{"id":"3ca04561-15f4-4aac-8cd0-a443bba2bb8e","arxiv_id":"2505.08338","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims an infinite-dimensional de Branges space can be associated with a semi-infinite Jacobi matrix in the limit circle case, via the semi-infinite connecting operator.","lead":"This paper tries to extend a known construction of de Branges spaces, special spaces of analytic functions used in inverse spectral theory, to dynamical systems given by semi-infinite Jacobi matrices. It compares the resulting connecting operator with classical Hankel matrices from moment problems, but a central estimate relies on a false orthogonality statement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proofs of Theorems 7 and 8 and the infinite-dimensional construction B_∞^A rest on a false isometry: T_k are not orthogonal with respect to dν = χ_{(-1,1)}(λ)/√(1−λ²)dλ, so the central claims are not established as written.","rationale":"The reader's weakest assumption identifies the same false orthogonality statement that is the principal defect. Independent computation confirms T_1 and T_3 have nonzero L²(dν) inner product despite the paper's claim. The finite-dimensional machinery in Sections 4–5 and Lemma 1 appear sound and are not the issue. However, the main new results (Theorem 7, Theorem 8, and the construction of B_∞^A with kernel J∞_z) all depend on the incorrect ℓ²–L²(dν) isometry. The paper does not provide an alternative proof that avoids this step. Therefore rejection remains appropriate: the central claims are not established as written, although the paper contains useful background and a plausible framework. No ad hominem is intended; the critique is strictly about the missing/untrue orthogonality relation and the proof steps that rely on it.","tokens_in":13051,"tokens_out":4915,"duration_ms":43278,"concrete_test":"Compute the Gram matrix G_{mn}=∫_{-1}^1 T_m(λ)T_n(λ)(1−λ²)^{-1/2}dλ for 1≤m,n≤4; the entry G_{13}=−π/2 is nonzero, while G_{12}=G_{14}=0, so the asserted orthogonality fails. Then replace the false identity in Eq. (28) with the true Gram matrix and check whether the optimization max{∑G_{ij}... } equals the claimed ℓ² normalization; it does not, which invalidates the passage from Eq. (28) to Eq. (29).","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 6.1, before Eq. (28), the paper asserts that the polynomials T_k defined by T_{t+1}+T_{t-1}−λT_t=0, T_1=1, are orthogonal with respect to dν(λ)=χ_{(-1,1)}(λ)/√(1−λ²)dλ, and concludes ∥f∥²_{ℓ²}=∫|F(λ)|²dν for F(λ)=∑f_kT_k(λ). This is false. For T_1=1 and T_3=λ²−1, ∫ T_1T_3 dν = ∫_{-1}^1 (λ²−1)/√(1−λ²)dλ = −π/2 ≠ 0. The true orthogonality for these polynomials is on (−2,2) with weight √(4−λ²) (up to normalization), not on (−1,1) with the reciprocal square-root weight. This identity is load-bearing: Eq. (28) converts the ℓ² normalization of the control into the L²(dν) normalization of F and thereby derives the variational bound on β_T; Theorem 8 uses the same identity to pass from f^{(n)}→0 in ℓ² to F^{(n)}→0 in L²(−1,1;dν). Without it, the optimization in Eq. (28) is not equivalent to Eq. (29), the trace estimate yielding Theorem 7 does not follow, and the closability argument in Theorem 8 loses its premise. The definition of B_∞^A and its claimed reproducing kernel J∞_z, which depend on C[f,f]=∫FG dM and on the same ℓ²–L² link, are consequently unsupported. The failure is internal to the proof, not merely a disagreement with standard convention.