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Explicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators

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Pith's one-line read The paper proves the resolvent difference for one-limit-circle Sturm-Liouville operators is a rank-one kernel $k_\theta(z)^{-1}\langle w_z,\cdot\rangle w_z$, and derives explicit Bessel spectral shift functions and the lone negative…

desk verdict Solid, reusable paper: explicit Krein identities for singular Sturm-Liouville endpoints plus Bessel trace formulas and spectral shift functions; the Bessel application hinges on one standard cited spectral fact. read the letter →

arxiv 1908.05392 v1 pith:Z7P424BZ submitted 2019-08-15 math.SP

S. Blake Allan , Justin Hanbin Kim , Gregory Michajlyszyn , Roger Nichols , Don Rung This is my paper · ORCID
classification math.SP MSC 47A1047A5547A5647B10
keywords KreinresolventidentitysingularSturm-LiouvilleoperatorsBesselspectralshiftfunctionWeyl-TitchmarshsolutionLagrangebracketboundaryconditionbasesrank-onedifference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives explicit Krein resolvent identities for singular Sturm-Liouville operators when one endpoint is limit circle and the other is limit point, and also for two limit circle endpoints. In the one-limit-circle case it proves that the resolvent difference of any two self-adjoint extensions is a rank-one operator of the form $k_\theta(z)^{-1}\langle w_z,\cdot\rangle w_z$, where $w_z$ is the Weyl-Titchmarsh solution, so its trace is $\langle w_z,w_z\rangle/k_\theta(z)$. Applying this to the Bessel expression $-d^2/dx^2+(\nu^2-1/4)x^{-2}$ on $(0,\infty)$ for $\nu\in[0,1)$, it computes the trace of the resolvent difference explicitly and uses it to determine the spectral shift function for the Friedrichs extension versus any other self-adjoint realization. The resulting spectral shift functions locate the single simple negative eigenvalue that appears exactly when the realization is not nonnegative, and they recover the known characterization of nonnegative Bessel realizations.

What carries the argument

The load-bearing objects are boundary condition bases $\{\varphi,\psi\}$ at a limit-circle endpoint, normalized by $[\psi,\varphi](c)=1$, and the Lagrange bracket $[f,g](x)=f(x)(pg')(x)-(pf')(x)g(x)$, which provides finite boundary data at singular endpoints where ordinary boundary values may not exist. The argument also uses the Weyl-Titchmarsh solution $w_z$, the unique solution of $\tau y=zy$ square-integrable near the limit-point endpoint and normalized by $[w_z,\varphi_a](a)=1$. The scalar $k_\theta(z)=\cot(\theta)+[w_z,\psi_a](a)$ measures how the boundary condition at $a$ changes, and the rank-one kernel $\langle w_z,\cdot\rangle w_z$ carries the whole resolvent difference. In the two-limit-circle case the same machinery produces a $2\times2$ matrix $K_{\alpha,\beta}(z)$ or $K_{R,\eta}(z)$ and a rank-two (sometimes rank-one) correction.

What would settle it

For fixed $\nu\in(0,1)$ and $\theta\in(\pi/2,\pi)$, solve the Bessel eigenvalue equation at negative energy with the boundary condition $\cos(\theta)[y,\varphi_{0,\nu}](0)+\sin(\theta)[y,\psi_{0,\nu}](0)=0$ and compare the computed eigenvalue with $e_{\theta,\nu}$ from (5.39); a mismatch would disprove the spectral shift reconstruction. For $\nu=0$, the analogous comparison uses $e_{\theta,0}=-4e^{-2[\cot(\theta)+\gamma]}$ from (5.117).

