{"id":"21ad1844-1da3-4bb7-995b-f608b435301c","arxiv_id":"2412.14786","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Semigroup admissibility along abstract Sobolev scales is equivalent to polynomial resolvent growth, and this equivalence yields sharp wave transfer function bounds and energy decay rates for Neumann damping.","lead":"The paper proves that admissibility of control and observation operators for semigroups on Hilbert spaces can be measured on fractional-domain or interpolation scales, and that these time-domain properties are equivalent to high-frequency growth rates of associated resolvent and transfer functions. This yields sharp Neumann-to-Dirichlet wave transfer function asymptotics and new non-uniform energy decay rates for damped wave equations.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.5 is only sketched and depends on unproved external lemmas, so the headline energy-decay result in Theorem 1.6 is not yet a complete theorem.","rationale":"The reader's conditional verdict identifies the same load-bearing concern: the energy-decay application depends on Theorem 6.4 as a black box and on Theorem 7.5, whose proof is only sketched and rests on external lemmas. I independently checked the main abstract theorems and found the arguments coherent, with only minor typographical issues that do not affect the conclusions; for example, the second frequency-domain line in Theorem 5.4 is stated without the p factor from the displayed transfer function, but the equivalence is preserved after dividing by p, and a constant in Proposition 4.19 differs by a harmless term. The genuine gap is in Section 7.1: the spectral decomposition and resolvent growth estimates for the damped semigroup are imported rather than proved, and the paper itself flags the maximal-unboundedness case as delicate. Since this is exactly the basis for the headline rectangle decay theorem, the conditional verdict is appropriate, and no verdict change is needed.","tokens_in":74204,"tokens_out":29016,"duration_ms":195123,"concrete_test":"Produce a complete proof of Theorem 7.5 by importing [KW24, Lemmas 3.4, 3.6, 3.9, 3.11] and [CPS+23, Proposition 3.10] and verifying their hypotheses for the specific operator D = gamma^* b on a rectangle: check that 0 is isolated in the spectrum of A_D, that the Riesz projection yields a complement on which the energy is a norm, and that the resolvent bound (7.16) holds with the stated constants. If this verification succeeds, Theorem 1.6 follows; if any step fails, the decay exponent 2/3 + epsilon must be re-derived.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract transfer-function results (Theorems 3.2, 3.9, 3.12) appear internally sound after correcting minor notational slips; I found no fatal gap in the main equivalence argument. The load-bearing weak point is the energy-decay chain leading to Theorem 1.6. Section 7.1 presents Theorem 7.5 with a proof that is explicitly only a sketch. It relies on [KW24, Lemmas 3.4, 3.6, 3.9, 3.11] to show that 0 is an isolated eigenvalue of A_D when 0 is in the spectrum of L and L has compact resolvent, to construct a Riesz projection onto ker A_D with a complement on which the restricted semigroup is a contraction, and to establish the resolvent bound (7.16). These lemmas are not stated or proved in the manuscript, and the paper explicitly flags the case of maximal unboundedness as delicate without treating it. Proposition 7.7 then applies Theorem 7.5 to the wave equation with Neumann damping D = gamma^* b, and Theorem 7.9 uses the resulting rate with eta = 1/4 + epsilon. If any imported lemma fails for this boundary damping operator, the asserted decay o(t^{-2/3+epsilon}) is unsupported. In addition, Theorem 6.4 is used as a black box with fixed sharp exponents, and the paper does not verify in detail that every geometry covered by Theorem 7.9 satisfies the hypotheses of the cited [LT91]/[Tat98] results. Thus the advertised PDE application is conditional on external results whose hypotheses are not checked in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops an admissibility theory for strongly continuous semigroups on Hilbert spaces along two continuous Sobolev scales: fractional domains of the generator and quadratic interpolation spaces. The main abstract results (Theorems 3.2, 3.9, 3.12) show that η-admissibility in these scales is equivalent to distributional output/input regularity and, under left/right-invertibility or in the input-output case unconditionally, to high-frequency resolvent and transfer function growth of order O(|ω|^η) or O(|ω|^{η1+η2}). A second-order specialization (Section 5) and applications