Continued fraction expansions of Herglotz–Nevanlinna functions and generalized indefinite strings of Stieltjes type

Abstract We employ some results about continued fraction expansions of Herglotz–Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral problem through explicit formulas.


INTRODUCTION
Stieltjes continued fractions played a decisive role in the solution of the inverse spectral problem for Krein strings [11,23,[26][27][28]. A certain modification of these continued fractions is of the same relevance for generalized indefinite strings, a class of spectral problems introduced in [15], based on previous work in [29][30][31][32]. This kind of continued fractions arose in [29,30] in connection with indefinite analogues of moment problems, was further studied in [9,10], including its role as Padé approximants [7,8], and applied to conservative multi-peakon solutions of the Camassa-Holm equation in [13]. It is the purpose of this article to discuss under which conditions a general Herglotz-Nevanlinna function admits a continued fraction expansion of this form. This will be done in the first section, which is close to classical material in [1] and [29,30], but does not seem to be available in the desired form. Subsequently, we will then use these findings in the second section to characterize the spectral data of generalized indefinite strings of Stieltjes type, whose coefficients are supported on discrete sets (near the left endpoint). This kind of generalized indefinite strings is closely related to Hamburger Hamiltonians [25] for canonical systems and hence also connected to the classical moment problem [14]. In particular, the results provide a solution of the inverse spectral problem for generalized indefinite strings of Stieltjes type with explicit formulas for the solution in terms of the moments of the spectral measure. A special case of such an inverse problem for indefinite strings was recently solved in [17] by means of a somewhat different approach.

CONTINUED FRACTION EXPANSIONS OF HERGLOTZ-NEVANLINNA FUNCTIONS
Let be a Herglotz-Nevanlinna function, that is, the function is defined and analytic on ℂ∖ℝ, maps the upper complex half-plane into the closure of the upper complex half-plane and satisfies the symmetry relation ( ) * = ( * ), ∈ ℂ∖ℝ. (1.1) It is well known (see, for example, [35, 5.3 Nevanlinna Representation]) that such a function admits an integral representation of the form where is a real constant, is a non-negative constant and is a non-negative Borel measure on ℝ which is subject to the growth restriction ∫ ℝ ( ) 1 + 2 < ∞. (1. 3) The constants and , as well as the measure in this integral representation are uniquely determined by the function and may be recovered explicitly. As long as they exist, we will denote with 0 , 1 , … the moments of the measure , that is, we set = ∫ ℝ ( ). (1.4) It follows readily from expanding the integrand in the representation (1.2) that for each ∈ ℕ 0 , the function allows the asymptotic expansion (1.5) as | | → ∞ along the imaginary axis, provided the moments of the measure exist up to order 2 . Here, the real constants −2 and −1 are defined by (1.11) again, provided they exist. In order to avoid ambiguity, it should be pointed out that all these determinants have to be interpreted as equal to one when is zero. For future reference, we also state the useful relations which follow from Sylvester's determinant identity (see the Appendix) and hold as long as the respective determinants are well defined. More precisely, if the moments of the measure exist up to order 2 , then the relations (1.12) and (1.13) hold for all ∈ {1, … , }, whereas relation (1.14) holds for all ∈ {1, … , + 1}. Let us suppose for now that is a rational Herglotz-Nevanlinna function and denote with the number of poles of . Because the support of the measure coincides with the poles of , it is evident that all moments of the measure exist in this case. Moreover, it follows from (1.9) that the determinants Δ 0,0 , … , Δ 0, are positive, but Δ 0, is zero when > . Similarly, after computing that where the non-negative constants 0 , … , and the real constants 0 , … , are given by . (1.20) Proof. Suppose that is a rational Herglotz-Nevanlinna function. For each ∈ ℝ, we define the rational Herglotz-Nevanlinna function by † The moments and Hankel determinants corresponding to the function will be denoted in a natural way with an additional subscript . In particular, note that 0 coincides with our initial function and hence so do the associated quantities (which is why we will omit the additional subscripts in this case). As each of the Hankel determinants depends analytically on , we may conclude that the determinants Δ ,1,0 , … , Δ ,1, ( +1) are all non-zero as long as ≠ 0 is small enough. Indeed, even if