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Point mass perturbations of spectral measures

Using a generalization of the moment problem and the extremal properties of spectral measures corresponding to the selfadjoint extensions of a regular symmetric operator, we study point mass perturbations of spectral measures. We obtain general results for a wide class of operators and apply them to the analysis of point mass perturbations of spectral measures pertaining to Bessel and generalized Schr\"odinger operators.


Introduction
The motivation of this work stems from the intriguing fact that very small perturbations of a measure may destroy the density of polynomials in the space of square integrable functions with respect to that measure. More precisely: one can find a Borel measure µ such that the polynomials are dense in L 2 (R, µ), but they are no longer dense in L 2 (R, µ) when µ is obtained by adding a point mass to µ, that is µ = µ + δ λ , λ ∈ supp µ , (1.1) where δ λ denotes a discrete measure which gives weight only to the point λ. This instability phenomenon pertains to the classical Hamburger moment problem and a particular realization corresponds to the following reformulation. If µ is an N-extremal solution to an indeterminate moment problem, then µ is a non-Nextremal solution to a moment problem (see [1,Thm. 2.3.2] and [10,Prop. 4.1]).
Remarkably, the theory of the Hamburger moments is the theory of Jacobi operators, i. e. operators in l 2 (N) generated by semi-infinite tridiagonal symmetric matrices (see Example 6.1). In this context, the mentioned phenomenon is paraphrased as follows: if µ is the spectral measure of a canonical selfadjoint extension of a nonselfadjoint Jacobi operator, i. e. a selfadjoint restriction of its adjoint (see Section 2.1), then there is no canonical selfadjoint extension of any Jacobi operator having µ as its spectral measure. This paper is devoted to studying the situation illustrated above in a setting that allows us to consider not only nonselfadjoint Jacobi operators, but a wide class of regular symmetric operators with deficiency indices (1, 1), including regular and certain singular Schrödinger operators. Our approach is based on Krein's theory of symmetric operators [20][21][22], the theory of generalized extensions due to Naȋmark [32] and a generalization of the moment problem originally proposed by Livsič [28]. This theoretical scaffold allows us to prove several results on point mass perturbations of spectral measures presented in Section 5. The central theorem (Theorem 5.4) is general enough to be applied to the analysis of point mass perturbation of spectral measures related to generalized Schrödinger operators with measures and Bessel operators. This analysis led to Theorems 6.2, 6.4, and 6.5.
The spectrum of a regular Schrödinger operator defined by any selfadjoint boundary conditions has the following asymptotic behavior λ n = cn 2 + O(1), as n → ∞ , (1.2) (see [33]). From this formula, by a straightforward argument (see Remark 6.3), one concludes that if µ is the spectral measure of a regular Schrödinger operator given by selfadjoint boundary conditions, then µ, defined as in (1.1), cannot be the spectral measure of any regular Schrödinger operator with any selfadjoint boundary conditions no matter which potential function is considered. It is worth remarking that the results of Section 5 provide a way for dealing with point mass perturbation of spectral measures for a wide range of operators even in the case when asymptotics of the kind of (1.2) are not available. This is illustrated in Examples 6.2 and 6.3. Let us elaborate on the theoretical framework developed in this paper which is of interest in itself irrespective of the results given in Sections 4 and 6. The classical theory of moments tells us that to account for all solutions to the moment problem, one has to recur to Naȋmark's theory of generalized extensions [31,32] since the solutions are the generalized spectral measures (see Definition 2.5) of the corresponding Jacobi operator. Here we remark that, despite the classicality of the theory, relatively recent developments [7,45] make use of [32] to construct solutions to the classical moment problem with the measure having certain predefined properties. Our approach makes use not only of Naȋmark's theory, but also Krein's theory of symmetric operators with deficiency indices (1,1) [20][21][22] to study the set of measures related to an arbitrary regular symmetric operator not necessarily entire [15]. This allows us to characterize the spectral measures associated with the operator (Theorem 3.4) and describe the extremal properties of measures pertaining to this operator's selfadjoint extensions (Theorem 3.6). The next step in our method is the use of the functional model for regular symmetric operators [38,39] to map any regular symmetric operator into the multiplication operator in a de Branges spaces. This reformulation leads us directly to the generalized moment problem.
There are various ways of generalizing the moment problem. Here, we have in mind a proposal of generalization that goes back to Livsič [28] for whom the moment problem consists in finding a measure ρ such that the inner product in a Hilbert space of functions is expressed through the inner product in L 2 (R, ρ). A related setting of the generalized moment problem is found in the introduction of [44]. As pointed out in [36,Chap. 6], de Branges treated this problem using a different terminology when the Hilbert space of functions is a de Branges space [9]. In this work, we also consider the generalization of [28] with the underlying Hilbert space being a de Branges space, but in contrast to [9] and [36,Chap. 6], we focus on the extremal properties of the solutions to the generalized moment problem.
Let us outline the material of this paper. Section 2 presents classical results that are included for the sake of completeness and to introduce the notation. Section 3 study the extremal properties of measures associated with regular symmetric operators. A short survey on de Branges spaces theory and the generalized moment problem are the subject of Section 4. In this section, we also present the functional model for regular symmetric operators by means of which we connect the results on de Branges spaces to our class of operators. In Sec-tion 5, we prove the main results related to perturbation of spectral measures that results when adding a mass at a single point. Here we also touch upon the fact that certain functions which are dense in L 2 (R, µ) are no longer dense in L 2 (R, µ) (see (1.1)). These results may have several applications to the inverse spectral analysis of operators as is shown in [10] for Jacobi operators. Finally, in Section 6, we illustrate the results of this work. The first example is used to put our results in the context of the classical moment problem. The other two examples present nontrivial results pertaining differential operators which include regular and singular Schrödinger operators.

