{"id":"e77a9d53-8fb1-442f-96c1-174432edfbf7","arxiv_id":"2506.23119","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The discrete bi-Laplacian on Z has sharp |t|^{-1/4} decay, and with decaying potentials and no embedded positive eigenvalues, the continuous spectral part of the evolution still decays at the same rate.","lead":"This paper proves that the discrete fourth-order Laplace operator on the integer lattice has the same |t|^{-1/4} decay rate as the continuous case, and extends this to Schrödinger and beam equations with decaying potentials. It also classifies threshold resonances and gives sharp dispersion estimates for the perturbed operators.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem is conditional on an unproved no-eigenvalue hypothesis; this hypothesis is load-bearing for Corollary 2.5 and is not verified for the stated potential class.","rationale":"The reader's weakest assumption identifies the same load-bearing point I find. The theorem is explicitly conditional, and the absence of positive eigenvalues is a strong spectral hypothesis that the paper does not establish for its potential class. This is not a fatal flaw: many dispersive estimates are conditional on similar spectral assumptions, and the internal proof appears coherent. However, because the abstract states the decay estimates without this caveat, and because Corollary 2.5 depends on the assumption in a way that is only sketched, I would keep the CONDITIONAL verdict. No further verdict change is needed.","tokens_in":69605,"tokens_out":14493,"duration_ms":158915,"concrete_test":"Prove or disprove the Birman-Schwinger equivalence used implicitly in Corollary 2.5: for |V(n)|≲⟨n⟩^{-β} with β>1 and µ∈(0,2), show that if (U+vR±_0(µ⁴)v)f=0 for some nonzero f∈ℓ², then ϕ:=R±_0(µ⁴)v f is a nonzero element of ℓ² and satisfies (H-µ⁴)ϕ=0. If the implication holds, the no-eigenvalue assumption is exactly the right hypothesis and the remaining issue is verifying it for the potential class; if it fails, Corollary 2.5 has a proof gap independent of the verification problem. As a secondary numerical check, test the explicit compactly supported potentials V1,V2 in §1.2 for eigenvalues in (0,16) to see whether the hypothesis is satisfied in the paper's own examples.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2 assumes H has no positive eigenvalues in I=(0,16), and this assumption is used exactly once, through Corollary 2.5, to conclude that M±(µ)=U+vR±_0(µ⁴)v is invertible on ℓ² for every µ∈(0,2). The invertibility of M±(µ) is then essential for the resolvent identity (2.11), for the expansions in Theorem 1.8, and hence for all kernel estimates in Section 4. The paper does not prove absence of embedded eigenvalues for any admissible V in the class |V(n)|≲⟨n⟩^{-β} with β>15; Remark 1.3 cites only the δ-potential case [22] and explicitly states that more studies are needed. Thus the advertised decay estimate is not an unconditional statement about the stated potential class. If an admissible V had an eigenvalue at some µ0⁴∈(0,16), then (M±(µ0))^{-1} would be unbounded, the expansion Theorem 1.8 would not apply near µ0, and the argument in Section 4 would not yield the uniform |t|^{-1/4} bound. This is a genuine limitation, though not an internal inconsistency: the theorem is correctly stated as a conditional result. The abstract, however, omits this hypothesis when announcing the result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies time-decay estimates for the fourth-order discrete Schrödinger operator H = Δ² + V on ℓ²(Z) and for the associated beam evolution. For V ≡ 0, it proves the sharp bound ∥e^{-itΔ²}∥_{ℓ¹→ℓ^∞} ≲ |t|^{-1/4}, matching the continuous one-dimensional bi-Schrödinger rate and contrasting with the slower |t|^{-1/3} rate of e^{itΔ} (Theorems 3.1 and 3.5, with a Knapp-type Strichartz sharpness argument). For nonzero potentials with |V(n)| ≲ ⟨n⟩^{-β}, the main theorem (Theorem 1.2) asserts that if H has no positive eigenvalues in I=(0,16), then for β>15 (0 regular), β>19 (first-kind resonance at 0), or β>27 (second-kind resonance at 0), one has ∥e^{-itH}P_ac(H)∥_{ℓ¹→ℓ^∞} ≲ |t|^{-1/4}, together with the beam estimate ∥cos(t√H)P_ac(H)∥ + ∥sin(t√H)/(t√H)P_ac(H)∥ ≲ |t|^{-1/3}. The proof