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On $m$-order logarithmic Schr\"odinger operator

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read m-order logarithmic operators yield Taylor expansions for fractional powers of Schrödinger operators in L^p

desk verdict The paper offers a spectral construction of m-order logs for Schrödinger operators and uses them for Taylor expansions of fractional powers in L^p, but the L^p transfer may need more justification on the potential assumptions. read the letter →

arxiv 2606.21056 v1 pith:NO5I67FR submitted 2026-06-19 math.AP

Jorge J. Betancor , Estefanía Dalmasso , Pablo Quijano , Lourde Rodríguez-Mesa This is my paper · ORCID
classification math.AP
keywords SchrödingeroperatorlogarithmicfractionalpowersTaylorexpansionspectralmeasuresemigroupL^pconvergence
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 defines the m-order logarithm of the Schrödinger operator L_V using its spectral measure on a suitable subspace of L^2(R^d) and extends the definition to Lipschitz functions via the semigroup generated by L_V. It then applies these log^m L_V operators to establish Taylor expansions for the fractional powers L_V^s and L_V^{-s} with respect to the order s in (0,1), where the series converge in the L^p(R^d) norm for 1 < p < ∞. A sympathetic reader would care because the result supplies an explicit way to track how these fractional operators vary continuously with their order parameter s.

What carries the argument

The m-order logarithmic operator log^m L_V, constructed from the spectral measure of L_V and extended via its generated semigroup to Lipschitz functions, which is then used to derive the expansions in the fractional order s.

What would settle it

A concrete counter-example would be a nonnegative potential V for which the claimed Taylor series for L_V^s fails to converge in L^p for some s close to 0 or 1 and some p between 1 and ∞.

Watch

Extended reading notes

Core claim

By defining log^m L_V via the spectral measure of the self-adjoint operator L_V and extending it through the semigroup {T_t^V}, the authors prove Taylor expansions for L_V^s and L_V^{-s} in the variable s ∈ (0,1) that converge in L^p(R^d) for 1 < p < ∞.

Load-bearing premise

The Schrödinger operator L_V must be self-adjoint on a suitable subspace of L^2(R^d) so that a spectral measure exists and can be used to define the logarithmic operators.

Editorial extensions

If this is right

  • The expansions hold for both positive and negative fractional powers.
  • Convergence is obtained in every L^p space with 1 < p < ∞.
  • The construction applies to a wide class of nonnegative potentials V.
  • The definition of log^m L_V extends from the spectral measure to Lipschitz functions via the semigroup.

Reading between the lines

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

  • Truncating the Taylor series could furnish practical approximations to fractional powers for concrete potentials.
  • The same logarithmic construction might be attempted for other self-adjoint operators to obtain order expansions outside the Schrödinger setting.
  • If the spectral theory carries over, analogous expansions could be studied on domains with boundary conditions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript defines the m-order logarithmic Schrödinger operator log^m L_V (m ∈ ℕ) for nonnegative potentials V via the spectral measure of the self-adjoint operator L_V on a subspace of L^2(R^d), extends the definition to Lipschitz functions using the semigroup {T_t^V} generated by L_V, and employs these operators to establish Taylor expansions of the fractional powers L_V^s and L_V^{-s} (s ∈ (0,1)) with convergence in L^p(R^d) for 1 < p < ∞.

Significance. If the L^p-convergence statements hold, the work would supply a functional-calculus route to fractional powers of Schrödinger operators that is expressed through logarithmic operators; this could be useful for PDE analysis involving such operators. The construction rests on standard spectral and semigroup tools, but its novelty hinges on the rigor of the L^p extension.

major comments (2)
  1. [Abstract] Abstract: the central claim asserts L^p-norm convergence of the Taylor series for L_V^s (s ∈ (0,1)). The construction begins with the L^2 spectral theorem and the semigroup extension of log^m L_V; no L^p-boundedness of the resulting operators or explicit control of the Taylor remainder in L^p norm is indicated. For general nonnegative V this transfer from L^2 to L^p typically requires additional assumptions (e.g., Kato-class conditions yielding Gaussian bounds) that are not stated.
  2. [Abstract] Abstract (extension step): the semigroup {T_t^V} is invoked to extend log^m L_V to Lipschitz functions, yet the text gives no indication that these extensions are bounded on L^p(R^d) or that the functional calculus they induce preserves the L^p topology needed for the remainder term in the Taylor expansion of L_V^s to vanish in L^p.
minor comments (1)
  1. The class of admissible potentials V is described only as 'certain nonnegative potentials'; a precise statement (e.g., local integrability or Kato-class membership) would clarify the setting in which the L^p results are claimed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the L^p aspects require explicit clarification and will revise the manuscript accordingly to state the necessary assumptions on V and detail the boundedness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim asserts L^p-norm convergence of the Taylor series for L_V^s (s ∈ (0,1)). The construction begins with the L^2 spectral theorem and the semigroup extension of log^m L_V; no L^p-boundedness of the resulting operators or explicit control of the Taylor remainder in L^p norm is indicated. For general nonnegative V this transfer from L^2 to L^p typically requires additional assumptions (e.g., Kato-class conditions yielding Gaussian bounds) that are not stated.