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Extended reading notes

Core claim

The central claim is Theorem 3.4: for a singular Sturm-Liouville expression with exactly one limit circle endpoint $a$, a fixed reference extension $T_0$, and another extension $T_\theta$ parametrized by a boundary condition basis $\{\varphi_a,\psi_a\}$, for $z\in\rho(T_0)\cap\rho(T_\theta)$, $(T_\theta-zI)^{-1}-(T_0-zI)^{-1}=k_\theta(z)^{-1}\langle w_z,\cdot\rangle w_z$, with $k_\theta(z)=\cot(\theta)+[w_z,\psi_a](a)$, where $w_z$ is the Weyl-Titchmarsh solution. Hence the resolvent difference is rank one and its trace is $\langle w_z,w_z\rangle/k_\theta(z)$. For the Bessel expression the paper evaluates all ingredients in terms of Bessel and Hankel functions, obtaining the trace formulas (5.35) for $\nu\in(0,1)$ and (5.116) for $\nu=0$, and then inverts the Stieltjes transform to obtain the spectral shift functions (5.42)-(5.46) and (5.117). These formulas show which realizations are nonnegative and expose the single negative eigenvalue $e_{\theta,\nu}$ for non-nonnegative realizations.

Load-bearing premise

The reconstruction assumes the reference extension $T_0^{(\nu)}$ has spectrum exactly $[0,\infty)$ with no eigenvalues, and that both operators are bounded below with trace-class resolvent difference; if hidden negative spectrum or an embedded eigenvalue existed, the eigenvalue conclusions would change.

Editorial extensions

If this is right

  • For any singular Sturm-Liouville pair with one limit-circle endpoint, the full resolvent difference is a single explicit rank-one operator, so trace identities and spectral shift functions follow without separately constructing Green's functions.
  • For Bessel operators with $\nu\in(0,1)$, the Friedrichs extension and any other realization are resolvent comparable, and their spectral shift function is given by the arctangent formulas (5.42)-(5.46), which simultaneously characterize $\theta\in[0,\pi/2]$ as exactly the nonnegative realizations.
  • For $\nu=0$, the unique nonnegative realization is $\theta=0$, and every other realization has exactly one simple negative eigenvalue $e_{\theta,0}$; in the special case $\nu=1/2$, the trace identity for the Neumann minus Dirichlet Laplacian on the half-line follows.
  • The explicit negative eigenvalues $e_{\theta,\nu}$ and $e_{\theta,0}$ are determined directly from $\theta$, $\nu$, and the gamma function, so the spectrum of any real Bessel realization is exactly $[0,\infty)$ plus at most one negative eigenvalue.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same rank-one identity should apply to other singular one-limit-circle Sturm-Liouville expressions, such as radial Schrödinger operators with Coulomb-like singular terms, wherever the Weyl-Titchmarsh solution is known in closed form.
  • Because the resolvent difference is literally a rank-one term, a cheap numerical test of the spectral shift reconstruction is to evaluate $\langle w_z,w_z\rangle/k_\theta(z)$ at a few complex energies and compare the predicted negative eigenvalue with a direct shooting method.
  • The logarithmic term in $k_{\theta,0}(z)$ for $\nu=0$ suggests that the critical case may share features with other operators having borderline long-range potentials, beyond the Bessel example treated here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper derives explicit Krein resolvent identities for singular Sturm–Liouville operators, distinguishing the cases of one and two limit-circle endpoints. For one limit-circle endpoint, the resolvent difference between a reference extension T_0 and any other extension T_θ is shown to be a rank-one operator built from the Weyl–Titchmarsh solution w_z, with an explicit trace formula (Theorem 3.4). For two limit-circle endpoints, analogous rank-two and rank-one identities are proved for separated and coupled boundary conditions (Theorems 4.4–4.7), and the regular-interval identities of [9] are recovered as special cases. The main application is the Bessel expression with ν∈[0,1): the authors compute the resolvent difference and its trace relative to the Friedrichs extension, then invert the trace formula to obtain explicit spectral shift functions and to locate the unique negative eigenvalue of non-nonnegative realizations (Propositions 5.1, 5.4, 5.8, 5.9).

