{"id":"0f704112-841b-4530-a79e-fb7d5a8ff729","arxiv_id":"2509.05178","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a symmetric operator invariant under a bounded invertible K, its Friedrichs and Krein-von Neumann extensions are automatically K-invariant, and K-invariant Sturm-Liouville extensions are characterized through Schroeder's equation.","lead":"This math paper shows which self-adjoint and maximally dissipative extensions of operators remain invariant under a generalized symmetry K, then applies this to Sturm-Liouville operators. Generalists may care because the symmetry conditions turn into the classical Schroeder and Julia functional equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hypothesis 3.4 (Eq. 3.10) gives the p and q Sturm-Liouville coefficient relations reversed relative to a direct K*τK computation; as printed Theorem 3.5 is false, although the abstract core is sound and the corrected formulas restore the section.","rationale":"Agree with the reader. The abstract core is solid: I checked the domain-invariance arguments in Theorem 2.7 (sequences for D(S_F), D(S_K)), Lemma 2.5, Lemma 2.10, and the B-extension criterion in Theorem 2.13; no gap emerged. The single load-bearing problem is the Sturm-Liouville bridge. A direct K*τK computation shows the coefficient conditions in (3.10) are inverted: p must be multiplied by φ'(φ^{-1}) and q divided by it. Because the printed hypothesis is used to prove Theorem 3.5, the theorem is false as stated; the counterexample with φ(x)=x/2 gives K*τK=(1/4)τ while satisfying (3.10). The corrected formulas (as already used in Remark 3.6, Eq. 3.14, and the examples) restore the argument. This is localized and correctable, so the reader's CONDITIONAL verdict is appropriate; the main claims of Section 2 stand.","tokens_in":23548,"tokens_out":29780,"duration_ms":293715,"concrete_test":"Algebraic check: compute K*τK explicitly for ψ=φ^{-1} using (3.6), and match coefficients of f'' and f; this yields the p and q relations above, opposite of (3.10). Then evaluate the family φ(x)=x/2, A=1, r=1, q=0 on (0,∞): with p=c/x (which satisfies printed (3.10)), verify K*τK f=(1/4)τf; with p=cx (which satisfies the corrected relation), verify K*τK f=τf. If the direct computation reproduces (1/4)τf, the printed Hypothesis 3.4 must be corrected before Theorem 3.5 and the examples can be used.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Load-bearing concern: Theorem 3.5's sufficient conditions (Hypothesis 3.4, Eq. 3.10) are inverted and, read literally, do not imply K-invariance. Substituting Kf=A(x)f(φ(x)) and the adjoint (3.6) into K*τK and matching coefficients gives, with ψ=φ^{-1}(x), p(x)=A(ψ)^2 φ'(ψ)p(ψ) and q(x)=A(ψ)/φ'(ψ)[A(ψ)q(ψ)-(A^{[1]})'(ψ)]. The printed (3.10) divides where p should multiply and multiplies where q should divide, and the proof's display (3.11) repeats this. This is not a harmless typo: on (0,∞), take r=1, q=0, A=1, φ(x)=x/2, C=1. The printed p-condition is p(x)=2p(2x), so p=c/x satisfies it; direct computation gives K*τK f = (1/4)τf, so T_min is not K-invariant. With the corrected p-condition p(x)=½p(2x), e.g. p=cx, one obtains K*τK f=τf. Thus Theorem 3.5 and the examples depending on (3.10) fail as printed; the abstract Theorems 2.7–2.18 are unaffected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of K-invariance for a densely defined closable operator S with respect to a bounded, boundedly invertible operator K, defined by K^* S K = S. The main abstract results are: the adjoint and closure of a K-invariant operator are K-invariant; the Friedrichs and Krein–von Neumann extensions of a nonnegative K-invariant symmetric operator are always K-invariant; the Friedrichs extension of a K-invariant sectorial operator is K-invariant; and, for strictly positive operators, the K-invariance of extensions described by the Birman–Krein–Vishik–Grubb parameter B is characterized by a domain condition and a projected commutator condition. The paper then applies this framework to Sturm–Liouville operators with K of the form (Kf)(x)=A(x)f(φ(x)), deriving sufficient conditions on p,q,r, characterizing K-invariant boundary conditions, and giving several worked examples including a Bessel-type Schrödinger