combines three components: a limiting absorption principle via Mourre theory with the conjugate operator A satisfying iA = NP − P*N (Section 2 and Appendix A), full asymptotic expansions of the free and perturbed resolvent at the degenerate threshold 0 and the second threshold 16, including a resonance classification in weighted ℓ² spaces (Theorems 1.7 and 1.8), and Van der Corput estimates for the four kernel components of Stone's formula (Sections 3 and 4).","tokens_in":69874,"tokens_out":29770,"duration_ms":291027,"significance":"If the conditional results are accepted, this is a significant contribution to discrete dispersive estimates: it is, to my knowledge, the first limiting absorption principle and threshold analysis for a higher-order discrete Schrödinger operator with potentials, and it provides a complete resonance classification at both thresholds of Δ². The free decay rate is derived parameter-free and its sharpness is proved by a rigorous Knapp-type argument rather than asserted; the β-thresholds in Theorem 1.2 are exactly matched to the order of the resolvent expansions in Lemma 5.1, which is a sign of internal consistency. The proofs are detailed and largely self-contained, with the Mourre-theory input from [30] and the discrete LAP input from [33] clearly identified. The principal weakness is the conditional nature of Theorem 1.2; this is acknowledged in Remark 1.3, but it is not reflected in the abstract.","major_comments":[{"comment":"The abstract announces the decay estimate for H = Δ² + V under 'suitable decay conditions on V' without stating the spectral hypothesis, but the result actually proved in Theorem 1.2 is conditional on the assumption that H has no positive eigenvalues in I=(0,16). This hypothesis is load-bearing: it is used in Corollary 2.5 (where the invertibility of M±(µ) is deduced from the absence of eigenvalues in I and Theorem 2.4) to obtain the resolvent identity (2.11), which in turn feeds the expansions in Theorem 1.8 and all the kernel estimates in Section 4. If an admissible potential had an eigenvalue at µ₀⁴ ∈ (0,16), the expansion of (M±(µ))^{-1} would not be available near µ₀ and the uniform |t|^{-1/4} bound would not follow from the given proof. The abstract and the announcement in Section 1.1 should therefore state the no-positive-eigenvalue hypothesis explicitly, so that the advertised claim matches the theorem.","section":"Abstract; Theorem 1.2; Corollary 2.5"},{"comment":"The no-positive-eigenvalue hypothesis is not verified for any potential in the allowed class beyond V ≡ 0 and the δ-potentials treated in [22]; Remark 1.3 explicitly says that 'more studies are needed' to establish the absence of positive eigenvalues for higher-order discrete operators, and the resonance examples in Section 1.2 are not checked for embedded eigenvalues. Since the invertibility of M±(µ) for every µ ∈ (0,2) is equivalent, through the resolvent identity, to the absence of eigenvalues at µ⁴, the advertised statement 'under suitable decay conditions on V' is stronger than what is proved. I recommend that the authors either (a) prove the absence of positive eigenvalues for a natural subclass of the allowed potentials (e.g., sign-definite potentials, potentials of sufficiently small norm, or compactly supported potentials), or (b) keep the conditional form but add an explicit paragraph collecting all cases where the hypothesis is known to hold, and state the theorem in exactly that conditional form in the abstract.","section":"Remark 1.3; Corollary 2.5"},{"comment":"The beam estimate (1.13), one of the two headline results, is not proved. The paper states (Section 1.3 and the opening of Section 4) that it suffices to prove (1.12) for e^{-itH}P_ac(H) and that (1.13) 'follows similarly' for e^{-it√H}P_ac(H), because the difference between (1.14) and (1.15) lies only in the power of µ in the exponent. The change of phase from tµ⁴ to tµ² is not purely notational for the stationary-phase analysis: for the beam, the µ³ factor cancels the µ^{-3} singularity of the free resolvent at µ=0, while the |t|^{-1/3} rate is produced by interior inflection points of the dispersion relation (the s=±2 cases of the free phase 2−2cosθ), a regime that does not occur in the |t|^{-1/4} analysis of Section 4.2. Since (1.13) is claimed for the full perturbed operator, please supply the detailed proof, or at least a complete reduction of each kernel component K±j for the phase tµ², including the treatment of the interior region [µ₀, 2−µ₀].","section":"Section 4 (opening); Eq. (1.13)"}],"minor_comments":[{"comment":"The displayed threshold conditions in (4.5) read '11, 0 is the resonance of H' and '15, 0 is the eigenvalue of H'; both should refer to the point 16, since K±3 is the component localized near µ=2 and the surrounding subsection is devoted to the threshold 16.","section":"Theorem 4.1(iii)"},{"comment":"There are several typos: the Abstract has 'a complete characterizations' (should be 'a complete characterization'), Section 1.1 has 'mathematics physics' (should be 'mathematical physics'), and Section 2.2 has 'Theroem 2.4' (should be 'Theorem 2.4').","section":"Abstract; Section 1.1; Section 2.2"},{"comment":"The differentiability claims ('in the same sense, the (5.1) can be differentiated N+4 times in µ'; similarly 'N+2 times' for (5.3)) are incomplete as stated: each differentiation multiplies the kernel by powers of |n−m|, so the required weight s must be larger for the differentiated expansion, and the size of the remainder after k derivatives should be stated. This matters because the bound (1.27) for the remainders in Theorem 1.8 involves ∂µΓ, so the weight and remainder statements in Lemma 5.1 should be precise.","section":"Lemma 5.1"},{"comment":"The proof of negative definiteness of the quadratic form (5.46)–(5.48) is terse: the inequalities ⟨g₁,h₁⟩ ≤ 1/4 and ⟨g₂,h₂⟩ ≤ ∥v′∥⁴/64 are asserted as consequences of (5.41) without derivation, and the strictness caveat 'both inequalities are strict if ⟨h₂,g₁⟩≠0' is used in a crucial way. Please expand this step to make the '⊆' direction transparent.","section":"Lemma 5.2(3)"},{"comment":"In Definition 1.1(II), the resonance class at 16 allows solutions in W1/2(Z) that are not in ℓ²(Z); since W1/2(Z) = ∩_{s>1/2}ℓ^{2,−s} contains non-ℓ² functions (e.g., constants), the distinction between resonance and eigenvalue at 16 is meaningful, but a short remark reminding the reader of the inclusion chain ℓ² ⊆ W₀ ⊆ W1/2 ⊆ W3/2 would prevent confusion.","section":"Definition 1.1; Section 1.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically strong and largely self-contained, and I agree with the reader's overall assessment (soundness around 7, no circularity). The conditional structure of the main theorem is acknowledged in Remark 1.3, and the free-case results are unconditional and sharp. My three major comments are all addressable without changing the architecture of the proof: reword the abstract to state the spectral hypothesis, collect the known cases of the no-eigenvalue hypothesis (or verify it for a natural subclass), and supply the proof of the beam estimate (1.13). The typo in Theorem 4.1(iii) and the imprecision in Lemma 5.1 should be corrected. I would support acceptance after a major revision addressing these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper to know: it proves the sharp |t|^{-1/4} decay for e^{-itΔ²} on ℤ and, under a no-embedded-eigenvalue assumption, the same rate for Δ²+V with a full classification of resonances at thresholds 0 and 16. That fills a real gap: higher-order discrete Schrödinger operators have been largely untouched, and the two-threshold structure (0 degenerate, 16 non-degenerate) makes this genuinely more intricate than the continuous bi-Schrödinger case. The proofs are long but structured: LAP via Mourre theory with an explicit conjugate operator, free resolvent expansions in weighted spaces, and Van der Corput estimates for the oscillatory kernels. The sharpness argument for the free case via Knapp examples and Strichartz is a nice bonus. This is honest, serious work with no fitted parameters or circular reasoning that I can see.