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies discrete-time dynamical systems associated with finite and semi-infinite Jacobi matrices and proposes to associate special Hilbert spaces of analytic functions, specifically de Branges spaces, with these systems. For finite matrices the construction was developed in the authors' earlier work, and the present paper reviews that construction, connects it with classical moment problems and Hankel matrices, and then attempts to extend it to the semi-infinite (limit circle) case. The main new claims are Theorem 7, a lower bound for the minimal eigenvalue of the finite connecting operators C_T in the indeterminate case; Theorem 8, closability of the quadratic form C[·,·] in the limit circle case and in a bounded case; and the construction of an infinite-dimensional space B∞_A with reproducing kernel J∞_z(λ)=Σ p_n(z)p_n(λ), claimed to be a de Branges space. The paper also contains a useful comparison of properties of the connecting operator matrices and classical Hankel matrices, including the relation C_T=Λ_T S_T Λ_T^* and a lemma linking boundedness of the largest eigenvalue of C_T to the limit point case.","tokens_in":13471,"tokens_out":7704,"duration_ms":76866,"significance":"If the central construction were correct, the paper would give a dynamical, control-theoretic realization of de Branges spaces for semi-infinite Jacobi matrices, going beyond the finite-dimensional case treated in the authors' earlier work, and would establish a new connection between the connecting operators of the boundary control method and the classical moment problem. Some parts of the paper are valuable independently: the review of finite-dimensional results is coherent, the relation C_T=Λ_T S_T Λ_T^* is explicit and useful, and Lemma 1 is a clean and correct argument using the same relation to detect the limit point case. However, the new infinite-dimensional claims rest on a false orthogonality identity for the polynomials T_k, and this error is load-bearing in the proofs of Theorems 7 and 8 and in the construction of B∞_A. As a result, the principal advertised results are not established as written, and the error is internal to the derivation rather than a matter of convention or presentation.","major_comments":[{"comment":"The proof asserts, immediately before Eq. (28), that the polynomials T_k satisfying T_{t+1}+T_{t-1}-λT_t=0 with T_1=1 are orthogonal with respect to dν(λ)=χ_{(-1,1)}(λ)/√(1-λ²)dλ, so that ∥f∥²=Σ|f_k|²=∫|F(λ)|²dλ/√(1-λ²). This identity is false. For T_1=1 and T_3=λ²-1, ∫_{-1}^{1} T_1(λ)T_3(λ)dλ/√(1-λ²)= -π/2, not 0. The true orthogonality for these polynomials holds on (-2,2) with weight √(4-λ²), not on (-1,1) with the reciprocal square-root weight. Since this identity is used to convert the ℓ² normalization of the control f into the L²(dν) normalization of F and hence to obtain the variational problems (28) and (29), the optimization problem is not equivalent as claimed, and the trace estimate leading to the lower bound (26) is not established.","section":"Section 6.1, proof of Theorem 7"},{"comment":"The closability argument uses the same false identity to pass from f^{(n)}→0 in ℓ² to F^{(n)}→0 in L²(-1,1;dλ/√(1-λ²)). This step is essential: without it, the contradiction argument for the limiting function F in the definition of closability has no premise, and the conclusion that F=0 does not follow. The subsequent line 'F=0 in L²(-1,1;dρ), so F=0 on (-1,1)' also mixes up the measures; the intended measure appears to be dν, not dρ. Even after correcting this notational slip, the key convergence assertion remains false because the polynomials T_k are not orthogonal with respect to dν. Thus the closability of the quadratic form C[·,·] in the limit circle case (Theorem 8(a)) is not proved.","section":"Section 6.1, proof of Theorem 8"},{"comment":"The space B∞_A is defined as {Σ f_k T_k(λ) : C[f,f]<∞} with scalar product C[f,g]=∫F(λ)G(λ)dM, and it is asserted that the conditions of Theorem 1 verifying that B∞_A is a de Branges space are 'trivially checked'. This is not a proof. One must show that B∞_A is a Hilbert space of entire functions, that the map from controls f with C[f,f]<∞ to functions is injective and complete with respect to the C-norm, that the integral representation equals the dynamic norm for all such f, and that the reproducing kernel is indeed given by J∞_z(λ)=Σ p_n(z)p_n(λ). These properties depend on the same faulty ℓ²-L²(dν) link and on limiting arguments that are not supplied. Hence the central infinite-dimensional construction, which is the