to Neumann boundary control of the wave equation (Section 6) yield bounds for the Neumann-to-Dirichlet transfer function, and Section 7 derives energy decay rates for a damped wave equation on a rectangle under a non-uniform Hautus test. The paper includes detailed self-contained appendices on Hilbert-valued Sobolev spaces and interpolation.","tokens_in":74426,"tokens_out":6703,"duration_ms":57026,"significance":"If correct, the abstract Theorems 3.2, 3.9, and 3.12 provide a clean dictionary between time-domain regularity on abstract Sobolev scales and high-frequency growth of resolvents and transfer functions; the second-order theorems and the wave estimates in Theorems 6.5 and 6.8 are natural and the exponents are sharp in the smooth case. The abstract part is supported by comparatively complete proofs, and the appendices on vector-valued Sobolev spaces are a useful and mostly self-contained reference. The advertised PDE application, however, is the least complete part: Theorem 7.5 is only sketched and depends on external lemmas, so Theorem 1.6 should currently be read as conditional. I would not recommend rejection, since the abstract core appears sound and the gaps are readily fixable.","major_comments":[{"comment":"The proof of Theorem 7.5 is labeled a sketch and delegates the spectral decomposition, the Riesz projection onto ker A_D, the contraction property of the restricted semigroup, and the resolvent bound (7.16) to [KW24, Lemmas 3.4, 3.6, 3.9, 3.11]. The manuscript explicitly flags maximal unboundedness of D as delicate and does not treat it. Since Theorem 7.5 is the load-bearing step behind Proposition 7.7 and Theorem 7.9, and hence behind Theorem 1.6, the advertised energy decay is conditional on unproved external lemmas and on hypotheses not checked for D = γ^*b. Please either state and prove the imported lemmas, verify their hypotheses for the Neumann damping operator, or clearly separate the conditional application from the unconditional abstract results.","section":"7.1 (Theorem 7.5)"},{"comment":"The compatibility condition for initial data is inconsistent between statements: Theorem 1.6 requires ∂_n w0 = -b^2 w1, while Proposition 7.7 requires ∂_n w0 = -b w1, and the abstract equation (7.19) has damping term γ^* b^2 γ. This is not cosmetic, since it changes the class of initial data for which the decay estimate is claimed. Please correct the statement and, if the two versions correspond to different normalisations of b, say so explicitly.","section":"7.2 (Proposition 7.7 vs Theorem 1.6)"},{"comment":"The sharp regularity theorem is imported as a black box with fixed loss exponents η = 1/3, 1/4, 1/6 and η = 1/4 + ε for rectangles. Theorems 6.5, 6.8, and Corollary 6.9 depend on it, and Proposition 7.7 and Theorem 7.9 inherit this dependence. The manuscript does not spell out the exact hypotheses under which [Tat98] applies to every geometry covered by Theorems 6.5 and 6.8, especially the unbounded-domain cases, and for the rectangle the use of [LT91, Theorem A] with η = 1/4 + ε is not accompanied by a verification that the hypotheses hold for the rectangle with an arbitrary open boundary observation set. Since the exponents in (1.7) and the decay rate in Theorem 1.6 are direct consequences of this input, please either state the full regularity theorem needed or add a careful remark explaining why the cited results apply in each case.","section":"6.1 (Theorem 6.4 and its use)"}],"minor_comments":[{"comment":"In the definition of X_B^1, equation (3.17), the infimum is taken over u ∈ Y but the intended space for the control is U; as written the norm is not defined correctly.","section":"3.3 (Remark 3.11)"},{"comment":"There are several typographical issues: 'strongly continous' in the abstract, the broken word 'transf er' in the title line, and 'it is to tempting' in Remark 4.5.","section":"Global"},{"comment":"In the proof of Lemma B.2, equation (B.5) uses H^{-s}(0,T;E) for the norm of χφ_n; this should presumably be H^s(0,T;E), and the displayed estimate appears to be a typo inherited from the duality notation.","section":"B (Lemma B.2)"},{"comment":"The sentence 'The letter C indicates various spaces of continuous...' conflicts with the later use of C for observation operators; consider renaming the generic constant in Section 3 to avoid confusion.","section":"2.1"},{"comment":"The phrase 'a suitable unbounded domain' is too vague, since Theorems 6.5 and 6.8 depend on the geometric hypotheses that make Theorem 6.4 valid; please make the precise class of domains explicit in the theorem statements.","section":"1.2 (Theorem 1.5)"}],"recommendation":"major_revision","confidential_remarks":"The