Δ 1, does vanish for some ∈ {1, … , ( + 1) − 1} (note here that Δ 1, ( +1) is certainly non-zero), then this holds because the derivative where = ( + 1), the non-zero real constants ,1 , … , , are given by (1.23) We note that the continued fraction has to be interpreted as when is zero and that the alternative expression in (1.22) is obtained by using relation (1.12). Put differently, the continued fraction expansion means that, upon defining the rational Herglotz-Nevanlinna functions ,0 , … , , recursively via where the constants , and are given by (1.19) and (1.20); note that is zero here. Otherwise, when Δ 1, ( )+1 vanishes, one has ( + 1) = ( ) + 2 and using the second expression in (1.22), we see that where the constant is given by (1.19b). Moreover, taking (1.23) into account, we also get in this case. Utilizing relation (1.12) with = ( ) + 1 as well as = ( ) + 2, in which the Δ 1, ( )+1 terms vanish, we infer that the last limit is equal to After plugging in relation (1.14) with = ( ) + 2, this becomes where the constant is given by (1.19a). In particular, this shows that is positive as relation (1.12) with = ( ) + 1 shows that the numerator on the left-hand side is not zero. Now observe that in the current case, the functions , ( ) satisfy Concluding, we note that the functions converge pointwise to our initial function by definition and thus we see from (1.25) that where the constants 0 and 0 are given by (1.19). In view of (1.27) and (1.26), this shows that the function admits the claimed continued fraction expansion. □ If none of the determinants Δ 1,0 , … , Δ 1, is zero (in view of (1.15), this happens when the function has no negative or no positive poles for example), then the expansion in Proposition 1.2 reduces to a Stieltjes continued fraction. More precisely, in this case we have ( ) = − 1 for all ∈ {1, … , + 1} and thus it follows from the formulas in (1.19a) that the constants 1 , … , vanish.
It is not difficult to see that any continued fraction of the form in Proposition 1.2 is a rational Herglotz-Nevanlinna function in turn. We also note that the mere fact that every rational Herglotz-Nevanlinna function can be expanded in such a way is much simpler to prove (see [12, Lemma B]), whereas working out explicit formulas for the constants takes more effort.
Remark 1.3. The constants in the continued fraction in Proposition 1.2 can also be expressed in different ways. In view of relation (1.13), the positive constants 1 , … , may be written in the form Apart from this, relation (1.12) shows that we have for ∈ {1, … , } as long as Δ 1, ( )+1 is not zero. On the other side, if Δ 1, ( )+1 is zero, then relation (1.14) allows us to write In particular, these expressions make it clear that the constant is not zero if and only if Δ 1, ( )+1 vanishes and also that + | | > 0 for all ∈ {1, … , }.
Before we proceed to non-rational Herglotz-Nevanlinna functions, let us first provide two auxiliary results. In order to state them, let 1 and 2 be Herglotz-Nevanlinna functions and denote with 1 and 2 the corresponding measures in the respective integral representations. Proof. We begin with the case when 2 ( )∕ → for some positive as | | → ∞ along the imaginary axis. If the moments of the measure 2 exist up to order 2 , then from Theorem 1.1 we infer that as | | → ∞ along the imaginary axis for some real constants 2,−1 , … , 2,2 . This implies ) as | | → ∞ along the imaginary axis. Since the first term on the right-hand side is a rational function, we conclude from Theorem 1.1 that the moments of the measure 1 exist up to order 2 + 2. Conversely, if the moments of the measure 1 exist up to order 2 + 2, then we have ) as | | → ∞ along the imaginary axis for some real constants 1,1 , … , 1,2 +2 and thus ) as | | → ∞ along the imaginary axis. Like before, we may conclude again that the moments of the measure 2 exist up to order 2 by invoking Theorem 1.1. Now let us assume that 1 ( )∕ → 0 and 2 ( )∕ → 0 as | | → ∞ along the imaginary axis. If the moments of the measure 2 exist up to order 2 , then from Theorem 1.1 we infer that ) as | | → ∞ along the imaginary axis for some real constants 2,−1 , … , 2,2 . Since 1 ( )∕ → 0 as | | → ∞ along the imaginary axis, we conclude that 2,−1 is not zero and thus we have ) as | | → ∞ along the imaginary axis. As the first term on the right-hand side is a rational function, we infer from Theorem 1.1 that the moments of the measure 1 exist up to order 2 . The converse direction follows by symmetry. ( 1 such that the functions for ∈ {1, 2} admit the continued fraction expansion ( ) = 0 + 0 + 1 for some ∈ ℕ, non-negative constants 0 , … , −1 , real constants 0 , … , −1 and positive con- as | | → ∞ along the imaginary axis, where the integer ∈ ℕ 0 is given by and is equal to one when and zero otherwise.