advertised result of the paper, is unsupported.","section":"Section 6.1, definition of B∞_A and the de Branges space claim"}],"minor_comments":[{"comment":"The polynomials T_k are called Chebyshev polynomials of the second kind, but with T_0=0, T_1=1 they are the shifted Chebyshev polynomials U_{k-1}(λ/2), not the standard U_k(λ). This nomenclature is likely the source of the wrong orthogonality interval and weight used in Section 6.1.","section":"Section 4, Eq. before Proposition 1"},{"comment":"The notation 'L²(-1,1;dρ)' should be 'L²(-1,1;dν)' where dν(λ)=dλ/√(1-λ²), and the statement 'F=0 on (-1,1)' should be stated in the measure sense.","section":"Section 6.1, proof of Theorem 8"},{"comment":"There is a typo: 'consequntly' should be 'consequently'.","section":"Remark 5"},{"comment":"The quantity l^{-1}(z)=Σ|p_k(z)|² appears as a pointwise series; the paper does not discuss convergence of this series for real z, which is relevant because the integral in (26) is over (-1,1).","section":"Section 6.1, Theorem 7 statement"}],"recommendation":"reject","confidential_remarks":"The paper is honest in its presentation and builds on the authors' prior work without circularity. The decisive issue is a concrete mathematical error in a central identity, not a disagreement over interpretation or a stylistic flaw. The error cannot be repaired locally: replacing the measure would change the theorems, and the dynamic definition of the polynomials does not allow one to choose a different orthogonality measure. I therefore recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nQuick take: the paper's genuinely new machinery—semi-infinite connecting operator, the dynamic eigenvalue bound, the determinacy criterion, and the proposed B∞_A space—does not hold together as written, because a load-bearing orthogonality claim about Chebyshev polynomials is false.\n\nThe authors extend their earlier finite-dimensional boundary-control construction to semi-infinite Jacobi matrices in the limit circle case. The setup is clear: they define C formally, relate it to the Hankel matrix S via C_T = Λ_T S_T Λ_T^*, and use the connecting operator to build a space of analytic functions. Lemma 1, which derives the limit point case from a uniform upper bound on the eigenvalues of C_T, is independent of the flawed step and looks sound. The comparison with Berg-Szwarc and Yafaev on Hankel matrices is a useful framing. The finite-dimensional de Branges construction in Section 5 repeats their prior work and is fine.\n\nThe problem is in Section 6.1, just before Eq. (28). The proof of Theorem 7 asserts that the polynomials T_k, defined by T_{t+1}+T_{t-1}−λT_t=0 with T_1=1, are orthogonal with respect to dν = χ_{(−1,1)}(λ)/√(1−λ²)dλ. That is false. These are Chebyshev polynomials of the second kind (up to scaling of the argument) and they are orthogonal on (−2,2) with weight √(4−λ²), not on (−1,1) with the reciprocal square-root weight. Concretely, ∫ T_1(λ)T_3(λ)dν = -π/2. The identity ∥f∥²_{ℓ²} = ∫|F|²dν fails, so the variational reformulation in Eq. (28)–(29) does not follow, and the trace bound on β_T has no proof. Theorem 8 uses the same ℓ²-to-L² step to prove closability, so that proof also collapses. The B∞_A construction depends on the same relation, so the claimed de Branges space and reproducing kernel are unsupported. The final 'conditions of Theorem 1 are trivially checked' is too quick given that the norm is not established.\n\nThis is not a minor blemish; the main abstract claims rest on it. But the paper is not absurd: the program is coherent, the authors engage the literature honestly, and the error is identifiable and might be repairable with a different expansion or by handling the non-orthogonality.\n\nWho is this for? Anyone working in inverse spectral theory for Jacobi matrices and de Branges spaces, especially the BC-method community. It deserves a serious referee—the flaw should be caught and the revision path evaluated—but as submitted I would not accept it.