Preliminaries
Let H be a separable Hilbert space. The inner product in it is denoted by ·, · where we agree that the left-hand-side argument is anti-linear. We denote by ⊕, ⊖, and ∔ the orthogonal sum, the orthogonal complement, and the direct sum of linear spaces, respectively.
Let A be a not necessarily densely defined linear closed symmetric operator. If A is not densely defined, then A * is a multivalued linear operator (linear relation). In this case, the symmetry condition, A ⊂ A * , remains valid (as subspaces of H ⊕ H). The deficiency indices n + (A) and n − (A) of A are defined as the dimension of H ⊖ ran(A − zI), when z ∈ C + := {z ∈ C : Im z > 0} and z ∈ C − := {z ∈ C : Im z < 0}, respectively. If A has equal deficiency indices, then it has canonical selfadjoint extensions (that is, selfadjoint restrictions of A * ), with some of these extensions being proper linear relations if dom(A) is not dense in H [16,Prop. 5.4].
What follows is a brief revision of Naȋmark's theory of selfadjoint extensions to a larger space [31,32] (see also [2,Appendix 1] and [44]), as well as some results from Krein's theory on regular symmetric operators [15,22].

Generalized resolvents
hold true for any v, w, z ∈ C \ spec(A + ). Also, it is straightforward to establish that 3) The following assertion arises by combining the first identity in ( is a generalized resolvent of A [32] (see also [2,Appendix 1] and [44, § 3]).
Fix a canonical selfadjoint extension of a closed symmetric operator A with deficiency indices n + (A) = n − (A) = 1 and denote it by A ∞ . For w ∈ C + and z ∈ C \ spec(A ∞ ), define is the so-called Q-function (see [2, Appendix 1 Eq. 8]). R τ (z) is the resolvent of a canonical selfadjoint extension of A if and only if τ (z) ≡ c ∈ R∪{∞}. Moreover, once a canonical selfadjoint extension A ∞ is fixed, (2.5) induces a bijection between R and the set of all generalized resolvents of A (see [11,Thm. 6.2], also [19,23,24,32,43]). Equation (2.5) is known as the Krein-Naȋmark formula.
Definition 2.5. Let E A + denote the spectral family of A + given by the spectral theorem. Let X denote the σ-algebra of Borel sets on R. Given a selfadjoint extension A + of a symmetric operator A, the map is called a generalized spectral family of the operator A. Furthermore, given φ ∈ H \ {0}, we define We call σ A + ,φ the generalized spectral measure of A associated with φ and A + .
so σ A + ,φ is also the spectral measure of the selfadjoint extension A + associated with φ ∈ H and therefore it is a nonnegative finite scalar measure. On the basis of [2, Appendix I Thm. 2], the next result holds true.

Proposition 2.6. F is a generalized spectral family of
satisfies, for any φ ∈ H, the following conditions: (iv) If φ ∈ dom(A), then The next statement is proven on the basis of Proposition 2.6 (see [2, Appendix 1, pp. 141-143]).

Proposition 2.7.
There is a bijection between R and the generalized spectral family of any symmetric operator with deficiency indices (1,1). The bijection is given by the combination of (2.5) and the identity Moreover, the same bijection gives a one-to-one map between R ∪ {∞} and the spectral family corresponding to canonical selfadjoint extensions of A.