\n\nThe soft spot is exactly what the stress-test note lands on. The perturbed decay estimate is conditional on H having no positive eigenvalues in (0,16), and that hypothesis is load-bearing: it guarantees invertibility of M±(µ), hence the resolvent identity and all the expansions in Theorem 1.8. The paper does not prove absence of embedded eigenvalues for any potential in the stated class; Remark 1.3 says this explicitly and cites only the δ-potential example. The abstract, however, announces the decay estimate without mentioning this condition. That is an overstatement and should be corrected. There is also a minor typo in Theorem 4.1(iii), where the cases are labeled “0 is the resonance” and “0 is the eigenvalue” when it clearly means 16, and the decay conditions β>15,19,27 are probably not optimal (the authors say so themselves). None of this is fatal: the theorem is correctly stated as conditional inside the paper, and the proof structure is sound.\n\nWho gets value from this: anyone working on dispersive estimates for discrete Schrödinger or beam equations, or on threshold resolvent expansions on lattices. It deserves a serious referee. My recommendation: send it to review, and ask the authors to put the no-eigenvalue hypothesis in the abstract and fix the 0/16 typos. If the eigenvalue exclusion gets resolved later, the main results will stand.","headline":"Genuinely new decay estimates for discrete bi-Laplace operators, but the headline perturbed result is conditional on an unproved no-eigenvalue assumption that the abstract omits.","tokens_in":70376,"tokens_out":2068,"would_cite":true,"duration_ms":24856,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q41","47B39","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lattice fourth-order Schrödinger dynamics disperse at the continuous rate $|t|^{-1/4}$, with a complete resonance classification, after excluding positive eigenvalues.","keywords":["discrete bi-Laplace operator","decay estimates","resonance classification","limiting absorption principle","ell-1 to ell-infinity estimates","beam equation","threshold expansions"],"falsifier":"Find a potential $V$ with $|V(n)|\\lesssim\\langle n\\rangle^{-\\beta}$ (for example compactly supported) for which the difference equation $(\\Delta^2+V)\\phi=\\lambda\\phi$ has a nonzero $\\ell^2(\\mathbb{Z})$ solution for some $\\lambda\\in(0,16)$; such a discovery would make the hypothesis of Theorem 1.2 false for that potential and would break the invertibility of $M^\\pm(\\mu)$ on which the resolvent expansion and decay proof depend.","tokens_in":69423,"feed_emoji":"📉","tokens_out":10865,"duration_ms":111949,"temperature":0.7,"pith_summary":"The paper establishes that on the lattice $\\mathbb{Z}$ the fourth-order Schrödinger flow generated by the discrete bi-Laplacian $\\Delta^2$ decays like $|t|^{-1/4}$ from $\\ell^1$ to $\\ell^\\infty$, the same rate as for the continuous fourth-order operator on $\\mathbb{R}$, and that this rate is sharp for $V=0$. For perturbed operators $H=\\Delta^2+V$ with real decaying potentials, it proves the same $|t|^{-1/4}$ decay for the absolutely continuous part $e^{-itH}P_{ac}(H)$, and $|t|^{-1/3}$ decay for the associated discrete beam evolution, provided $H$ has no positive eigenvalues in $(0,16)$ and the potential decays fast enough depending on the resonance type at $0$. The proof works by proving a limiting absorption principle for $H$, expanding the boundary resolvent near the two thresholds $0$ and $16$, and classifying all resonance types there; this is the first analysis of its kind for a higher-order discrete Schrödinger operator. A sympathetic reader would care because one basic expectation of lattice dispersion—that discreteness slows decay down—is shown to fail for fourth-order operators, and the result supplies the full resonance-typed picture needed for scattering and nonlinear applications.","feed_headline":"Bi-Laplacian on Z decays like the continuous case: |t|^{-1/4}","feed_subtitle":"For Δ²+V on ℤ with decaying V, the absolutely continuous part decays at |t|^{-1/4} and the beam flow at |t|^{-1/3}.","key_machinery":"The engine of the proof is Stone's formula written in the spectral variable $\\mu$ with $\\lambda=\\mu^4$, so that the decay problem becomes a family of oscillatory integrals in $\\mu$ over $(0,2)$. The free resolvent boundary values $R^\\pm_0(\\mu^4)$ have the explicit kernel $\\frac1{4\\mu^3}(\\pm i a_1(\\mu)e^{\\mp i\\theta_+|n-m|}+a_2(\\mu)e^{b(\\mu)|n-m|})$, whose singularities near $\\mu=0$ and $\\mu=2$ are expanded by powers of $\\mu$ and $(2-\\mu)^{1/2}$; perturbation is organized through the effective operator $M^\\pm(\\mu)=U+vR^\\pm_0(\\mu^4)v$, whose invertibility (guaranteed by the no-positive-eigenvalue assumption) converts $R^\\pm_V(\\mu^4)$ into $R^\\pm_0(\\mu^4)-R^\\pm_0(\\mu^4)v(M^\\pm(\\mu))^{-1}vR^\\pm_0(\\mu^4)$. The resonance classification is encoded in finite-codimensional projections $S_j$ and $\\widetilde S_j$ built from the moments of $v$, which decide how many powers of $\\mu$ are lost in the Neumann expansion. A standard oscillatory-integral estimate with phases $(2-2\\cos\\theta)^2$ having a fourth-order critical point at $\\theta=0$ converts each expansion into the $|t|^{-1/4}$ bound, while the same phase at $\\mu=2$ after conjugation by $J\\phi(n)=(-1)^n\\phi(n)$ produces half-integer powers and the beam-flow estimates.","core_discovery":"On the paper's own terms, the central claim is that $H=\\Delta^2+V$ on $\\ell^2(\\mathbb{Z})$, with real $V$ obeying $|V(n)|\\lesssim\\langle n\\rangle^{-\\beta}$ and with no positive eigenvalues in $I=(0,16)$, satisfies $$\\|$e^{{-itH}}$P_{ac}(H)\\|_{\\$ell^{1}$\\to\\ell^\\infty}\\lesssim|t|^{-1/4}$$ and $$\\|\\cos(t\\sqrt H)P_{ac}(H)\\|_{\\$ell^{1}$\\to\\ell^\\infty}+\\left\\|\\frac{\\sin(t\\sqrt H)}{t\\sqrt H}P_{ac}(H)\\right\\|_{\\$ell^{1}$\\to\\ell^\\infty}\\lesssim|t|^{-1/3},$$ and that for $V=0$ the $|t|^{-1/4}$ exponent is optimal. The required decay of $V$ is $\\beta>15$, $19$, or $27$ according as $0$ is regular, a first-kind resonance, or a second-kind resonance; near $16$, the requirements are $\\beta>7$, $11$, or $15$ in the regular, resonance, and eigenvalue cases. These estimates cover all resonance types at both thresholds, with a complete characterization of the resonance spaces in weighted $\\ell^2$ spaces.","pith_inferences":["Editorial inference: the same machinery should transfer to other higher-order difference operators whose symbol has a degenerate critical point, with the decay exponent presumably set by the order of the first nonvanishing derivative of the symbol.","Editorial inference: because the no-positive-eigenvalue hypothesis is only verified in the paper for a $\\delta$-potential, a natural next step is to prove absence of embedded positive eigenvalues for compactly supported or short-range potentials, or to find a counterexample; the theorem's range of validity expands or contracts accordingly.","Editorial inference: the half-integer power expansions at $\\mu=2$ suggest that the threshold $16$ behaves like a nondegenerate edge for the conjugated operator $JHJ$, which may make endpoint or weighted estimates at the top of the spectrum behave differently from the bottom threshold.","Editorial inference: the Strichartz sharpness from localized data suggests the discrete bi-Schrödinger equation lies in the same admissible family as the continuous fourth-order equation, so nonlinear well-posedness theory on $\\mathbb{Z}$ could be developed along the same admissibility range."],"forward_implications":["For $V=0$ the $\\ell^1\\to\\ell^\\infty$ rate $|t|^{-1/4}$ is sharp, and it yields Strichartz estimates in the admissible range $1/q+1/(4r)\\le 1/8$, with a counterexample built from narrowly localized data excluding any faster rate.","For $H=\\Delta^2+V$, the absolutely continuous part of the fourth-order Schrödinger flow decays at the same $|t|^{-1/4}$ rate even when $0$ is a first- or second-kind resonance or $16$ is a resonance or eigenvalue; only the required decay of $V$ changes.","The beam-flow combination $\\cos(t\\sqrt H)P_{ac}(H)+(\\sin(t\\sqrt H)/(t\\sqrt H))P_{ac}(H)$ has decay $|t|^{-1/3}$, matching the second-order discrete wave and beam decay.","The threshold classification is complete in the sense that $S_3\\ell^2=\\{0\\}=\\widetilde S_2\\ell^2$ for $\\beta>9$, so the zero-energy eigenvalue case is excluded and all remaining zero and sixteen resonant types are captured by the projections listed in Theorem 1.7.","The discrete Schrödinger and beam equations therefore have solutions whose continuous spectral parts disperse exactly as their continuous counterparts, with discrete eigenvalue parts contributing no decay or exponential growth."],"supporting_citations":[{"why":"Supplies the baseline sharp $|t|^{-d/3}$ decay for