\n\nBest regards.","headline":"The paper's new semi-infinite de Branges construction and its main theorems rest on a false Chebyshev orthogonality, so the central claims fail as written; the finite-dimensional part and Lemma 1 are sound.","tokens_in":13981,"tokens_out":3966,"would_cite":false,"duration_ms":35934,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B36","44A60","46E22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A semi-infinite Jacobi matrix in the limit circle case determines an infinite-dimensional de Branges space built from its connecting operator.","keywords":["Jacobi matrices","semi-infinite Jacobi matrices","de Branges spaces","boundary control method","moment problems","Hankel matrices","connecting operator","limit circle case"],"falsifier":"Compute $\\int_{-1}^{1} T_1(\\lambda)T_3(\\lambda)\\,\\frac{d\\lambda}{\\sqrt{1-\\lambda^2}}$ with $T_1=1$ and $T_3=\\lambda^2-1$. The integral equals $-\\pi/2$, not $0$, so the claimed orthogonality of the $T_k$ with respect to $d\\nu$ is false; consequently the norm identity $\\|f\\|_{\\ell^2}^2=\\int |\\sum f_kT_k|^2 d\\nu$ fails, and the proof of the lower bound (26) in Theorem 7 needs a different justification.","tokens_in":12865,"feed_emoji":"📐","tokens_out":9052,"duration_ms":82253,"temperature":0.7,"pith_summary":"This paper tries to extend the authors' boundary-control construction of de Branges spaces from finite Jacobi matrices to semi-infinite ones. In the limit circle (indeterminate) case, where the Jacobi operator has deficiency indices $(1,1)$ and the associated moment problem has many measures, the paper defines a Hilbert space $B_{\\infty}^{A}$ of entire functions $F(\\lambda)=\\sum_{k=1}^{\\infty} f_k T_k(\\lambda)$ by declaring the formal connecting operator $C$ to provide the norm, $[F,F]_{B_{\\infty}^{A}}=C[f,f]$. The claim is that this space has the reproducing kernel $J_z^{\\infty}(\\lambda)=\\sum_{n=1}^{\\infty} p_n(z)p_n(\\lambda)$ and satisfies the axioms of de Branges space theory, so it is a genuine infinite-dimensional de Branges space attached to the Jacobi matrix. This matters because it gives a dynamical, control-theoretic route to de Branges spaces for semi-infinite Jacobi matrices, in parallel with the classical moment-problem and Hankel-matrix analysis, and it connects the minimal eigenvalues of the finite connecting operators with the limit point/limit circle dichotomy.","feed_headline":"Limit-circle Jacobi matrices yield infinite de Branges spaces","feed_subtitle":"The same boundary-control recipe that works for finite matrices survives the infinite limit.","key_machinery":"The central object is the connecting operator $C_T$: the Gram matrix of the reachable set $\\{u^f_{\\cdot,T}\\mid f\\in F_T\\}$ of the discrete dynamical system (2), with entries $\\{C_T\\}_{ij}=\\sum_{k=0}^{T-\\max\\{i,j\\}} r_{|i-j|+2k}$ built from the response vector $(r_0,r_1,\\ldots)$; its infinite counterpart $C=(W)^*W$ supplies the norm on $B_{\\infty}^{A}$. The bridge to the classical moment problem is the identity $C_T=\\Lambda_T S_T \\Lambda_T^*$, where $S_T$ is the truncated Hankel matrix and $\\Lambda_T$ is the change of basis from monomials to the Chebyshev polynomials of the second kind $T_k(\\lambda)$. The proof of Theorem 7 converts the variational formula for $\\beta_T$ into a Gram-matrix problem for the polynomials $p_k$, using the claimed orthogonality of the $T_k$ against $d\\nu(\\lambda)=\\chi_{(-1,1)}(\\lambda)/\\sqrt{1-\\lambda^2}\\,d\\lambda$; Theorem 1, the de Branges classification criterion, is then the test that certifies $B_{\\infty}^{A}$ as a de Branges space.","core_discovery":"On the paper's own terms, the discovery is that the finite-time construction of de Branges spaces via reachable sets survives the passage $T\\to\\infty$ exactly in the indeterminacy regime. The main object is the semi-infinite connecting operator $C=(W)^*W$, whose finite truncations $C_T$ are Gram matrices of the reachable set of the discrete wave equation (2). Theorem 7 asserts that in the limit circle case the minimal