{"id":"c2910f05-36ca-4d8a-beb9-89167ff8ddb9","arxiv_id":"2506.12650","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For reflection-symmetric single-well potentials, the paper derives lower bounds on the tunneling splitting for each excited level, not just the ground state, in all dimensions.","lead":"Using reflection-symmetric single-well potentials, this math paper proves new exponential lower bounds on quantum tunneling between two wells for excited states, in any dimension. The result extends well-known ground-state tunneling estimates to higher energy levels, which underlies electron hopping in solids and quantum device modeling.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reflection-symmetry hypothesis U v=v in §4 is vacuous under the paper’s compact-support/disjoint-well assumptions; for d>2a reflection about x1=d/2 maps supp v into a disjoint ball, forcing v≡0. Corollary 5.2 therefore has no nontrivial instance.","rationale":"The reader’s weakest assumption is exactly the one I find decisive. I verified the support calculation directly: reflection of B_a(0) about x_1=d/2 is B_a(d e_1), and since d>2a these two balls are disjoint, so requiring v to equal its reflected copy is inconsistent with (1.3) unless v≡0. This empties the paper’s headline result. I do not see a reading of §4 that avoids the contradiction: c=d/2 and d>2a are explicit, and U is explicitly reflection about the bisecting plane, not about the origin. The 1D note’s evenness condition A_-=±A_+ suggests the intended symmetry is about the origin, and replacing U by x↦−x would remove the contradiction, but that replacement is not what is written. A secondary reservation is that even with the symmetry repaired, Theorem 5.1 is not fully proved in the text; its proof is a sketch and the energy-estimates section contains a '??' placeholder. The paper does contain real, checkable ideas: the integration-by-parts formula (3.1) and the exact 1D formula (2.2) are clean and appear correct, and the overall strategy of proving (1.9) by Schur complement plus Rouché is plausible. But those components do not support Corollary 5.2 as stated, because the symmetry hypothesis that feeds the lower bound has no nonzero compactly supported instance. I therefore concur with the reader’s verdict and recommend no change.","tokens_in":14611,"tokens_out":6508,"duration_ms":78649,"concrete_test":"Take ν=1, a=1, d=4, and let v be any nonzero function with supp(v)⊂[−1,1]. The reflection U of §4 acts by Uv(x)=v(4−x); for x∈[−1,1], 4−x∈[3,5], so Uv(x)=0, whereas v is nonzero on [−1,1]. Hence Uv=v cannot hold. The same computation in general dimensions gives supp(Uv)⊂B_a(d e_1), disjoint from supp(v), so a nonzero reflection-symmetric v is impossible under (1.3). If the authors reply that the intended condition is evenness about the origin, the test should be to rerun Theorem 4.1 under v(−x_1,x_⊥)=v(x_1,x_⊥), checking that (4.3) and the parity choice of φ_j still yield the exponential lower bound (4.2) with no additional hidden sign condition on A(p_⊥).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing defect is in §4. The paper fixes supp(v)⊆B_a(0) and d>2a in (1.2)–(1.3), then defines U by (Uf)(x_1,x_⊥)=f(d−x_1,x_⊥) and assumes Uv=v. For every x∈B_a(0) we have x_1∈[−a,a] and d−x_1∈[d−a,d+a]⊂(a,∞) because d>2a; more generally (d−x_1)^2+|x_⊥|^2>a^2, so Uv(x)=0 on B_a(0). Equality Uv=v therefore forces v≡0 on B_a(0), hence v≡0. This is not a technicality about sign or normalization: Theorem 4.1 and Corollary 5.2 have no nonzero admissible v as written. What is consistent with the stated support is evenness v(−x_1,x_⊥)=v(x_1,x_⊥), the symmetry used implicitly in the 1D warmup; reflection about the bisecting plane swaps the two wells and cannot be a symmetry of one compactly supported well. A further independent gap is that Theorem 5.1, needed to convert lower bounds on |ρ_j| into splitting lower bounds, is only sketched, and the energy-estimates section contains an unresolved 'assumption??' placeholder before invoking Theorem 6.4. The vacuity of the symmetry hypothesis alone already removes the paper’s advertised new lower bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the double-well Schrödinger operator H_{λ,d}=P^2+λ^2(v(x)+v(x-d)), with a compactly supported single-well potential v satisfying supp(v)⊆B_a(0) and d>2a. Its goal is to prove lower bounds on the excited-state tunneling splitting Δ_j=E_j^+-E_j^- as the well separation d tends to infinity. The strategy is to express the hopping coefficient ρ_j as a boundary integral on the bisecting hyperplane x_1=d/2, prove a d-dependent exponential lower bound on |ρ_j| under a reflection-symmetry assumption on v, and