abstract core of the paper is strong and the appendices are careful; the main risk is that the headline PDE application is presented as a theorem while its proof is a sketch relying on [KW24] and on sharp regularity results from [LT91]/[Tat98]. I would urge the editor to require that the application be made either complete or explicitly conditional, and that the compatibility-condition inconsistency in Section 7.2 be resolved before publication. The citation pattern appears appropriate and does not by itself raise concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the paper's core contribution — the equivalence between eta-admissibility measured on fractional-domain versus interpolation Sobolev scales, distributional output regularity, and high-frequency resolvent/transfer-function growth — is genuinely new for 0 < eta < 1 and appears correctly proven. The main abstract theorems (3.2, 3.9, 3.12) are proved in detail, with a serious appendix on vector-valued Sobolev spaces. For anyone working on admissibility theory or transfer-function estimates, this is a useful and well-made contribution.\n\nSecond, the advertised application to wave energy decay (Theorem 1.6, via Theorem 7.5) is not yet a complete theorem. Section 7.1 presents Theorem 7.5 as a proof sketch, delegating the key spectral decomposition and resolvent estimates to lemmas in [KW24] and [CPS+23]. Those lemmas are not stated or proved in the manuscript, and the paper itself flags the maximal-unboundedness case as delicate without treating it. The rectangle energy decay also imports Tataru's regularity theorem (Theorem 6.4) as a black box with fixed sharp exponents, without checking in detail that every geometry in Theorem 7.9 satisfies the hypotheses. If any imported lemma fails for the boundary damping operator used there, the asserted o(t^{-2/3+epsilon}) rate is unsupported.\n\nThat said, the soft spot is localized. The abstract equivalences are the heart of the paper, and they stand on their own. There is no circularity and no data fitting. The PDE transfer-function bounds in Theorems 6.5 and 6.8 follow cleanly once the sharp regularity theorem is accepted. The proof sketch in Section 7.1 sits at the end of the paper but carries the headline decay result, so it cannot be ignored.\n\nThis paper deserves a serious referee. The right referee should check whether the lemmas cited from [KW24] and [CPS+23] actually cover the operators used here, and should push for a fuller proof or a clearly stated import of the spectral decomposition. The abstract material alone justifies publication if the sketch can be tightened or explicitly reduced to published theorems.\n\nFor a reading group: yes, bring it. It is a strong example of turning time-domain regularity into frequency-domain estimates, and the proof techniques are instructive.","headline":"Solid new abstract equivalences for eta-admissibility, but the headline wave-energy decay theorem rests on a sketched proof and imported lemmas that are not verified in the manuscript.","tokens_in":75028,"tokens_out":2015,"would_cite":true,"duration_ms":19841,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47D06","34G10","93B28","35L90","35L05","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Admissibility in abstract Sobolev scales is equivalent to high-frequency resolvent and transfer function growth.","keywords":["admissibility","strongly continuous semigroups","Sobolev scales","quadratic interpolation","transfer function growth","wave equation","Neumann boundary control","non-uniform stability"],"falsifier":"Compute the operator norm $\\|\\gamma((1+i\\lambda)^2-\\Delta_N)^{-1}\\gamma^*\\|_{\\mathcal L(L^2(\\partial\\Omega))}$ on a smooth bounded domain at a sequence of large real $\\lambda$; if its growth exceeds $O(|\\lambda|^{-1/3})$, the wave transfer-function asymptotics and the rectangle energy decay built on them are false.","tokens_in":73918,"feed_emoji":"🌊","tokens_out":15430,"duration_ms":110789,"temperature":0.7,"pith_summary":"This paper establishes exact equivalences between time-domain regularity properties of linear semigroup systems and high-frequency growth rates of their resolvents and transfer functions. For control and observation operators that are $\\eta$-admissible — meaning $L^2$ inputs produce final states in a negative-order Sobolev space, or solutions from smoother initial data have $L^2$ outputs — the paper shows that admissibility measured on a fractional-domain scale, on a quadratic interpolation scale, or through distributional outputs is the same condition, and that this condition implies a resolvent bound growing like $O(|\\omega|^\\eta)$. When