eigenvalue $\\beta_T$ of $C_T$ is bounded below by a positive constant, whose reciprocal is bounded by $\\int_{-1}^{1} \\ell^{-1}(\\lambda)\\,d\\lambda/\\sqrt{1-\\lambda^2}$ with $\\ell^{-1}(\\lambda)=\\sum_{k=0}^{\\infty}|p_k(\\lambda)|^2$; Theorem 8 asserts that the quadratic form $C[f,f]$ is closable in $\\ell^2$ in the limit circle case and also for bounded Jacobi matrices with absolutely continuous spectral measure. With closability and the existence of the kernel $J_z^{\\infty}(\\lambda)=\\sum_{n=1}^{\\infty} p_n(z)p_n(\\lambda)$ in hand, the paper defines $B_{\\infty}^{A}$ as the linear manifold of series with $C[f,f]<\\infty$ and claims that the two de Branges axioms of Theorem 1 hold, so $B_{\\infty}^{A}$ is a de Branges space. The proof of the lower bound in Theorem 7 uses the claimed orthogonality of the $T_k$ with respect to the Chebyshev weight.","pith_inferences":["Editorial consequence: if the Chebyshev orthogonality used in Theorem 7 fails, the quantitative lower bound on $\\beta_T$ needs a different proof; the space $B_{\\infty}^{A}$ itself may still be a de Branges space, but the stated eigenvalue estimate and its connection to the classical Hankel result would have to be reworked.","Editorial suggestion: a concrete test of the construction is to compute $B_{\\infty}^{A}$ for the free Jacobi operator ($a_k=1$, $b_k=0$), where $C_T=I_T$; the resulting space should coincide with the classical de Branges space of the associated canonical system, and identifying its Hermite–Biehler function would settle whether the infinite limit reproduces the known object.","Editorial speculation: the same boundary-control recipe could be tested on multidimensional discrete wave equations, where finite speed of propagation should still give finite-dimensional reachable sets; the authors mention this direction in passing.","Editorial observation: the paper itself notes that $\\beta_T$ alone cannot distinguish the limit point case, so the load-bearing content of the construction lies in the closability of $C[\\cdot,\\cdot]$ and the existence of the reproducing kernel, not in a sharp spectral criterion from $\\beta_T$."],"forward_implications":["In the limit circle case, the connecting-operator norm gives an intrinsic construction of a de Branges space for a semi-infinite Jacobi matrix, with no need to pass to a canonical system first.","The reproducing kernel of this space is the classical kernel $\\sum_{n=1}^{\\infty}p_n(z)p_n(\\lambda)$ from indeterminate moment theory, giving that kernel a dynamical interpretation as the limit of finite-time reproducing kernels $J_z^T(\\lambda)=\\sum_{n=1}^{T}p_n(z)p_n(\\lambda)$.","The minimal eigenvalues $\\beta_T$ of the finite connecting operators stay bounded away from zero in the limit circle case, the dynamic counterpart of the Berg–Chen–Ismail small-eigenvalue result for Hankel matrices; the paper also proves a converse direction in Lemma 1: boundedness of the maximal eigenvalues $\\gamma_T$ forces the limit point case.","The quadratic form $C[f,f]$ is closable in the limit circle case and for bounded absolutely continuous spectra, so $B_{\\infty}^{A}$ is a genuine Hilbert space and the identification with a de Branges space is legitimate."],"supporting_citations":[{"why":"Supplies the definition of de Branges spaces, the Hermite–Biehler framework, and Theorem 1, the classification criterion used to certify $B_{\\infty}^{A}$.","marker":"[8]"},{"why":"Introduces the finite-dimensional boundary-control construction of de Branges spaces that the paper extends to $T\\to\\infty$.","marker":"[10, 11]"},{"why":"Gives the dynamic inverse problem for Jacobi matrices and the formulas for the response vector and connecting operator.","marker":"[12]"},{"why":"Establishes the relation between connecting operators and classical moment problems, motivating the comparison with Hankel matrices.","marker":"[14]"},{"why":"The small-eigenvalue theorem for Hankel matrices in the indeterminate case that Theorem 7 is designed to parallel.","marker":"[5]"},{"why":"The