Regular symmetric operators and gauges
Definition 2.8. A closed symmetric nonselfadjoint operator is said to be completely nonselfadjoint if it is not a nontrivial orthogonal sum of a symmetric and a selfadjoint operators.
Definition 2.9. The complex number z is in the set of points of regular type of a closed operator T when there exists c z > 0 such that The complement of the set of points of regular type of T is called the spectral kernel of T .

Definition 2.10.
A closed operator T is regular when the set of points of regular type is the whole complex plane.
A regular, closed symmetric operator is necessarily completely nonselfadjoint since regularity implies that the spectral kernel is empty and, therefore, the operator cannot have selfadjoint parts. On the other hand, there are many completely nonselfadjoint operators that are not regular [41,Sec. 2]. We note that the dimension of ker(A * − zI) is constant for all z ∈ C whenever A is regular. The next assertion is well known for densely defined symmetric operators [3,12,15]. A proof for the case under discussion in this work can be found in [39,Prop. 2.4]. (ii) Every real number is part of the spectrum of one, and only one, canonical selfadjoint extension of A.
(iii) The spectra of the canonical selfadjoint extensions of A are pairwise interlaced.
Note that item (i) above means that every selfadjoint extension of A is a simple operator [2, Sec. 69]. Also, item (iii) above (together with the fact that under extensions the spectral kernel does not decrease) implies that A is regular if and only if the spectra of any two selfadjoint extensions of A do not intersect.
For any A ∈ S(H), let κ ∈ H be such that for some z 0 ∈ C. For this vector κ, consider the set The following result is proven in [ Definition 2.14. We call the vector κ, whose existence is established in Proposition 2.13, a spectral gauge of A ∈ S(H).

Lemma 2.15. A spectral gauge of A ∈ S(H) is a generating vector (as defined in [2, Sec. 69]) for any canonical selfadjoint extension of A.
Proof. Let A γ be any canonical selfadjoint extension of A and κ a spectral gauge of A. Since κ is not in ran(A−xI) for any x ∈ R, κ has a nonzero projection onto the one-dimensional space ker( , it follows that κ has a nonzero projection onto every eigenspace of A γ since its spectrum is discrete.
The following assertion is stated in [ for any φ ∈ H and ψ given in (2.4).

Extremality of measures
Inside V κ (A), we single out the set of extremal measures The reason for using word "extremal" above will be explained in Theorem 3.6 below. In the context of this definition, Proposition 2.7 implies the following.

Corollary 3.2.
For any spectral gauge κ of A ∈ S(H), there is a bijection between R and V κ (A). Moreover, this bijection gives a one-to-one mapping between R ∪ {∞} and V ext κ (A). The bijection referred above depends on the choice of A ∞ (see Section 2.1), but regardless of this choice, all measures arising from spectral families corresponding to canonical selfadjoint extensions of A are obtained when τ runs through R ∪ {∞}. Note that, as a consequence of Proposition 2.12, one has: where A(z), B(z), C(z), D(z) are meromorphic functions on C, analytic on R.
Proof. Reasoning as in [15, Ch. 42, Sec. 7.1], one obtains the stated formula by inserting the right-hand side expression from (2.5) into κ, R τ (z)κ and considering the functions: where ψ is given in (2.4). One verifies from their definition that A(z), B(z), D(z) can be analytically extended to any point z outside the set given in (2.8). The function C(z) has a priori poles in spec(A ∞ ), but by an argument along the lines of [15, Ch. 2 Secs. 7.3 and 7.4], one concludes that C(z) can be analytically extended to any point z outside the set given in (2.8). [42,Thm. 4.9]. Note that, for the Herglotz function −1/τ , [42,Eq. 4.36] is indeed an expression for κ, R −1/τ (z)κ .

Remark 3.5. For Jacobi operators, the functions A(z), B(z), C(z), D(z) coincide with the ones given in [1, Eq. 2.28] and
Reasoning along the lines of the proof of [42,Thm. 4.17], one obtains the following assertion from Theorem 3.4. Theorem 3.6. Let κ be a spectral gauge of A ∈ S(H) and λ a real number. If

