n } n≥0 is orthonormal with respect to the measure ρ, however, it does not necessarily form a basis in L 2 (R; ρ) (if the moment sequence is positive but not strictly positive, (2.2) allows to define exactly N polynomials {P n } N −1 n=0 and these polynomials serve as an orthonormal basis in L 2 (R; ρ)). Moreover, the polynomials P n satisfy the three-term recurrence relations b n−1 P n−1 (z) + a n P n (z) + b n P n+1 (z) = zP n (z), n ≥ 0, (2.5) upon setting P −1 ≡ 0 for notational simplicity. Hereby, the coefficients in (2.5) are given by b −1 = 0 and b n = R λP n (λ)P n+1 (λ) ρ(dλ) = Δ 0,n−1 Δ 0,n+1 Δ 0,n , n ≥ 0, (2.6) as well as by The recurrence relations (2.5) naturally generate the following Jacobi (tridiagonal) matrix which defines a minimal (closed) symmetric operator in 2 (Z ≥0 ). This operator is either self-adjoint or has deficiency indices (1, 1). In the former case, the matrix is said to be in the limit point case and in the latter it is said to be in the limit circle case. The next result is well known (see [1] for example). Moreover, it follows readily thatã n = a n andb n = b n for all n ≥ 0 in this case.
Let us next introduce the polynomials of the second kind: Notice that the polynomials Q n do not actually depend on the choice of ρ if the moment problem is indeterminate. Using (2.2), the polynomials Q n can be expressed via the moment sequence through , n ≥ 0, (2.11) where R n,0 ≡ 0 and One can now characterize determinate Hamburger moment problems in terms of the Jacobi coefficients as well as the orthogonal polynomials (see [1] for example). (iv) There exists λ ∈ R such that the series n≥0 |P n (λ)| 2 diverges.

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The Classical Moment Problem Page 7 of 30 23 Theorem 2.3 above establishes a connection between strictly positive sequences {s k } k≥0 with s 0 = 1 and semi-infinite Jacobi matrices. This correspondence can also be described in another important way. Upon denoting for all n ≥ 0, let us define the function m n on C\R by The rational function m n is a Herglotz-Nevanlinna function, that is, it is analytic, maps the upper complex half-plane into the closure of the upper complex half-plane and satisfies the symmetry relation Let us mention in this context that (2.7) implies the following identity Furthermore, this trace is equal to the sum of the roots of the polynomial P n+1 . The function m n admits the asymptotic expansion (see [ Since the polynomials of the first and the second kind satisfy the recurrence relations in (2.5), m n admits the continued fraction expansion [1, § I.4.2]: For this reason, it makes sense to identify any semi-infinite Jacobi matrix (2.8) with the formal continued fraction Clearly, its n-th order convergent is precisely the rational function m n having the asymptotic expansion (2.18). It remains to notice that the following limit In any case, the limit function m admits an integral representation of the form for some finite positive Borel measure ρ 0 on R. The asymptotic expansion in (2.18) entails that the measure ρ 0 obtained in this way solves the Hamburger moment problem (1.1).

The Stieltjes Moment Problem and Krein-Stieltjes Strings
are non-negative (positive) for all n ≥ 1. In the following, we will also set Δ 1,0 := 1 for notational simplicity. If the moment sequence {s k } k≥0 is strictly double positive, then from the definition of the polynomials of the first kind in (2.2) we have for all n ≥ 0. Hence, upon setting for all n ≥ 0 and using the recurrence relations (2.5) with z = 0, we conclude that the coefficients of the Jacobi matrix (2.8) admit the representation a n = 1 l n for all n ≥ 0, where we set 1 ω−1 := 0 for notational simplicity. Notice that by (3.2) and (2.6) we then have for all n ≥ 0. Moreover, in this case the rational function m n defined in (2.15) admits the Stieltjes continued fraction expansion where we defined the additional determinants as well as , n ≥ 1. Notice that this transformation does not change the coefficients a n and b n in (3.4) (cf. Remark 2.4), however, it does change the continued fraction expansion (3.6).
It was observed by Krein [23] (see also [1,Appendix], [20, § 13]) that in the double positive case, the corresponding Jacobi matrix (2.8)  respectively, where we assume that N ∈ Z ≥0 ∪ {∞} and 0 =: In the case of infinitely many masses, we shall assume that lim n→∞ x n = L.
Otherwise, if the string carries only finitely many point masses N < ∞, then we shall assume that x N −1 < x N := L, that is, there is no point mass at the right endpoint.
If the ends of this string are fixed and it is stretched by a unit force, then small oscillations are described by the spectral problem associated with the corresponding Jacobi matrix J given by (2.8)  If the number of point masses N is finite, then (3.15) represents a rational Herglotz-Nevanlinna function. Otherwise, when there are infinitely many point masses, it was observed by Stieltjes that (3.15) converges for every z ∈ C\R ≥0 if and only if at least one of the sums n≥0 ω n = ω([0, L)) and n≥0 l n = L is infinite (or, equivalently, the Krein-Stieltjes string is singular). When the string (L, ω) is regular, the even order convergents q 2n (z)/p 2n (z) and the odd order convergents q 2n+1 (z)/p 2n+1 (z) of the continued fraction (3.15) still converge for every z ∈ C\R ≥0 , however, to different limits. Similar to the definition in Sect. 2, the limit Notice that it coincides with the dynamical compliance of the (dual) string (see [20,21] for further details).  where the fraction on the right-hand side has to be interpreted as zero when L is infinite (cf. [20, § 11] and also [11, § 5]). In particular, when zero is an isolated singularity of m, then it is a pole if and only if L is finite. , that is, the corresponding measure ω is supported on a finite set. (iv) Notice that the Jacobi matrix (2.8) can be written in the form (3.4) only if the polynomials of the first kind do not vanish at z = 0, or, equivalently Δ 1,n = 0 for all n ≥ 0. In particular, the principal Weyl-Titchmarsh function admits an expansion (3.15) only if the above condition holds true; compare [13].
As in the previous section, one is again able to characterize determinate Stieltjes moment problems in terms of the corresponding Krein-Stieltjes strings. Remark 3.6. The equivalence (i) ⇔ (iii) is due to Stieltjes [30] and the connection with strings together with the equivalence (i) ⇔ (ii) was observed by Krein [23] (see also [20]).