operator on the half-line. The abstract theorems are carefully argued. The Sturm–Liouville section contains a load-bearing error in the printed coefficient conditions: the transformation rule for p is inverted, so Theorem 3.5 and the examples relying on Hypothesis 3.4 fail as printed, although the correction is local.","tokens_in":23911,"tokens_out":17784,"duration_ms":172098,"significance":"If the Section 3 coefficient condition is corrected, the paper makes a useful contribution. Theorem 2.7 is a clean and non-obvious result: nonnegative K-invariant symmetric operators have K-invariant Friedrichs and Krein–von Neumann extensions for every bounded invertible K, not only unitary K. The extension to sectorial operators and the characterization in Theorem 2.13 are also valuable. The Sturm–Liouville application connects invariance conditions to Schröder's and Julia's equations and provides explicit, nontrivial examples. A particular strength is that the abstract results are derived from stated definitions without fitted parameters or circular reasoning. However, the printed sufficient conditions in Hypothesis 3.4 are internally inconsistent with Remark 3.6 and with the paper's own examples, and this undermines the Sturm–Liouville portion until fixed.","major_comments":[{"comment":"The transformation rule for p is inverted. Let ψ=φ^{-1}(x). Substituting Kf=A(x)f(φ(x)) and the adjoint (3.6) into K^*τK and comparing the coefficient of f'' gives p(x)=A(ψ)^2 φ'(ψ) p(ψ). The printed condition divides by φ'(ψ). The same error appears in (1.1) and in the displayed expression (3.11), where the coefficient of d/dx inside the outer derivative is printed as p(ψ)A(ψ)^2/φ'(ψ) instead of p(ψ)A(ψ)^2 φ'(ψ). This is not a harmless typo: with r=1, q=0, A=1, φ(x)=x/2, C=1, the printed condition p(x)=2p(2x) admits p(x)=c/x, but direct computation gives K^*τK f = (1/4)τf, so T_min is not K-invariant. The q condition in (3.10) is, by contrast, correct.","section":"§3, Hypothesis 3.4 (Eq. (3.10)), Eq. (1.1), and Theorem 3.5 (Eq. (3.11))"},{"comment":"As a consequence of the inverted p-condition, Theorem 3.5 is false as stated, and the examples that claim to verify Hypothesis 3.4 do not: for Example 3.10, substituting p(x)=μx^2, A_c=(1+c)^{1/2}, φ_c(x)=(1+c)x/(1+cx) into the printed p-condition gives μ(1+c)^2 x^2/(1+c-cx)^4, which equals μx^2 only for c=0. The corrected condition p(x)=A^2 φ'(ψ)p(ψ) makes the example satisfy the hypothesis. Thus Theorems 3.8 and Corollary 3.9, and Examples 3.10–3.12, are valid only after correcting Eq. (3.10)/(3.11) as described. The abstract results in Section 2 are independent of this issue and appear sound.","section":"§3, Examples 3.10–3.12 and Theorems 3.5, 3.8"}],"minor_comments":[{"comment":"The boundedness estimate has a constant error: using r(x)=C r(φ^{-1}(x)), the change of variables in (3.7) gives the factor C, not C^{-1}. Since C is fixed, boundedness still follows; the displayed constant M=C^{-1} sup(A^2/φ') should be C sup(A^2/φ') (or any finite upper bound).","section":"§3, Lemma 3.3, Eq. (3.7)"},{"comment":"In the display after (2.18), the term ∥Kf_n - Kf_n∥^2_A should read ∥Kf_n - Kf_m∥^2_A.","section":"§2, Theorem 2.9 proof"},{"comment":"In the rewritten expression for K^{-1}ψ, the first summand is written as (Kf_0 + ...); it should be (K^{-1}f_0 + ...), as the subsequent sentence confirms.","section":"§2, Theorem 2.13 proof, Eq. (2.25)"},{"comment":"The symbols A and B are reused for the transformed interval endpoints, while A is already the coefficient in K. This overloads notation and is confusing; use different letters (e.g., α, β) for the endpoint values.","section":"§3, Liouville–Green transformation, Eq. (3.35)"},{"comment":"The paper should explicitly note that Eq. (1.1) and Eq. (3.10) need the p-condition corrected to p(x)=A(φ^{-1}(x))^2 φ'(φ^{-1}(x)) p(φ^{-1}(x)); otherwise Remark 3.6 and the examples will continue to contradict the stated hypothesis.","section":"§1 and §3"}],"recommendation":"major_revision","confidential_remarks":"The abstract core of the paper is sound and the Section 3 coefficient error is local and fixable. I recommend major revision rather than rejection. The authors should correct the p-transformation in (1.1), (3.10), and (3.11), re-run the examples, and carefully reconcile the statement of Hypothesis 3.4 with Remark 3.6. There is no reason to doubt the originality or value of the abstract framework; the manuscript's own examples already use the corrected condition, which indicates the intended statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The abstract core—K-invariance for bounded invertible K, Theorem 2.7 for Friedrichs and Krein–von Neumann extensions, Theorem 2.13 for the Birman–Krein–Vishik–Grubb parameterization, and the sectorial Friedrichs result—is carefully argued and looks correct. The Sturm–Liouville application as printed has a real error: the coefficient conditions in (1.1) and Hypothesis 3.4 are reversed. Taken literally, Theorem 3.5 is false.\n\nWhat is genuinely new: the non-unitary generalization of µ-scale invariance is a meaningful step, the sectorial result is new, and the connection to Schröder’s and Julia’s equations is a nice hook. The Section 2 proofs are standard extension theory done cleanly, with no fitted parameters or circular reasoning. Credit for that.\n\nThe soft spot, in proportion: the p and q conditions in (3.10) are internally inconsistent with Remark 3.6 and the paper’s own examples. Direct computation gives p(x)=A(φ⁻¹)² φ′(φ⁻¹)p(φ⁻¹) and q(x)=A(φ⁻¹)/φ′(φ⁻¹)[A(...)q(...)−(A^[1])′(...)]; the printed version divides where p should multiply and multiplies where q should divide, and (3.11) repeats the error. A concrete counterexample: on (0,∞) with r=1, q=0, A=1, φ(x)=x/2, the printed p-condition is p(x)=2p(2x), whose solution p=c/x yields K*τK=(1/4)τ, so T_min is not K-invariant. The examples in 3.10–3.12 and Theorem 3.8 rely on the corrected version. This is a load-bearing typo in the application section, but the abstract results do not depend on it, so it is fixable in revision.\n\nThe citation pattern is fine; self-citations are background rather than load-bearing. The paper is honest about scope. This one is for extension-theory people and anyone interested in functional-equation symmetries of differential operators. I would send it to peer review—a serious referee will spot the condition error quickly, and the rest is worth evaluating. If the authors correct (3.10), I’d be happy to see it in print.","headline":"The abstract K-invariance results are solid and worth engaging; the Sturm-Liouville application has a real but fixable coefficient-conditions error.","tokens_in":24391,"tokens_out":4248,"would_cite":true,"duration_ms":39041,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34B24","47A05","47B25","47A10","47B44","47B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a nonnegative symmetric operator's K-invariance—the identity K*SK=S for a bounded, boundedly invertible K—is automatically inherited by its Friedrichs and Krein–von Neumann extensions.","keywords":["K-invariant operators","Friedrichs extension","Krein–von Neumann extension","maximally dissipative extensions","Sturm–Liouville operators","Schröder’s equation","Julia’s equation","self-adjoint extensions"],"falsifier":"Test Theorem 2.18 on Example 2.8: with K a scaling and S the Dirichlet Laplacian on (0,infinity), the theorem says only the Dirichlet (Friedrichs) and Neumann (Krein–von Neumann) extensions are K-invariant. Checking that a Robin condition mu in (0,infinity) cannot be invariant—since (Kf)'(0)=lambda^{1/2}f'(0) and (Kf)(0)=lambda^{-1/2}f(0) cannot match mu(Kf)(0) when lambda is not 1—is a direct verification; a failure would disprove the classification.","tokens_in":23447,"feed_emoji":"🔁","tokens_out":12615,"duration_ms":115504,"temperature":0.7,"pith_summary":"The paper introduces a notion called K-invariance for operators on Hilbert space: a bounded, boundedly invertible operator K leaves S invariant when K*SK=S. The central theorem is that if a nonnegative symmetric operator S has this property, then its Friedrichs extension and its Krein–von Neumann extension automatically have the same property. It then gives an exact condition, phrased in the Birman–Krein–Vishik–Grubb parametrization, for any other self-adjoint or maximally dissipative extension to be K-invariant. In the Sturm–Liouville setting, with K acting by a weighted change