A generalization of the moment problem
There are various ways of generalizing the moment problem. We use the generalization proposed by Livsič [28] (see also [44]). According to [28], the generalized moment problem consist in finding a measure ρ such that the inner product in a Hilbert space of functions is expressed through the inner product in L 2 (R, ρ). Our choice for the Hilbert space of functions is a de Branges space. In [36,Chap. 6], the same choice is done for the generalized moment problem and the treatment is made by means of canonical systems whose theory contains the one of de Branges spaces (see [37]). In contrast to [36,Chap. 6], our final goal is the extremal properties of measures. (A2) for every non-real zero w of f (z), the function f (z)(z−w)(z−w) −1 belongs to B and has the same norm as f (z); (A3) the function f # (z) := f (z) also belongs to B and has the same norm as f (z).
Alternatively, a de Branges space can be defined in terms of an Hermite-Biehler function, that is, an entire function e(z) such that |e(z)| > |e(z)| for all z ∈ C + [25,Ch. 7]. In that case, the de Branges space is sometimes denoted B(e) and one has |e(x)| 2 dx .
It is noteworthy that the operator S is in S(B) [ Since A ∈ S(H), one has dim ker(A * − zI) = 1 for all z ∈ C. Using this fact, one can prove the following assertion (see [ There is a standard way of constructing a function ξ A (z) with the desired properties: Pick a canonical selfadjoint extension A γ of A ∈ S(H) and let h γ (z) be a real entire function whose zero set (counting multiplicities) equals spec(A γ ); the existence of this function follows from classical theorems on entire functions, see [30,Ch. 7 Sec. 2]. Then, define  [41]. Clearly, the linear space Φ A H is turned into a Hilbert space by defining The resulting Hilbert space is a de Branges space which is henceforth denoted by B A [39, Prop. 2.14].

Remark 4.4. If
The reproducing kernel in the de Branges space Note that k(z, w) is anti-entire with respect to its second argument.

Remark 4.5.
It is not difficult to verify that k(·, w) ∈ ker(S * − wI) for all w ∈ C and moreover where S γ is a canonical selfadjoint extension of S given in Definition 4.2 and h(w) is a real entire function having zeros at spec(S γ ) (cf. (2.4) and (4.2)). Proof. By our definition of spectral gauge, ξ(z), κ = 0 for any z ∈ R. The second part of the statement follows from noticing that k(·, w) ∈ ker(S * − wI) for all w ∈ C.
The following definition coincides with [36, Def. 6.1]. We came to it motivated by a generalization of the moment problem discussed in [28] (see also [44]). Several results in [9] pertain to this definition.
for every f, g ∈ B.
As (4.1) shows, the Lebesgue measure multiplied by 1/ |e(·)| 2 is always a solution to the generalized moment problem for B(e). Below, it will be established that there are other solutions to this generalized moment problem. Proof. Fix t ∈ R and define which is a sesquilinear form in B ×B. This form is bounded due to the inequality It turns out that F (t) corresponds to a generalized spectral family in the sense of Proposition 2.6. Indeed, (i)-(iii) are verified directly from the definition. As regards (iv), one verifies Thus, for any real Borel set ∂, since m is a spectral gauge of S by Lemma 4.6. Lemma 4.9. Let m ∈ B be such that m(x) = 0 when x ∈ R and a, b ∈ R such that a < b. If F is a generalized spectral family of S, then for any f ∈ B.
Proof. According to Remark 4.5, since m(z) is a spectral gauge of S, the righthand side of (2.9) in this case is The assertion then follows from Proposition 2.16.
is a solution to the generalized moment problem for B.
Proof. By the hypothesis and Lemma 4.9, for any real Borel set ∂. It follows from Lemma 4.9 that Putting the last two equalities together, one obtains For finishing the proof take the limit when a → −∞ and b → +∞. where E γ is the spectral family of S γ , then for any f in L 2 (R, σ) and λ ∈ spec(S γ ).
Proof. Using the spectral theorem, From the last expression the result follows due to the properties of the reproducing kernel. Proof. Since ρ is a solution to the generalized moment problem, ΨB is contained in L 2 (R, ρ), where Ψf = f ↾ R and Ψ is isometric. The hypothesis {f ↾ R : f ∈ B} = L 2 (R, ρ) therefore means that Ψ is unitary. Let A ρ be the operator of multiplication by the independent variable in L 2 (R, ρ) defined in the maximal domain. Since A ρ is selfadjoint, the same holds for and for any function where f and g are the images of f and g under the canonical map. On the basis of Lemma 4.11, f (t) = f(t)m(t) and g(t) = g(t)m(t) with the equalities in the Since this equality holds for any element in L 2 (R, ρ), one concludes that η(∂) = σ(∂) for any real Borel set ∂ and hence σ is extremal, that is, σ ∈ V ext m (S). Let us prove the other direction. If σ ∈ V ext m (S), then σ is the spectral measure of a canonical selfadjoint extension of S which is known to be simple. By the canonical map (see [2, Sec. 69]), B is unitarily equivalent to L 2 (R, σ). In turn, since m is zero-free on the real axis, L 2 (R, σ) is unitarily equivalent to L 2 (R, ρ). On the other hand, the Hilbert spaces B and ΨB ⊂ L 2 (R, ρ) are unitarily equivalent due to the fact that ρ is a solution to the moment problem. Therefore, ΨB cannot be properly contained in L 2 (R, ρ).  and, on the other hand, one has the following interpolation formula for every f ∈ B, where the convergence is in the sense of Hilbert space, hence uniform in compact subsets of C. Thus, if ρ γ is given by (4.3), then every f in L 2 (R, ρ γ ) satisfies (4.4) and, through the interpolation formula (4.5), it is mapped into a function in B whose restriction to the real axis is the function f . The identity implies that Ψ is norm-preserving. Conversely, if ρ is an extremal solution, then supp ρ coincides with spec(S γ ) for some γ as a consequence of Theorem 4.12. Thus, for any f in L 2 (R, ρ) whose zero set is spec(S γ ) \ {λ}, one has which shows that the weight c of the measure ρ at λ should be equal to 1/k(λ, λ).