the discrete Laplacian against which the fourth-order result is compared.","marker":"[51]"},{"why":"Supplies the free resolvent kernel for $\\Delta$ on $\\mathbb{Z}$ used to write $R^\\pm_0(\\mu^4)$ explicitly.","marker":"[33]"},{"why":"Supplies the continuous one-dimensional bi-Schrödinger analysis whose kernel-expansion lemmas are adapted to the lattice.","marker":"[52]"},{"why":"Supplies the multiple-commutator resolvent-smoothness theorem behind the limiting absorption principle.","marker":"[30]"},{"why":"Provides the threshold-resolvent expansion formalism used for the Neumann expansions of $(M^\\pm(\\mu))^{-1}$.","marker":"[31]"},{"why":"Provides the abstract Strichartz theorem that turns the free decay estimate into sharp Strichartz estimates and proves sharpness.","marker":"[37]"},{"why":"Gives the $\\delta$-potential example where absence of positive eigenvalues in $(0,16)$ is known.","marker":"[22]"},{"why":"Supplies the discrete conjugate-operator regularity facts used in proving the required smoothness of $H$ with respect to the conjugate operator.","marker":"[5]"}],"fun_headline_variants":["Discrete bi-Laplacian matches continuous decay: |t|^{-1/4} on Z","Sharp decay for bi-Laplacian with potential: |t|^{-1/4} even with resonances","Discrete beam equation: |t|^{-1/3} decay from bi-Laplacian","Resonance types for bi-Laplacian on Z classified; decay |t|^{-1/4}"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that $H=\\Delta^2+V$ has no positive eigenvalues in the continuous-spectrum interval $(0,16)$; the paper does not prove this for the general decaying potentials it allows, citing only the $\\delta$-potential case and calling for further study.","fun_headline_variants_meta":{"raw":{"variants":["Discrete bi-Laplacian matches continuous decay: |t|^{-1/4} on Z","Sharp decay for bi-Laplacian with potential: |t|^{-1/4} even with resonances","Discrete beam equation: |t|^{-1/3} decay from bi-Laplacian","Resonance types for bi-Laplacian on Z classified; decay |t|^{-1/4}"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000719,"raw_usage":{"total_tokens":3397,"prompt_tokens":1285,"completion_tokens":2112,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":901,"completion_tokens_details":{"reasoning_tokens":2007}},"tokens_in":901,"tokens_out":2112,"duration_ms":17140,"temperature":1.0,"reasoning_tokens":2007,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:48:09.516388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a potential $V$ with $|V(n)|\\lesssim\\langle n\\rangle^{-\\beta}$ (for example compactly supported) for which the difference equation $(\\Delta^2+V)\\phi=\\lambda\\phi$ has a nonzero $\\ell^2(\\mathbb{Z})$ solution for some $\\lambda\\in(0,16)$; such a discovery would make the hypothesis of Theorem 1.2 false for that potential and would break the invertibility of $M^\\pm(\\mu)$ on which the resolvent expansion and decay proof depend.","supporting_citations":[{"cited_title":"Stefanov and P","cited_arxiv_id":null,"evidence_quote":"Supplies the baseline sharp $|t|^{-d/3}$ decay for the discrete Laplacian against which the fourth-order result is compared."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the free resolvent kernel for $\\Delta$ on $\\mathbb{Z}$ used to write $R^\\pm_0(\\mu^4)$ explicitly."},{"cited_title":"Soffer, Z","cited_arxiv_id":null,"evidence_quote":"Supplies the continuous one-dimensional bi-Schrödinger analysis whose kernel-expansion lemmas are adapted to the lattice."},{"cited_title":"Jensen, E","cited_arxiv_id":null,"evidence_quote":"Supplies the multiple-commutator resolvent-smoothness theorem behind the limiting absorption principle."},{"cited_title":"Jensen and G","cited_arxiv_id":null,"evidence_quote":"Provides the threshold-resolvent expansion formalism used for the Neumann expansions of $(M^\\pm(\\mu))^{-1}$."},{"cited_title":"Hiroshima and J","cited_arxiv_id":null,"evidence_quote":"Gives the $\\delta$-potential example where absence of positive eigenvalues in $(0,16)$ is known."},{"cited_title":"Boutet de Monvel and J","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete conjugate-operator regularity facts used in proving the required smoothness of $H$ with respect to the conjugate operator."}],"review_version":1}