the semigroup is a group, the converse holds as well. In the presence of both a control and an observation operator, the paper proves that an input-output Sobolev regularity gain of order $\\eta_1+\\eta_2$ is equivalent to the transfer function growing like $O(|\\omega|^{\\eta_1+\\eta_2})$, with no invertibility assumption. These results matter because they convert boundary-control regularity questions for PDEs into frequency-domain estimates; the paper uses them to derive sharp high-frequency bounds for the Neumann-to-Dirichlet wave transfer function and, from these, non-uniform energy decay rates for a damped wave equation on a rectangle.","feed_headline":"Transfer function growth equals Sobolev regularity loss","feed_subtitle":"New equivalences turn boundary-control regularity into checkable frequency-domain bounds and sharp wave decay rates.","key_machinery":"The argument runs on two continuous Hilbert scales built from the generator $A$: the fractional-domain scale $\\{X^{\\mathrm{fr}}_s\\}$ of domains of fractional powers of $\\mu-A$, and the quadratic interpolation scale $\\{X^{\\mathrm{in}}_s\\}$. Although these spaces can differ, the paper shows that admissibility measured on either scale is the same condition. The frequency-domain step uses the integral representation of fractional powers together with a resolvent-growth cancellation identity, which lets resolvent bounds on $X^{\\mathrm{fr}}_\\eta$ be traded for bounds on $X$ with growth $O(|\\omega|^\\eta)$. The time-domain machinery uses Besov-type characterisations of vector-valued Sobolev spaces, extension by reflection, and an elliptic-regularity lemma for the operator $L = -(d/dt)^2 - 2(d/dt) + 4$ to pass from finite intervals to the whole line. Duality between control and observation flows through the identification $X_{-\\theta} \\simeq (X^d_\\theta)^*$ given by the Riesz map.","core_discovery":"The central discovery is that, for a strongly continuous semigroup on a Hilbert space, how much Sobolev regularity is lost at the input or gained at the output is tightly tied to how fast the resolvent or transfer function grows at high frequency. Concretely, for an $A$-bounded control operator $B$, the following are equivalent for each $\\eta \\in [0,1]$: $L^2$ inputs produce final states in the fractional-domain space $X^{\\mathrm{fr}}_{-\\eta}$; the same holds with the quadratic interpolation space $X^{\\mathrm{in}}_{-\\eta}$ (which can differ from $X^{\\mathrm{fr}}_{-\\eta}$ in general); and inputs that are $\\eta$-smoother in the Sobolev sense produce final states in the original space. Each of these conditions is equivalent to the resolvent bound $\\|(\\sigma+i\\omega-A)^{-1}B\\|_{\\mathcal L(U,X)} = O(|\\omega|^\\eta)$, and conversely when the semigroup is right-invertible; the observation-operator analogue is dual. For systems with both a control $B$ and an observation $C$, the paper proves that $Cx \\in H^{-\\eta_1-\\eta_2}(0,T;Y)$ for every $L^2$ input $u$ holds exactly when $\\|C(\\sigma+i\\omega-A)^{-1}B\\|_{\\mathcal L(U,Y)} = O(|\\omega|^{\\eta_1+\\eta_2})$, without needing invertibility. Applied to the wave equation with Neumann boundary control, these equivalences yield the optimal growth $O(|\\lambda|^{2\\eta-1})$ for the Neumann-to-Dirichlet transfer function, with $\\eta$ the loss exponent coming from sharp interior and boundary regularity theory.","pith_inferences":["The proved equivalences suggest a practical numerical admissibility test: fit the slope of $\\|(\\sigma+i\\omega-A)^{-1}B\\|$ on a vertical line at high frequencies; the measured exponent should equal the Sobolev loss of the input-to-state map, and deviations would expose modelling errors.","The same framework should extend to non-self-adjoint generators without difficulty, since the Sobolev scale only needs a resolvent; the paper's Hilbert-space assumption is used essentially in the Fourier/Plancherel steps, so a Banach-space version would need a genuinely different proof.","One could combine the transfer-function growth bounds with wavenumber-explicit Helmholtz solvers to test numerically whether the sharp boundary-regularity loss exponents are optimal on non-smooth domains, a question the paper leaves open.","The frequency-domain characterisation offers an 'admissibility diagnostic' for model reduction: systems whose transfer function grows faster than $O(|\\omega|)$ are not classically admissible, and the Sobolev-scale bookkeeping here quantifies exactly how far they are."],"forward_implications":["For any semigroup system satisfying the hypotheses, a resolvent growth bound $O(|\\omega|^\\eta)$ is equivalent to a concrete time-domain