Canonical Systems
Let us first briefly review some facts about canonical systems as far as they are needed in this section; for more details we refer the reader to [5,17,31,32]. In order to set the stage, let H be a locally integrable, real, symmetric and non-negative definite 2 × 2 matrix function on [0, ∞). Furthermore, we shall assume that H is trace normed, that is, for almost all x ∈ [0, ∞), and also exclude the cases when for almost all x ∈ [0, ∞). A matrix function H with all these properties is called a Hamiltonian and associated with such a function is the canonical first order system with a complex spectral parameter z. We introduce the fundamental matrix solution U of the canonical system (4.3) as the unique solution of the integral equation (4.4) The Weyl-Titchmarsh function m of the canonical system (4.3) is now defined by As a Herglotz-Nevanlinna function, it admits an integral representation of the form for some constants c 1 , c 2 ∈ R with c 1 ≥ 0 and a positive Borel measure ρ on R with Note that the coefficient c 1 of the linear term can be read off the Hamiltonian H immediately (see [32,Lemma 2.5]); It is a fundamental result of de Branges [5] (see also [32,Theorem 2.4]) that indeed every Herglotz-Nevanlinna function arises as the Weyl-Titchmarsh function of a unique canonical system (4.3).

Theorem 4.1. (de Branges) For every Herglotz-Nevanlinna function m there is a Hamiltonian H such that m is the Weyl-Titchmarsh function of the canonical system (4.3). Upon identifying Hamiltonians which coincide almost everywhere on [0, ∞), this correspondence is also one-to-one.
Let us also mention that in the case when H(x) = H 0 for almost all x ≥ L with some L ∈ (0, ∞), straightforward calculations show that the corresponding Weyl-Titchmarsh function m is given by Hence, we can consider canonical systems (4.3) on any finite interval [0, L), the function (4.9) will be called the principal Weyl-Titchmarsh function and it coincides with the Weyl-Titchmarsh function of the canonical system whose Hamiltonian function is given by Hamiltonians on a finite interval are called regular and singular otherwise.

Hamburger Hamiltonians
Following [18,19], let us now introduce a special class of Hamiltonians. To this end, fix some L ∈ (0, ∞], an N ∈ Z ≥0 ∪ {∞} and let L := { k } N −1 k=0 and Θ := {θ k } N k=0 be real sequences such that θ 0 = π 2 and k > 0, θ k < θ k+1 < θ k + π, (4.11) for all k ∈ {0, . . . , N − 1}. We then set We also assume that and θ N ∈ πZ if N < ∞, and where the matrix H θ is defined by Hamiltonians of the above form are called Hamburger Hamiltonians [18,19]. Notice that the requirement (4.11) implies that every interval (x k−1 , x k ) is maximal H-indivisible of type θ k . Before we formulate the main results from [18,19], we need the following well-known fact (see, e.g., [ If U n is the fundamental matrix solution of the system  if θ n ∈ πZ, and m n (z) = cot(θ n ) + 1 Proof. Noting that the fundamental matrix-solution U n is given by for all x ∈ [x n−1 , x n ] and z ∈ C, we get Hence straightforward calculations show that which readily establishes the claim.