of variable, the invariance conditions reduce to classical functional equations such as Schröder's equation, yielding explicit invariant operators and a full classification of invariant boundary conditions. The interest is that K need not be unitary, so the symmetry is a genuine similarity, and the two canonical extremal extensions are robust under it.","feed_headline":"Extremal self-adjoint extensions inherit every K-invariance","feed_subtitle":"Friedrichs and Krein–von Neumann extensions automatically share any bounded invertible symmetry.","key_machinery":"The central object is the weighted composition operator K, defined in the Sturm–Liouville section as (Kf)(x)=A(x)f(phi(x)); in the abstract section it is simply an arbitrary bounded operator with bounded inverse satisfying K*SK=S. What does the work: (i) the graph-limit description of the Friedrichs and Krein–von Neumann extensions, since K and K^{-1} preserve both the domain and the energy form when S is K-invariant; (ii) the Birman–Krein–Vishik–Grubb parametrization S_B of all nonnegative self-adjoint and maximally dissipative extensions through an auxiliary operator B on ker(S*), where K-invariance is equivalent to D(B)=D(K*BK) and P_{D(B)}K*BK|_{D(B)}=B; and (iii) in the Sturm–Liouville","core_discovery":"On the paper's own terms, the discovery is Theorem 2.7: every nonnegative symmetric operator S with K-invariance K*SK=S has K-invariant Friedrichs and Krein–von Neumann extensions. The proof uses the graph-limit characterizations of these two extremal extensions: an approximating sequence from D(S) that is Cauchy in the energy form stays of the same type under left multiplication by K or K^{-1}, because K*SK=S forces both K and K^{-1} to preserve D(S) and the form. The paper goes on to characterize all K-invariant extensions S_B in the Birman–Krein–Vishik–Grubb picture by a condition on the auxiliary operator B, and it specializes this to the finite-defect case, obtaining that root vectors o","pith_inferences":["The abstract framework is likely to transfer to boundary-triple parametrizations of extensions, where the condition on the auxiliary operator B becomes a concrete boundary-matrix condition; this would extend the classification beyond the parametrization used here.","Because the coefficient conditions reduce to Schröder's and Julia's equations, one can generate whole families of K-invariant Sturm–Liouville operators from a single solution by the power and periodic constructions sketched in the paper; a systematic catalogue of such potentials would be a testable by-product.","The result that the Friedrichs extension of a K-invariant sectorial operator is K-invariant suggests the same extremal-extension inheritance may hold for other distinguished extensions in the sectorial case, although the paper does not pursue this.","The explicit half-line example indicates that non-unitary K-invariances are not exotic: natural Schrödinger operators with a Bessel-type singularity carry them, and the invariant boundary condition for the Krein–von Neumann extension is exactly computable."],"forward_implications":["Any nonnegative K-invariant symmetric operator automatically has K-invariant Friedrichs and Krein–von Neumann extensions; this holds for every bounded invertible K, not just unitaries.","In the Birman–Krein–Vishik–Grubb parametrization, an extension S_B is K-invariant iff its auxiliary operator B satisfies D(B)=D(K*BK) and P_{D(B)}K*BK|_{D(B)}=B; this condition is checkable in concrete examples.","When the defect is finite, a K-invariant extension must have ker B containing all root spaces of K restricted to ker(S*) for eigenvalues of modulus not equal to 1; with one-dimensional defect, either exactly two or all maximally dissipative extensions are K-invariant.","Sturm–Liouville operators built from coefficient functions solving the functional equations (3.10) are K-invariant, and the K-invariant self-adjoint extensions are exactly those whose boundary conditions satisfy Theorem 3.8; in particular the Krein–von Neumann extension is characterized in Corollary 3.9.","There exist nontrivial K-invariant Schrödinger operators on the half-line, such as the Bessel-type example, with the invariance given by a