so that = 0 + 0 + ,1 by assumption. A computation then shows that as | | → ∞ along the imaginary axis for all ∈ {1, … , − 1}, which yields as | | → ∞ along the imaginary axis. Since this entails that as | | → ∞ along the imaginary axis, it remains to note that With the help of these two lemmas, we are now ready to add another item to the equivalence in Theorem 1.1; a continued fraction expansion of the function . Proof. Assume first that admits such a continued fraction expansion. We define the rational Herglotz-Nevanlinna function 1 by replacing̃in the continued fraction on the right-hand side of (1.38) with the function given by It then follows from Lemma 1.5 that (1.40) as | | → ∞ along the imaginary axis. In view of Theorem 1.1, this already guarantees that the moments of the measure exist up to order 2 .
For the converse direction, we will use induction. To this end, let us first consider the case when is zero, that is, we assume that the measure is finite, so that for some Herglotz-Nevanlinna function with ( )∕ → 0 as | | → ∞ along the imaginary axis. This allows us to write for some non-negative constant 1 and a Herglotz-Nevanlinna functioñ, which is the claimed expansion. Now let ∈ ℕ and suppose that the assertion holds for all lesser integers. As before, we may write like in (1.41) for some non-negative constant 1 and a Herglotz-Nevanlinna functioñ with̃( )∕ → 0 as | | → ∞ along the imaginary axis. In the case when = 1 and 1 ≠ 0, this is the required expansion. Otherwise, we conclude from Lemma 1.4 that the moments of the measurẽcorresponding to the functioñexist up to order Since the functioñthen admits a continued fraction expansion of the claimed form, we readily see that so does (it only remains to note that 1 + | 1 | > 0 in this expansion since ( )∕ → 0 as | | → ∞ along the imaginary axis). □ As in the rational case, we are able to provide explicit formulas for the constants in the continued fraction expansion in terms of the Hankel determinants. To this end, let us suppose that is a non-rational Herglotz-Nevanlinna function such that the moments of the measure exist up to order 2 for some ∈ ℕ 0 . Under these assumptions, the determinants Δ 0, are well defined for all ∈ {0, … , + 1} and positive in view of (1.9), as the measure must not be supported on a finite set. The determinants Δ 1, on the other hand exist at least for ∈ {0, … , }. We define ∈ ℕ as the number of non-zero elements of the sequence Δ 1,0 , … , Δ 1, and introduce the function such that ( ) is the smallest integer ∈ {0, … , } for which the sequence Δ 1,0 , … , Δ 1, has precisely non-zero elements. One observes that the increasing function is defined in such a way that Δ 1, (1) , … , Δ 1, ( ) enumerates all non-zero members of the sequence Δ 1,0 , … , Δ 1, . As it follows from relation (1.12) that there are no consecutive zeros in the sequence Δ 1,0 , … , Δ 1, , we may conclude that  where the non-negative constants 0 , … , −1 and the real constants 0 , … , −1 are given by (1.19) and the positive constants 1 , … , are given by (1.20). Furthermore, if the determinant Δ 1, is zero, then ( ) = − 1 and the functioñsatisfies Proof. Suppose that is a non-rational Herglotz-Nevanlinna function such that the moments of the measure exist up to order 2 for some ∈ ℕ 0 . According to Theorem 1.6, the function admits a continued fraction expansion of the form (1.38). With this notation, we define the rational Herglotz-Nevanlinna function 1 by replacing̃in the continued fraction on the right-hand side of (1.38) with the function given by (1.39) so that (1.40) as | | → ∞ along the imaginary axis. By virtue of the expansion (1.7) in Theorem 1.1, this shows that the numbers −2 , −1 , 0 , … , 2 corresponding to the functions and 1 coincide and hence so do the Hankel determinants Δ 0,0 , … , Δ 0, +1 , Δ 1,0 , … , Δ 1, , Δ 2,0 , … , Δ 2, , Δ −1,0 , … , Δ −1, +1 and Δ −2,0 , … , Δ −2, +2 . In particular, we find that our current function is a restriction of (or coincides with) the corresponding function (1.16) for 1 . Since the continued fraction expansion of the rational function 1 is unique (due to the preconditions that the constants 1 , … , are positive and that + | | > 0 for all ∈ {1, … , − 1}), we may obtain expressions for the constants in this expansion in terms of the Hankel determinants by comparison with Proposition 1.2. After possibly redefining the integer (because the one gets from Theorem 1.6 is potentially greater than the number of non-zero elements of the sequence Δ 1,0 , … , Δ 1, , whereas (1.37) guarantees that it is not less), the constant and the functioñin an appropriate way, we see that the expansion (1.38) can be brought into the claimed form. □ We now turn to the situation when is a non-rational Herglotz-Nevanlinna function such that all moments of the measure exist. In this case, all the Hankel determinants are well defined and the function extends to an increasing function such that Δ 1, (1) , Δ 1, (2) , … enumerates all non-zero elements of the sequence Δ 1,0 , Δ 1,1 , … (of which there are infinitely many as the sequence does not contain any consecutive zeros).