closability result for Hankel operators and moment problems that Theorem 8 mirrors for the connecting operator.","marker":"[7]"},{"why":"Provides the convergence of spectral measures $d\\rho_N\\to d\\rho_{\\infty,h}$ used to select the measure in the limit circle case.","marker":"[2]"},{"why":"Supplies the moment-problem background, the description of self-adjoint extensions, and the reproducing kernel (9).","marker":"[19]"}],"fun_headline_variants":["Semi-infinite Jacobi matrices carry de Branges spaces in limit circle","Limit-circle regime extends de Branges reachable-set construction","Infinite de Branges spaces from limit-circle Jacobi matrices","Finite-time de Branges recipe survives the infinite limit","Indeterminacy regime yields closable forms and de Branges spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the polynomials $T_k$ defined by $T_{t+1}+T_{t-1}-\\lambda T_t=0$, $T_1=1$, are orthogonal with respect to the Chebyshev weight $d\\nu(\\lambda)=\\chi_{(-1,1)}(\\lambda)/\\sqrt{1-\\lambda^2}\\,d\\lambda$, so that $\\|f\\|_{\\ell^2}^2=\\int_{-1}^{1}|\\sum f_kT_k(\\lambda)|^2 d\\nu$; this identity is what converts $\\ell^2$ norms of controls into $L^2$ norms in the proof of Theorem 7.","fun_headline_variants_meta":{"raw":{"variants":["Semi-infinite Jacobi matrices carry de Branges spaces in limit circle","Limit-circle regime extends de Branges reachable-set construction","Infinite de Branges spaces from limit-circle Jacobi matrices","Finite-time de Branges recipe survives the infinite limit","Indeterminacy regime yields closable forms and de Branges spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000397,"raw_usage":{"total_tokens":2086,"prompt_tokens":960,"completion_tokens":1126,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":1037}},"tokens_in":576,"tokens_out":1126,"duration_ms":8767,"temperature":1.0,"reasoning_tokens":1037,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:58:42.163349+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\int_{-1}^{1} T_1(\\lambda)T_3(\\lambda)\\,\\frac{d\\lambda}{\\sqrt{1-\\lambda^2}}$ with $T_1=1$ and $T_3=\\lambda^2-1$. The integral equals $-\\pi/2$, not $0$, so the claimed orthogonality of the $T_k$ with respect to $d\\nu$ is false; consequently the norm identity $\\|f\\|_{\\ell^2}^2=\\int |\\sum f_kT_k|^2 d\\nu$ fails, and the proof of the lower bound (26) in Theorem 7 needs a different justification.","supporting_citations":[{"cited_title":"Hilbert space of entire functions.Prentice-Hall, NJ.1968","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of de Branges spaces, the Hermite–Biehler framework, and Theorem 1, the classification criterion used to certify $B_{\\infty}^{A}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the dynamic inverse problem for Jacobi matrices and the formulas for the response vector and connecting operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the relation between connecting operators and classical moment problems, motivating the comparison with Hankel matrices."},{"cited_title":"Small eigenvalues of large Hankel matrices: The indetermi- nate case.Mathematica Scandinavica,91, 67–81, 2002","cited_arxiv_id":null,"evidence_quote":"The small-eigenvalue theorem for Hankel matrices in the indeterminate case that Theorem 7 is designed to parallel."},{"cited_title":"Closable Hankel Operators and Moment Problems.Integral Equations and Operator Theory92, no","cited_arxiv_id":null,"evidence_quote":"The closability result for Hankel operators and moment problems that Theorem 8 mirrors for the connecting operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the convergence of spectral measures $d\\rho_N\\to d\\rho_{\\infty,h}$ used to select the measure in the limit circle case."},{"cited_title":"Schm¨ udgen The moment problem.Springer International Publishing, Cham,2017","cited_arxiv_id":null,"evidence_quote":"Supplies the moment-problem background, the description of self-adjoint extensions, and the reproducing kernel (9)."}],"review_version":1}