admissibility condition, so admissibility can be verified from frequency-domain data alone.","The Neumann-to-Dirichlet wave transfer function satisfies $\\|\\gamma((1+i\\lambda)^2-\\Delta_N)^{-1}\\gamma^*\\|_{\\mathcal L(L^2(\\partial\\Omega))} = O(|\\lambda|^{2\\eta-1})$ and the $H^1(\\partial\\Omega)$-valued version grows like $O(|\\lambda|^{2\\eta})$, with the sharp exponents from boundary regularity theory.","The wave equation on a rectangle with Neumann damping supported on an arbitrary open portion of the boundary loses energy at rate $o(t^{-2/3+\\varepsilon})$ for every fixed $\\varepsilon>0$.","For the Schrödinger equation on a half-space with Neumann boundary data, $L^2$ boundary data produce $L^2$ solutions whose Dirichlet trace lies in $L^2(0,T;L^2(\\partial\\Omega)) \\cap H^{-1/2}(0,T;H^1(\\partial\\Omega))$, showing that a natural transfer from waves to Schrödinger does not reverse.","The abstract results apply to second-order systems of the form $\\ddot w + Lw = Pu$ with observation $Q_0w + Q_1\\dot w$, yielding resolvent conditions for the quadratic pencil $p^2+L$ that control both classical and distributional admissibility."],"supporting_citations":[{"why":"supplies the classical admissibility framework and the resolvent-boundedness characterisations (including the converse under left or right invertibility) that the new results extend.","marker":"[TW09]"},{"why":"contributes the resolvent-growth cancellation technique and the integral representation of fractional powers used in the proof of the observation-operator theorem.","marker":"[LS01]"},{"why":"characterises shift-invariant bounded operators on $L^2$ by bounded holomorphic transfer functions, a step needed in the proof of the transfer-function theorem.","marker":"[Wei94]"},{"why":"provides the quadratic interpolation-space theory (reiteration, duality, norm-preserving interpolation) on which the continuous scale $\\{X^{\\mathrm{in}}_s\\}$ is built.","marker":"[CWHM15]"},{"why":"supplies the Sobolev-space technical results, including extension by reflection and interpolation of $H^s$ spaces, used throughout the proofs.","marker":"[LM68]"},{"why":"states the sharp interior and boundary regularity for the wave equation under Neumann control that feeds the transfer-function estimates and energy-decay application.","marker":"[Tat98]"},{"why":"provides the rectangle regularity result with loss exponent $\\eta = 1/4+\\varepsilon$ used for the rectangular wave application.","marker":"[LT91]"},{"why":"gives the wavenumber-explicit Helmholtz a priori estimate used to upgrade the transfer-function bound to $H^1(\\partial\\Omega)$-valued operator norms.","marker":"[Spe14]"},{"why":"connects polynomial resolvent growth to energy decay rates of contraction semigroups, turning the resolvent estimates into the rectangle decay bound.","marker":"[BT10]"},{"why":"establishes exact observability of the Schrödinger group on a rectangle, used to verify the non-uniform Hautus test in the damping application.","marker":"[TT09]"}],"fun_headline_variants":["Sobolev loss = transfer growth: a tight equivalence","High-frequency transfer growth reveals Sobolev loss","Sharp wave decay from transfer growth equivalence","Transfer growth mirrors Sobolev regularity loss"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharp wave and rectangle results inherit a black-box boundary-regularity theorem with fixed loss exponents, and the abstract equivalences require the semigroup to be left- or right-invertible before resolvent growth can be converted back into time-domain admissibility.","fun_headline_variants_meta":{"raw":{"variants":["Sobolev loss = transfer growth: a tight equivalence","High-frequency transfer growth reveals Sobolev loss","Sharp wave decay from transfer growth equivalence","Transfer growth mirrors Sobolev regularity loss"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000362,"raw_usage":{"total_tokens":2041,"prompt_tokens":1118,"completion_tokens":923,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":734,"completion_tokens_details":{"reasoning_tokens":864}},"tokens_in":734,"tokens_out":923,"duration_ms":7667,"temperature":1.0,"reasoning_tokens":864,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:55:07.206259+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the operator norm $\\|\\gamma((1+i\\lambda)^2-\\Delta_N)^{-1}\\gamma^*\\|_{\\mathcal L(L^2(\\partial\\Omega))}$ on a smooth bounded domain at a sequence of large real $\\lambda$; if its growth exceeds $O(|\\lambda|^{-1/3})$, the wave transfer-function asymptotics and the rectangle energy decay built on them are false.","supporting_citations":[],"review_version":1}