Remark 4.15.
Assume that A ∈ S(H) and take the measure ρ γ as defined in Theorem 4.14 for the corresponding de Branges space B A . The multiplication by the independent variable in L 2 (R, ρ γ ) is unitarily equivalent to a canonical selfadjoint extension A γ of A. Note that one can write It is worth remarking that in the Sturm-Liouville theory (when A is associated with a regular Sturm-Liouville difference expression) the measure ρ γ given above is usually defined via the Weyl function and is called the spectral measure of the selfadjoint operator A γ (see details in Example 6.2).

Point mass perturbations of measures
Lemma 5.1. Let B be a de Branges space with inner product ·, · B . Given a > 0 and λ ∈ R, define g, f ∼ := g, f B + a g(λ)f (λ), f, g ∈ B. (5.1) Then the linear manifold B equipped with the inner product ·, · ∼ is a de Branges space.
implying the assertion. Now we prove (A1)-(A3) of Definition 4.1. Since it follows that point evaluation is continuous with respect to the ∼-norm.
From the equality Finally, suppose w ∈ C \ R is a zero of f ∈ B and define g(z) := (z − w)(z − w) −1 f (z). Then, g ∈ B and Lemma 5.1 has the following corollary. moment problem for B, then ρ + aδ λ (a > 0, λ ∈ supp ρ) is a solution to the generalized moment problem for a de Branges space B ∼ having the same elements as B but with inner product given by (5.1).

Corollary 5.2. Let B be a de Branges space. If ρ is a solution to the generalized
Proof. The statement follows from the equality

Theorem 5.3. Let S be the operator of multiplication by the independent variable in a de Branges space B and m be a function in B not vanishing in
where S is the multiplication operator in the de Branges space B ∼ given in Corollary 5.2.
Proof. By hypothesis, ρ given in Theorem 4.10 is a solution to the generalized moment problem for B. By Corollary 5.2, ρ+ aδ λ is a solution to the generalized moment problem for B ∼ and Theorem 4.8 implies that On the other hand, Corollary 3.3 yields a measure σ λ ∈ V ext m (S) such that σ λ {λ} > 0. The function ρ λ related to σ λ as in Theorem 4.10 is a solution to the generalized moment problem for B and recurring again to Corollary 5.2, one concludes that ρ λ + aδ λ is a solution to the generalized moment problem for B ∼ . Thus, in view of Theorem 4.8, Since λ ∈ supp(σ λ ) \ supp(σ), one has Assume that σ + a |m(λ)| 2 δ λ is extremal. If σ λ + a |m(λ)| 2 δ λ is extremal, then a contradiction arises from Corollary 3.3. If σ λ + a |m(λ)| 2 δ λ is not extremal, then, using Theorem 3.6, one obtains an extremal measure giving weight to λ so again we have a contradiction by Corollary 3.3. Therefore σ + a |m(λ)| 2 δ λ is not in V ext m ( S). Remark 5.5. Before proving the assertion, there are two points to comment. First, the measure ρ+aδ λ is not necessarily a solution of the generalized moment problem for B ′ . Second, the particular case when in the hypothesis ρ is an extremal solution has interesting applications (see Corollary 5.7 and Section 6).
Proof. Assume that ρ + aδ λ is an extremal solution to the generalized moment problem for B ′ . Thus, By hypothesis, Therefore the inner products of B and B ′ are related as in (5.1). Now, since ρ and ρ + aδ λ generate via Theorem 4.8 spectral measures of the operators of multiplication in B and B ′ , one obtains a contradiction from Theorem 5.3.