Proof. It suffices to refer to Theorem 1.6 and Corollary 1.7, but let us note that the constants in the continued fraction expansion in (1.45) are independent of . □ Since the last result in this section will not be needed in the following, we shall state it without a proof. In fact, a convenient way to verify it uses one of the forthcoming results from the next section. In this case, the non-negative constants 0 , 1 , … and the real constants 0 , 1 , … are given by (1.19) and the positive constants 1 , 2 , … are given by (1.20).

GENERALIZED INDEFINITE STRINGS OF STIELTJES TYPE
Let ( , , ) be a generalized indefinite string so that ∈ (0, ∞], is a real distribution † in −1 loc [0, ) and is a non-negative Borel measure on [0, ). We consider the corresponding spectral problem of the form where is a complex spectral parameter. Of course, this differential equation has to be understood in a distributional sense (we refer to [15] for more details).

Definition 2.1.
A solution of (2.1) is a function ∈ 1 loc [0, ) such that for some complex constant . In this case, the constant is uniquely determined and will henceforth always be denoted with ′ (0−) for apparent reasons.
Associated with the generalized indefinite string ( , , ) is the corresponding Weyl-Titchmarsh function defined on ℂ∖ℝ by where ( , ⋅ ) is a non-trivial solution of the differential equation (2.1) satisfying and vanishing at the right endpoint when is finite. The function is a Herglotz-Nevanlinna function and contains all the spectral information of the differential equation (2.1). In fact, the measure in the integral representation (1.2) for this function is a spectral measure for an underlying self-adjoint linear relation; see [15,Section 5]. All other quantities derived from the function will be denoted and used in the same way as they were introduced in the previous section. It was shown in [15, Theorem 6.1] that the mapping ( , , ) ↦ establishes a one-to-one correspondence between generalized indefinite strings and Herglotz-Nevanlinna functions. In the following, we are going to use our findings from the last section to characterize those Herglotz-Nevanlinna functions that correspond in this way to generalized indefinite strings that begin with † We denote with 1 loc [0, ) and 1 c [0, ) the function spaces given by a discrete part. To be more precise, we consider generalized indefinite strings ( , , ) of the form for some integer ∈ ℕ 0 , increasing points 1 , … , +1 in (0, ), real weights 0 , … , and nonnegative weights 0 , … , , where denotes the unit Dirac measure centered at a point ∈ [0, ). For such coefficients, the solutions ( , ⋅ ) of the differential equation ( as long as the fractions on the right-hand side are well defined. Here we have set 0 equal to zero for simplicity, so that the left-hand side of (2.7) becomes the Weyl-Titchmarsh function when is zero. In particular, these considerations show that the function admits a continued fraction expansion of the form (1.18), and hence is rational, when the coefficients and are supported on a finite set. The converse of this statement holds true as well.

Proposition 2.2.