non-unitary weighted change of variable."],"supporting_citations":[{"why":"Supplies the characterization of the Krein–von Neumann extension used in the proof of Theorem 2.7.","marker":"[1]"},{"why":"Supplies the characterization of the Friedrichs extension used in the proof of Theorem 2.7.","marker":"[10]"},{"why":"Is the source the paper follows in Proposition 2.6 for these two characterizations.","marker":"[4]"},{"why":"Provides the parametrization of all nonnegative self-adjoint and maximally dissipative extensions (Proposition 2.12), the setting for Theorems 2.13–2.18.","marker":"[14]"},{"why":"Introduces the mu-scale invariant operators that motivate allowing non-unitary K; the paper extends their extremal-extension result.","marker":"[19]"},{"why":"Studies symmetry-preserving extensions in the unitary case, which the paper generalizes by dropping unitarity.","marker":"[16]"},{"why":"Identifies the Krein–von Neumann extension with coupled boundary conditions in the regular/quasi-regular case, used in Corollary 3.9.","marker":"[11]"},{"why":"Supplies the Sturm–Liouville framework and the boundary-condition parametrization used in Theorem 3.7 and Theorem 3.8.","marker":"[13]"}],"fun_headline_variants":["Extremal extensions always preserve K-invariance","K-invariance survives in Friedrichs and Krein–von Neumann","K-symmetric operators keep symmetry in extremal extensions","Self-adjoint extremes inherit K-symmetry automatically"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"For the Sturm–Liouville part, everything depends on the sufficient coefficient conditions in Hypothesis 3.4 actually making the minimal operator K-invariant; the abstract extension theorems themselves only assume K is bounded and boundedly invertible.","fun_headline_variants_meta":{"raw":{"variants":["Extremal extensions always preserve K-invariance","K-invariance survives in Friedrichs and Krein–von Neumann","K-symmetric operators keep symmetry in extremal extensions","Self-adjoint extremes inherit K-symmetry automatically"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000648,"raw_usage":{"total_tokens":2822,"prompt_tokens":766,"completion_tokens":2056,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":1997}},"tokens_in":510,"tokens_out":2056,"duration_ms":15063,"temperature":1.0,"reasoning_tokens":1997,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:33:50.579228+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Theorem 2.18 on Example 2.8: with K a scaling and S the Dirichlet Laplacian on (0,infinity), the theorem says only the Dirichlet (Friedrichs) and Neumann (Krein–von Neumann) extensions are K-invariant. Checking that a Robin condition mu in (0,infinity) cannot be invariant—since (Kf)'(0)=lambda^{1/2}f'(0) and (Kf)(0)=lambda^{-1/2}f(0) cannot match mu(Kf)(0) when lambda is not 1—is a direct verification; a failure would disprove the classification.","supporting_citations":[{"cited_title":"Ando and K","cited_arxiv_id":null,"evidence_quote":"Supplies the characterization of the Krein–von Neumann extension used in the proof of Theorem 2.7."},{"cited_title":"Freudenthal, ¨Uber die Friedrichssche Fortsetzung halbbeschr¨ ankter Hermitescher Operatoren, Kon","cited_arxiv_id":null,"evidence_quote":"Supplies the characterization of the Friedrichs extension used in the proof of Theorem 2.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the source the paper follows in Proposition 2.6 for these two characterizations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the parametrization of all nonnegative self-adjoint and maximally dissipative extensions (Proposition 2.12), the setting for Theorems 2.13–2.18."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the mu-scale invariant operators that motivate allowing non-unitary K; the paper extends their extremal-extension result."},{"cited_title":"Ibort, F","cited_arxiv_id":null,"evidence_quote":"Studies symmetry-preserving extensions in the unitary case, which the paper generalizes by dropping unitarity."},{"cited_title":"Fucci, F","cited_arxiv_id":null,"evidence_quote":"Identifies the Krein–von Neumann extension with coupled boundary conditions in the regular/quasi-regular case, used in Corollary 3.9."}],"review_version":1}