Remark 5.6.
There is an alternative proof of Theorem 5.4. Indeed, assume that ρ + aδ λ is an extremal solution of the moment problem for B ′ and consider a function g ∈ L 2 (R, ρ + aδ λ ) whose zero set is supp ρ. Thus, g L 2 (R,ρ+aδ λ ) = 0 and, since Ψ is surjective and norm preserving, the function Ψ −1 g is a nonzero element of B ′ and therefore a nonzero element of B. But we get a contradiction since the norm of the restriction to the real line of Ψ −1 g has zero norm in L 2 (R, ρ).

Corollary 5.7. Let ρ be a solution to the generalized moment problem for a de Branges space
Proof. The hypothesis implies that ρ is an extremal solution, i. e. ΨB = L 2 (R, ρ). As in Corollary 5.2, let B ∼ be the de Branges space having the same elements of B with the inner product given by (5.1). Then ΨB ∼ , which contains A, is not dense in L 2 (R, ρ + aδ λ ) as a consequence of Theorem 5.4.

Remark 5.8.
Note that a function is square-integrable with respect to ρ if and only if it is square-integrable with respect to ρ + aδ λ . Nevertheless, there functions in the equivalence class of zero in L 2 (R, ρ) which have nonzero norm in L 2 (R, ρ).

The classical moment problem
According to [1, Thm. 2.1.1], for a given real sequence {s k } ∞ k=0 , there exists a Borel measure ρ such that if and only if, for all k ∈ N, The measure is said to be a solution to the moment problem given by the sequence {s k } ∞ k=0 . We refer to all sequences {s k } ∞ k=0 satisfying (6.1) and normalized so that s 0 = 1 as sequences of moments. To any sequence of moments there corresponds one and only one Jacobi matrix where {q k } ∞ k=1 is a sequence of real numbers and {b k } ∞ k=1 is a sequence of positive numbers (see [1, paragraph after Eq. 1.8] and [42, p. 93]). Given a separable Hilbert space H and an orthonormal basis {δ k } ∞ k=1 , with any matrix (6.2), one can uniquely associate a closed symmetric operator [2,Sec. 47]. This operator, denoted by A and called minimal Jacobi operator, has either deficiency indices n + (A) = n − (A) = 0 or n + (A) = n − (A) = 1 [5, Ch. 7, Thm. 1.1]. When the Jacobi operator has deficiency indices (0, 0) the matrix (6.2) is said to be in the limit point case, otherwise the matrix (6.2) is said to be in the limit circle case. Due to the bijection between Jacobi matrices and sequences of moments, the limit point and limit circle dichotomy corresponds to the determinate and indeterminate dichotomy for sequences of moments.
It is established in [1, Ch. 4] that s k = δ 1 , A k δ 1 for k = 0, 1, . . . For a Jacobi operator A ∈ S(H), the function ξ A defined in (4.2) is given by ξ A (z) = ∞ k=1 P k−1 (z)δ k , where P k is the k-th orthogonal polynomial of the first kind associated with (6.2) (see [ Note first that, since the finite linear combinations of the basis {δ k } ∞ k=1 are dense in H, the polynomials are dense in B A . This is a peculiarity of de Branges spaces generated by Jacobi operators. Also, Proof. Assume that ρ is a solution to the generalized moment problem for B A . Then, in view of (6.3) and (6.4), one has which means that ρ is a solution to the moment problem given by {s k } ∞ k=0 . Now, suppose that ρ is a solution to the moment problem given by {s k } ∞ k=0 . Hence, by (6.3), one has R t k dρ(t) = δ 1 , A k δ 1 for k = 0, 1, . . .

One then verifies that if R(t) is a polynomial, then
But, due to (6.4), For completing the proof one uses the fact that the polynomials are dense in B A and the polarization identity.
In the context of the classical moment problem, Theorem 5.3 corresponds to [10, Prop. 4.1(a)], which says that if ρ is an extremal solution to an indeterminate moment problem, then ρ + aδ λ (a > 0, λ ∈ supp ρ) is not extremal although is a solutions of a moment problem. Thus, according to Corollary 5.7, the density of the polynomials in L 2 (R, ρ) no longer holds in L 2 (R, ρ + aδ λ ). Note that the sequences of moments for which ρ and ρ + aδ λ are different, therefore the corresponding Jacobi operators are different and the corresponding de Branges spaces are different. However, the associated de Branges spaces are set-wise equal, which is actually a consequence of the fact that the set of polynomials are dense in these two spaces, as well as the argument in the proof of Lemma 5.1.
We conclude this example by pointing out that it was probably first mentioned in [14,Remark 4.5 (iii)] that the set of points obtained by adding a point to the spectrum of a selfadjoint extension of a discrete Schrödinger operator in the limit circle case is no longer the spectrum of a selfadjoint extension of a discrete Schrödinger operator in the limit circle case (cf. [13,Sec. 7 Example 2] and [26,Chap. 2 Sec. 7]).