If the Weyl-Titchmarsh function is rational, then the generalized indefinite string ( , , ) has the form where the length , the increasing points 1 , … , in (0, ), the real weights 0 , … , and the nonnegative weights 0 , … , are given by where ∈ 2 loc [0, ) is the normalized anti-derivative of the distribution so that Indeed, the first identity (2.11) follows readily from the expressions in (2.9). For the second identity (2.12), we first compute that as the function is piecewise constant. After plugging in the expressions from (2.9), it remains to note that the first term in the sum on the right-hand side is given by whereas for ∈ {1, … , − 1}, the th term in the sum is given by More specifically, the last equation can be verified by using the relations (1.13) and (1.14) after distinguishing whether the determinant Δ 1, ( )+1 is zero or not.
We now proceed to add another item to the equivalences in Theorem 1.1 and Theorem 1.6, this time in terms of the corresponding generalized indefinite string. Proof. Assume first that the moments of the measure exist up to order 2 so that the function admits the continued fraction expansion (1.38) by Theorem 1.6. With the constants from this expansion, we consider all generalized indefinite strings (̃,̃,̃) that satisfỹ where the points 1 , … , are given by (2.10) and it is supposed that̃> . According to (2.7), the Weyl-Titchmarsh functions corresponding to these generalized indefinite strings admit continued fraction expansions of the form (1.38). It then follows from [15, Theorem 6.1] that there is one such generalized indefinite string such that the continued fraction expansion coincides precisely with the initial one for . We conclude that this generalized indefinite string is necessarily the same as ( , , ), which guarantees that ( , , ) has the claimed form.
For the converse direction, it remains to note that the recursion in (2.7) yields the continued fraction expansion (1.38) with the constants 1 , … , given by Corollary 2.5. If the Weyl-Titchmarsh function is not rational and such that the moments of the spectral measure exist up to order 2 for some ∈ ℕ 0 , then the generalized indefinite string ( , , ) has the form where the increasing points 1 , … , in (0, ), the real weights 0 , … , −1 and the non-negative weights 0 , … , −1 are given by (2.9). Furthermore, if the determinant Δ 1, is zero, then ( ) = − 1 and one has Proof. as | | → ∞ along the imaginary axis, of the Weyl-Titchmarsh function uniquely determines the generalized indefinite string ( , , ) near the left endpoint. This can be viewed as a variant of local inverse uniqueness results as in [5,19,20,33,38] or an instance of the principle that the asymptotics of the Weyl-Titchmarsh function are related to the behavior of the coefficients near the left endpoint [6, 16 22, 24 39].
We continue with the characterization of those generalized indefinite strings that give rise to spectral measures with finite moments of arbitrary order. In this case, the increasing points 1 , 2 , … in (0, ), the real weights 0 , 1 , … and the non-negative weights 0 , 1 , … are given by (2.9).
Proof. The claim follows immediately from Theorem 2.4 and Corollary 2.5. □ In order to make sure that the Weyl-Titchmarsh function corresponds to a generalized indefinite string ( , , ) whose coefficients are supported on discrete sets, we just need to introduce one additional condition on the spectral data.
Proof. This is a consequence of Corollary 2.7 and the relation ({0}) = −1 , which holds for arbitrary generalized indefinite strings; see [15,Equation (5.12)]. □ As already indicated before, this last result can be used to prove Corollary 1.9 from the previous section, where one should also recall [15, Proposition 6.2]. Remark 2.9. In conjunction with the solution of the inverse spectral problem for generalized indefinite strings in [15,Theorem 6.1], the characterization in Corollary 2.8 gives rise to a solution of the inverse spectral problem for generalized indefinite strings whose coefficients are supported on discrete sets. Since one has explicit formulas for the solution, this also yields a solution of the inverse spectral problem for the class of generalized indefinite strings for which the distribution is supported on a discrete set and the measure vanishes identically. More precisely, these indefinite strings are determined by the additional conditions that lim →∞ (i ) i = 0 (2.25) and that none of the Hankel determinants Δ 1,1 , Δ 1,2 , … is zero.

APPENDIX: SYLVESTER'S DETERMINANT IDENTITY
In this appendix, we are going to show how the relations (1.12), (1.13) and (1.14) as well as (1.21) can be derived from Sylvester's determinant identity [2, Section I], [18,Chapter II,§3]. To this end, let 0 , 1 , … be a sequence of complex numbers. For each ∈ ℕ with > 1 (the relations can be