Significance. If the formulas are taken with the intended bilinear pairing, the paper provides genuinely explicit, parameter-free Krein identities for singular endpoints and furnishes the first fully explicit spectral shift functions for the Bessel family in the regime ν∈[0,1), including the ν=0 case. The recovery of the regular theory from Section 4 and the reduction of the ν=1/2 case to the Dirichlet–Neumann spectral shift function are useful consistency checks. The algebraic structure of the proofs is detailed, and the trace formulas are concrete enough for direct verification; I independently spot-checked the ν=1/2 and ν=0 computations. The main reservation is that the notation for the pairing in the rank-one terms is inconsistent with the declared Hilbert-space inner product, which affects the literal statement of the principal theorems; this is fixable by a notational revision, not by new mathematics.

major comments (2)
  1. [§3, Eq. (3.13); also (3.15), (4.21), (4.52), (4.69), (4.108)] The symbol ⟨w_z,·⟩_(a,b) is used both as the Hilbert-space inner product declared in §1 (linear in the second argument, hence conjugate-linear in the first) and as the bilinear pairing f↦∫_a^b w_z f r dx. Under the declared inner product, the rank-one operator in (3.13) has kernel w_z(x) overline{w_z(y)}, and its Hilbert-space trace would be k_θ(z)^{-1}‖w_z‖², which is real and nonnegative. But the Bessel computations in (5.29) and (5.111), and the resulting trace formulas (5.35) and (5.116), use the bilinear expression ∫ w_z², which for non-real z is complex; for instance the ν=1/2, θ=π/2 case gives trace -1/(2z), not a positive real number. The proof of Lemma 3.2 and the derivation of (3.13) also rely on the boundary value ∫ w_z f, not ∫ overline{w_z} f. Thus, as written, the principal resolvent identities are not literally correct with the stated inner product, although they are correct if ⟨·,·⟩_(a,b) is read as the bilinear form (f,g)↦∫ f g r dx. Please introduce a distinct notation for this bilinear pairing and restate Theorems 3.4, 4.4–4.7, and the trace formulas using it; alternatively, reformulate the rank-one terms using the Hilbert inner product with w_\bar z in the first slot, where w_\bar z=overline{w_z}.
  2. [§5.2, Eq. (5.109)] The identity [w_{z,0},ψ_{0,0}](0)=ln(z^{1/2}/2)+γ−iπ/2 is stated with the parenthetical remark “a calculation, omitted here.” This bracket determines k_{θ,0}(z), the trace formula (5.116), and ultimately the spectral shift function (5.117) and the negative eigenvalue claim for the ν=0 family. Because this is a load-bearing step, the computation should be included in the paper or an explicit reference supplied. I verified independently from the J0/Y0 small-argument asymptotics that the stated value is correct, so this is a local gap rather than an error.
minor comments (5)
  1. [§5, Eq. (5.6)] The Bessel applications depend on the cited result from [13] that T_0^{(ν)} is the Friedrichs extension with purely absolutely continuous spectrum [0,∞) and empty point spectrum. Please state explicitly which boundary condition in the parametrization of [13] corresponds to θ=0 and confirm that no hidden negative eigenvalue occurs for any ν∈[0,1), so that the SSF normalization (5.9) and the eigenvalue conclusions via Lemma A.3 are unambiguous.
  2. [§5.1, after Eq. (5.7)] The assertion that T_θ^{(ν)} and T_0^{(ν)} are bounded from below is used to invoke the standard existence and uniqueness of the spectral shift function, but no proof or reference is given at that point. Since this is a standard consequence of finite deficiency indices and semiboundedness of the Friedrichs extension, a one-sentence citation would suffice.
  3. [Throughout] There are numerous typographical and formatting artifacts in the LaTeX source, such as “/greaterorequalslant”, “0<ε≪1” being typeset inconsistently, and missing overlines around conjugate quantities. These should be cleaned up in the final version.
  4. [Lemma 3.3] If the recommended change to a bilinear pairing is adopted, Lemma 3.3 must be supplemented with the corresponding statement for operators of the form A=(φ,·)ψ, where (φ,g)=∫ φ g r dx; the trace is then (φ,ψ)=∫ φ ψ r dx, not ⟨φ,ψ⟩_H.
  5. [Remark 4.9] The recovery of the regular-interval identities from [9] is convincing but somewhat compressed; explicitly displaying the index interchange in (4.138) would make the comparison easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the resolvent identities, trace formulas, and spectral shift functions are derived from explicit boundary-data computations and standard external spectral assumptions.