    Authors: The referee is correct that the abstract omits the conditions on V needed for L^p transfer. The L^2 construction via spectral theorem is complete, but to justify L^p convergence we will add the standing assumption that V belongs to the Kato class (ensuring Gaussian heat kernel bounds). Under this assumption the semigroup is bounded on L^p (1 ≤ p ≤ ∞) and the Taylor remainder can be controlled in L^p norm via the functional calculus. We will revise the abstract, add a remark in the introduction, and include a brief L^p remainder estimate. revision: yes

  2. Referee: [Abstract] Abstract (extension step): the semigroup {T_t^V} is invoked to extend log^m L_V to Lipschitz functions, yet the text gives no indication that these extensions are bounded on L^p(R^d) or that the functional calculus they induce preserves the L^p topology needed for the remainder term in the Taylor expansion of L_V^s to vanish in L^p.

    Authors: We acknowledge the need for explicit L^p boundedness of the extended operators. Under the Kato-class assumption on V (to be stated), the semigroup {T_t^V} extends to a bounded semigroup on L^p and the functional calculus for Lipschitz functions of L_V preserves this boundedness. Consequently the Taylor remainder vanishes in L^p. We will insert a short subsection after the semigroup definition explaining this L^p extension and topology preservation. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: definitions via spectral theorem and semigroup yield independent Taylor expansions in L^p

full rationale

The paper defines log^m L_V first via the spectral measure of the self-adjoint L_V on a subspace of L^2, then extends the definition to Lipschitz functions using the semigroup {T_t^V}. These operators are then applied to obtain Taylor expansions of L_V^s and L_V^{-s} for s in (0,1) with L^p convergence. No quoted step reduces the expansions to a fitted quantity, a self-citation chain, or a renaming of the input; the L^2-to-L^p passage is presented as a derived result rather than an identity by construction. The derivation remains self-contained against external functional-calculus benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The central claim rests on the existence of the spectral measure for a self-adjoint realization of L_V and on the standard properties of the generated semigroup; no free parameters or new postulated entities are introduced beyond the definition of log^m L_V itself.

assumptions (1)
  • domain assumption L_V admits a self-adjoint realization on a dense subspace of L^2(R^d) so that the spectral theorem applies
    Invoked to define log^m L_V via the spectral measure.
invented entities (1)
  • log^m L_V
    purpose: To serve as the coefficients in the Taylor expansion of fractional powers
    Newly defined functional-calculus object; no independent evidence outside the paper is supplied.

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Cite this review

Pith. "Pith review of On $m$-order logarithmic Schr\"odinger operator." pith.science (2026). https://pith.science/paper/NO5I67FR

@misc{pith2026260621056,
  author       = {Pith},
  title        = {Pith review of: On $m$-order logarithmic Schr\"odinger operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NO5I67FR}},
  note         = {Machine review of arXiv:2606.21056}
}
abstract

In this paper we study the logarithm of order $m$ of the Schr\"odinger operator $\mathcal L_V$ in $\mathbb R^d$, for certain nonnegative potentials $V$. First, the operator $\log^m\mathcal L_V$, $m\in \mathbb N$, is defined by using the spectral measure associated with the self-adjoint operator $\mathcal L_V$ on a suitable subspace of $L^2(\mathbb R^d)$. Then, the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$ allows us to extend the definition of $\log^m\mathcal L_V$ to a wider class of Lipschitz functions. By using logarithmic operators $\log^m\mathcal L_V$, $m\in \mathbb N$, we prove Taylor expansions for the fractional powers $\mathcal L_V^s$ and $\mathcal L_V^{-s}$ with respect to the order $s\in (0,1)$, where the convergence is understood in $L^p(\mathbb R^d)$, $1<p<\infty$.

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Reference graph

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