Schrödinger operators with measures
The following example is based on [4] (see also [29]). Let us consider the differential expression given informally by where µ is a signed Borel measure on [0, b]. Properly, for a function ϕ in AC[0, b] we define ϕ [1] The derivative of an element ϕ ∈ dom(τ µ ) has a (unique) representative for which (6.5) holds for every x ∈ [0, b]; in what follows ϕ ′ will denote this particular representative. From [4,Thm. 2.4] it follows that ϕ ′ may have discontinuities.
Let A denote the symmetric operator in L 2 (0, b) given by The boundary condition ϕ [1] (0) = 0 is just the usual one ϕ ′ (0) + hϕ(0) = 0 with h = −µ({0}). On the other hand, the boundary condition at b is consequence of (6.7) plus the fact that A is the closure of the minimal operator. Standard arguments yields that A has deficiency indices (1, 1) and ξ(z, ·) ∈ ker(A * − zI) for all z ∈ C. Since Green's identity holds true for this kind of operators [4,Thm. 2.2], its canonical selfadjoint extensions are defined in the usual way, The de Branges space associated to A is given by is an Hermite-Biehler function as shown in [4,Prop. 4.1]. Thus, since A is unitarily equivalent to the multiplication operator in B A , A ∈ S(L 2 (0, b)).
The solutions of the generalized moment problem for B A correspond to the generalized spectral measures of A. Let us consider here the extremal solutions of the generalized moment problem. According to Theorem 4.14 and Remark 4.15, any extremal solution is given by is a proper subset of L 2 (R, ρ).
Proof. According to [4,Thm. 4.4], the set of functions in the de Branges space B A coincides with the linear set for any signed Borel measure supported in [0, b]. Then, assertion (a) is a direct consequence of Theorem 5.4 while (b) follows from Theorems 4.12 and 5.3.