full rationale

The paper's central derivation is self-contained. Theorem 3.4 is proved directly: Lemma 3.2 computes the boundary bracket of the reference resolvent from the Green function representation, and the proof of the theorem verifies that the proposed rank-one correction maps into the domain of T_theta and satisfies the resolvent equation, so (3.13) is a theorem rather than an assumed ansatz. The trace formula (3.15) is the exact trace of a rank-one operator. In the Bessel application, the Weyl solution w_z, the scalar k_theta, and the inner product <w_z,w_z> are evaluated in closed form from Bessel/Hankel asymptotics; the trace formulas (5.35) and (5.116) are explicit computations, not fits to data. The spectral shift function is recovered by Stieltjes inversion of the logarithmic derivative of the computed function m, which is the standard uniqueness mechanism for the SSF; no parameter is tuned. The negative-eigenvalue conclusions use Lemma A.3 on SSF jumps, applied to the explicitly computed SSF. The main externally loaded input, Eq. (5.6) from [13] (Everitt-Kalf), asserts the reference spectrum; it comes from an external source and is not derived from the paper's outputs, so it is an assumption but not a circular one. Self-citations to [9] and [11] concern standard Sturm-Liouville facts with independent published proofs and do not carry the paper's conclusions by themselves.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the standard Sturm-Liouville machinery: endpoint classification, boundary condition bases, Green's function representations, and the explicitly cited spectral properties of the Bessel Friedrichs extension. No fitted parameters and no new postulated entities appear. The Bessel-specific evaluations of Wronskians and inner products are derived from classical Bessel function identities.

assumptions (6)
  • domain assumption Hypothesis 2.1: p,q,r real-valued, p,r>0 a.e., p^{-1},q,r∈L¹_loc on (a,b).
    Defines the class of Sturm-Liouville expressions studied; standard and stated at the outset.
  • standard math Weyl's alternative and the deficiency index theorem (Theorem 2.11) determine the number of self-adjoint extensions.
    Background from [11] used throughout to set up the one- or two-parameter families of extensions.
  • standard math Existence of boundary condition bases {φ_c,ψ_c} at a limit circle endpoint (Lemma 2.15) and the Naimark patching lemma.
    Needed to construct φ_{0,ν}, ψ_{0,ν} with the prescribed local behavior near 0 and vanishing near infinity.
  • standard math Green's function representation of (T_0-z)^{-1} with kernel (3.6), cited from [11, Theorem 7.1].
    The boundary data Lemma 3.2, which is central to the main resolvent identity, relies on this explicit kernel.
  • domain assumption For ν∈[0,1), the Bessel expression is limit circle at 0 and limit point at infinity (cited from [12]).
    Ensures the Bessel operator falls under the one-limit-circle theory of Section 3.
  • domain assumption The reference extension T_0^(ν) (Friedrichs) has spectrum [0,∞) and no eigenvalues, σ_p=∅ (Eq. (5.6), cited from [13]).
    Identifies ρ(T_0^(ν)) as C\[0,∞) and is used to normalize the spectral shift function on the negative half-line.

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Pith. "Pith review of Explicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators." pith.science (2026). https://pith.science/paper/Z7P424BZ

@misc{pith2026190805392,
  author       = {Pith},
  title        = {Pith review of: Explicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z7P424BZ}},
  note         = {Machine review of arXiv:1908.05392}
}
abstract

We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+(\nu^2-(1/4))x^{-2}$ on $(0,\infty)$ for values of the parameter $\nu\in[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.

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