Corollary 4.3. If N is finite, then the Weyl-Titchmarsh function m admits the continued fraction expansion
since by normalization θ N ∈ πZ, and then apply Lemma 4.2.
In particular, the Weyl-Titchmarsh function corresponding to a Hamburger Hamiltonian with finite N is a rational function that vanishes at ∞. The converse holds true as well.
where m 1 := −1/ m 0 is a rational Herglotz-Nevanlinna function. Therefore, there are constants ω 0 ∈ R and υ 0 ∈ R ≥0 such that m 1 (z) = ω 0 +υ 0 z + m 1 (z), where m 1 is a rational Herglotz-Nevanlinna function that vanishes at ∞. Since m 0 is bounded near ∞, at least one of the coefficients ω 0 or υ 0 is nonzero. Moreover, we have m 1 = q/ p, where the polynomials q and p do not have common zeros and deg( p) = deg( q) + 1 ≤ n − 1. Upon applying the same procedure to m 1 , we arrive at the representation (4.23) after finitely many iterations.

Connection with the Moment Problem
Notice that if θ k / ∈ πZ for all k ∈ {0, . . . , N}, then it follows from (4.5) and Corollary 4.3 that the Weyl-Titchmarsh function m admits the Stieltjes continued fraction expansion  (4.24) where the coefficients are given by This establishes a connection between canonical systems with such Hamburger Hamiltonians and continued fractions of the form (2.20), and thus also with Jacobi matrices. Indeed, taking (3.4) into account, the corresponding Jacobi coefficients (after some calculations) are given by where θ −1 := 0 for notational simplicity. Moreover, the second formula in (3.7) together with the first formula in (3.5) imply It was observed by Kac [18,19] that in fact (4.26) establishes a one-to-one correspondence between Hamburger Hamiltonians with N = ∞ and 0 = 1, semi-infinite Jacobi matrices and thus also strictly positive sequences {s k } k≥0 with s 0 = 1. More precisely, to this end we only need to set θ n := 0 (mod π) (4.28) in (4.27) if Δ 1,n = 0. Note that the lengths n are indeed positive for all n ≥ 0 since we have the inequality which follows upon evaluating (2.13) at zero and using (2.2) as well as (2.11) to compute the values of P n and Q n at zero.
for some finite positive Borel measure ρ 0 on R, which is a solution of the corresponding Hamburger moment problem.
Again, we are able to characterize determinate Hamburger moment problems in terms of the corresponding Hamiltonian (see [18,19]).

Generalized Indefinite Strings
Let us first briefly review some facts about generalized indefinite strings; for more details we refer the reader to [10][11][12]. To this end, fix some L ∈ (0, ∞], let ω ∈ H −1 loc ([0, L)) be a real-valued distribution on [0, L) and υ be a positive Borel measure on [0, L). We will first discuss the meaning of the differential equation where z is a complex spectral parameter. Of course, this equation has to be understood in a distributional sense: A solution of (5.1) is a function f ∈ H 1 loc ([0, L)) such that for all h ∈ H 1 c ([0, L)) and some constant Δ f ∈ C. In this case, the constant Δ f is uniquely determined and will henceforth always be denoted with f (0−) for apparent reasons. Of course, there are also several other ways of introducing for every x ∈ [0, L), one observes that a function f ∈ H 1 loc ([0, L)) is a solution of (5.1) if and only if one has for all x ∈ [0, L). Note that this formulation simply reduces to the usual integral equation (as used in, for example, [20, § 1], [27,Section 1], see also [10]) if ω is a Borel measure: For every z ∈ C, we introduce the fundamental system of solutions c(z, · ), s(z, · ) of the differential equation As a Herglotz-Nevanlinna function (see [11]), the function m has an integral representation of the form (4.6)-(4.7) again. Similarly to Krein strings, a triple (L, ω, υ) such that L ∈ (0, ∞], ω is a real-valued distribution in H −1 loc ([0, L)) and υ is a positive Borel measure on [0, L) is called a generalized indefinite string. Such a string (L, ω, υ) is called regular if the length L is finite, ω ∈ H −1 ([0, L)) and υ([0, L)) < ∞, that is, if where w ∈ L 2 loc ([0, L)) is the anti-derivative of ω specified by Otherwise, the string is called singular. Note that although the class of generalized indefinite strings contains the class of Krein strings, the notion of regularity does not coincide on this subset. However, the regularity of Krein strings corresponds to the indeterminacy of the Stieltjes moment problem whereas the regularity of generalized indefinite strings correlates with the indeterminacy of the Hamburger moment problem (see Theorem 5.7 below).
if k(j + 1) − k(j) = 2. It follows from (4.29) that there are no consecutive zeros within the sequence {Δ 1,k } ∞ k=0 , which ensures that the above quantities are well-defined. Proof. In view of Theorem 5.2, it suffices to prove the claim only for strictly positive sequences. Moreover, we just need to show that the map is surjective. However, this follows from the continued fraction expansion (5.17). Indeed, sending j to infinity there, we see that every Krein-Langer string can be identified with a formal infinite continued fraction of this type. It remains to use the one-to-one correspondence between continued fractions of this type and strictly positive sequences as well as noting that the coefficients therein are related via (5.18)-(5.20).
Remark 5.4. One can also prove Theorem 5.3 by combining Kac's Theorem 4.5 with the transformation connecting canonical systems with generalized indefinite strings; see [11,Section 6] and Sect. 6.