Remark 6.3.
We note that regular Schrödinger operators are included in Theorem 6.2 as they correspond to signed Borel measures µ that are absolutely continuous with respect to the Lebesgue measure, i.e., For these cases, assertion (a) of Theorem 6.2 can be also shown using an elementary argument based on the fact that, the eigenvalues of a regular Schrödinger operator with separated boundary conditions (Dirichlet case excluded) obey an asymptotic formula of the form for some c > 0. Clearly, the addition of a point to the spectrum amounts to shifting the enumeration by 1, producing an asymptotic formula with an additional non trivial term linear in n not present in (6.12). An analogous reasoning holds for the Dirichlet case. We remark that Theorem 6.2 holds for a wide class of perturbations of the regular Laplacian for which we do not have at our disposal an asymptotic formula of the kind of (6.12) for the spectra of the corresponding selfadjoint extensions.
The example presented here is also related via [35,Sec. 17] to the Gelfand-Levitan theory for half-line Schrödinger operators. Let us discuss this in detail. In [13] and [35,Sec. 17] the spectral measure of the half-line Schrödinger operator is defined as follows. Suppose in (6.11) that q ∈ L 1,loc ([0, ∞)) (with fixed boundary conditions at 0) and take into account that ξ(x, z) is the same for any right endpoint b ∈ R. A spectral measure of the half-line Schrödinger operator is a measure ρ such that the Parseval identity holds for every f = Φ A ϕ ∈ b>0 B A (see (6.9) and compare with [13,Eq. 2] and [35,Thm. 3.2]). The classical Weyl theory tells us that a Schrödinger expression has exactly one spectral measure if q is in the limit point case at infinity, otherwise it has an infinite set of spectral measures. We remark that in the limit circle case, the definition of spectral measure given above includes measures which do not correspond to selfadjoint operators in L 2 (0, b) (see in [13,Sec. 8] complementary conditions for a spectral measure to correspond to selfadjoint extensions). The central result in Gelfand-Levitan theory is a theorem on necessary and sufficient conditions for a measure to be a spectral measure of a half-line Schrödinger operator, that is, the necesssary and sufficient conditions on ρ such that there is a potential function q ∈ L 1,loc ([0, ∞)) ensuring that (6.13) holds. Due to the nonlocality of the conditions (see [13,Introduction]), there is a potential such that a point mass perturbation of a spectral measure is the spectral measure of the half-line Schrödinger operator with that potential and same boundary condition at 0 (see [13,Sec. 7 Example 2] and [26, Chap. 2 Sec. 7]).
We emphasize that the last statement does not contradict Theorem 6.2 since the definition of spectral measure in the Gelfand-Levitan theory is different than the one considered in this work. We believe that a connection between these different notions of a spectral measure can be established by resorting to the theory of chains of de Branges spaces developed in [48]. This remains to be done.
To conclude this example, let us revisit the assertion of Theorem 6.2(a) at the light of [4,Secs. 5 and 6]. Since now it is relevant to keep track of the signed measure defining the operators, let us denote by A(µ) the operator given in (6.8). By [4, Thms. 5.2, 6.1 and 6.2], we know that there is a bijective correspondence between the set of signed Borel measures µ in [o, b], and the set of even functions v on (−2b, 2b) of bounded variation such that v(0) = −µ({0}) and the operator By this correspondence, one has Proof. In view of Theorem 6.2(a), it only remains to prove the existence of a signed Borel measure ν such that, for the corresponding space B A(ν) , the equality holds. From the linearity of K v as a map on the function v, we can assume without loss of generality that ρ γ is given by (6.10) for a selfadjoint extension of A(0). Thus, According to [4, Thm. 6.1] (see above), the proof will be established, once we find an even function v on (−2b, 2b) of bounded variation such that However, using the notation introduced in Section 4, So K v must be equal to s ξ A(0) (λ), · ξ A(0) (λ). Therefore, using the fact that ξ A(0) (λ, ·) = cos( √ λ ·), we have which yields v(·) = s cos( √ λ ·).
Note that, since ν({0}) = v(0), the point mass perturbation of a extremal spectral measure of a regular Schrödinger operator is a (non extremal) spectral measure of a Schrödinger operator with measure-valued potential ν such that ν({0}) = 0.
The corresponding canonical selfadjoint extensions A γ are defined as usual, For every γ ∈ [0, π), the spectrum of A γ , beside of being simple and discrete, has at most a finite number of negative eigenvalues [18,Thm. 2.4]. By [18,Lemma 2.2], the eigenvalue equation τ ϕ = zϕ (z ∈ C) admits a solution ξ(z, x), real entire with respect to z, with derivative ξ ′ (z, x) also real entire. Moreover, ξ(z, x) also obeys boundary condition (6.15) whenever ν ∈ (0, 1). This in turn implies ξ(z, ·) ∈ ker(A * − zI) for all z ∈ C. Hence, the associated de Branges space is Also, the extremal solutions of the corresponding generalized moment problem are given by , (6.16) where A γ is any selfadjoint extension of A. We now have the following statement analogous to Theorem 6.2.
Theorem 6.5. Suppose ν > 0, and q ∈ L 1,loc (0, b) such that q ∈ L r (0, b) for some r ∈ (2, ∞]. Let A be the Bessel operator defined by τ q as in (6.8). Let A γ an arbitrarily chosen selfadjoint extension of A and ρ γ the corresponding spectral measure given in (6.16). If ρ = ρ γ + sδ λ with λ ∈ spec(A γ ) and s > 0, then: (a) There is no p ∈ L 1,loc (0, b) with p ∈ L r (0, b) for some r ∈ (2, ∞] such that ρ is the spectral measure corresponding to selfadjoint extensions of A generated by τ p . (b) The set of entire functions is a proper subset of L 2 (R, ρ), where J ν denotes the Bessel function of the first kind.
Proof. The assertions follow from Theorem 4.12, Theorem 5.3 and Corollary 5.4, this time combined with Theorem 4.2 of [40].
Remark 6.6. The requirement on q stated in Theorem 6.5 is a technical limitation related to the perturbative argument used in the proof of [40,Thm. 4.2].
The technique used there can be easily modified to include r = 2 but breaks down for r ∈ [1, 2).
We emphasize that our theory does not rule out the possibility that ρ is the spectral measure of a Bessel operator for some different value of the parameter ν. In this respect, it is worth noticing the following: Assuming r ≥ 2 and resorting to [18,Thm. 2.5], one can show that the eigenvalues of A γ obeys the asymptotic formula λ n = π 2 b 2 (n + κ ν ) 2 + O(n 1/2 ), n → ∞, κ ν := This implies that, contrary to the case of regular Schrödinger operators, the addition of a point to the spectrum is still compatible with (6.17) but for a different value of ν, namely, for ν = ν + 2.