then invoke a Schur-complement/Rouché argument to show Δ_j/(2|ρ_j|)→1 as d→∞. A one-dimensional warm-up gives an exact exponential formula for ρ_j.","tokens_in":14950,"tokens_out":27065,"duration_ms":304470,"significance":"If valid, the advertised lower bound would extend Agmon-type tunneling estimates from the ground state to excited states in all spatial dimensions, with an explicit exponential rate in d. The one-dimensional exact formula and the boundary-integral representation are clean and potentially useful. The proof of Theorem 4.1 is plausible after correcting the symmetry hypothesis, and the paper is honest about the sketchy nature of Theorem 5.1. However, as written the main theorem's symmetry hypothesis is vacuous, a central proposition has a sign error, and the proof of the key bridge theorem is incomplete; the paper therefore needs substantial revision.","major_comments":[{"comment":"The hypothesis Uv=v is vacuous for any nonzero admissible v. Since supp(v)⊆B_a(0) and d>2a, for x∈B_a(0) we have x_1∈[-a,a] and d-x_1≥d-a>a, so Uv(x)=v(d-x_1,x_⊥)=0; hence Uv=v forces v≡0 on B_a(0). Consequently Theorem 4.1 and Corollary 5.2 have no nontrivial instance. The reflection symmetry that is compatible with the setup is evenness about the origin, v(-x)=v(x), with Uφ_j=±φ_j understood as parity about x=0; the proof of (4.3) also requires this parity, not reflection about the bisecting plane.","section":"Section 4, Eq. (4.1)-(4.2)"},{"comment":"The proof of (5.1) is only a sketch and does not establish the key estimate (5.4). The Schur complement reduction is asserted, the correction terms are dismissed with 'All other terms follow a similar pattern', and the resolvent lower bound in Theorem 6.3 depends on Lemma 6.5, whose proof contains a literal 'assumption??' placeholder immediately before invoking Lemma 6.4 (referred to as Theorem 6.4). Lemma 6.6 likewise refers to 'Theorem 6.5' where Lemma 6.5 is meant. The energy estimates are therefore not proved, so the bridge from |ρ_j| lower bounds to Δ_j is missing.","section":"Theorem 5.1 and Section 6.1"},{"comment":"The sign in Eq. (3.1) is incorrect. Because P^2=-Δ, the Green identity gives ⟨P^2φ,ψ⟩_D = ⟨φ,P^2ψ⟩_D - ∫_{∂D}((∇φ)ψ - φ∇ψ)·n, and hence ρ_j = +∫_{∂D}((∇φ)ψ - φ∇ψ)·n. Equation (3.1) has the opposite sign and is inconsistent with the correct specialization (3.2), which has the sign used in the rest of the paper. The proof's line immediately after 'Now using the various integration by parts formulae' should also have a plus sign before the boundary integral.","section":"Proposition 3.1, Eq. (3.1)"}],"minor_comments":[{"comment":"The phrase 'set R=ε' should be 'set R=√ε' (or equivalent) in order to obtain the exponent -d√(-e_j+ε) from the proof of Theorem 4.1.","section":"Corollary 5.2"},{"comment":"The text 'we may assume e>γ' should read '|e|>γ', since e_j<0 and assumption 2 of Theorem 5.1 is |e_j|>γ_j.","section":"Section 6.2"},{"comment":"There are several incorrect cross-references: Lemma 6.5's proof refers to 'Theorem 6.4' (should be Lemma 6.4), Lemma 6.6 refers to 'Theorem 6.5' (should be Lemma 6.5), and Lemma 6.7 refers to 'Theorem 6.2' (should be Assumption 6.2).","section":"Section 6.1"},{"comment":"The sentence 'outside of the support of a' appears to have a typo; it should say 'outside of the support of v'.","section":"Section 4, proof of Theorem 4.1"}],"recommendation":"major_revision","confidential_remarks":"This manuscript appears to be an unfinished draft. In addition to the load-bearing problems above, the text contains placeholders and misnumbered theorem references, and the main theorem's hypothesis is inconsistent with the setup. The intended result may be salvageable, but a careful rewrite and a complete proof of Theorem 5.1 are needed before the paper can be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the short version. The 1D exact formula for the hopping coefficient (Theorem 2.1) is a clean, correct result and the main reason to read the paper. The higher-dimensional story, however, has a load-bearing flaw: the reflection symmetry assumed in Section 4 is impossible for any nonzero compactly supported v under the paper's own assumptions.\n\nWhat is new and good. Theorem 2.1 gives rho_j(d) = C_j exp(-d sqrt(-e_j)) for every excited state in one dimension, with C_j nonzero and d-independent. The proof via a Wronskian at the bisector is elegant. The boundary representation (3.2) is classical, and the authors say so. The use of Aronszajn's unique continuation to keep sign-changing excited states from making rho vanish is a reasonable idea. The fixed-lambda, large-d limit (1.9) is a natural extension of FSW22 and worth having if it can be proved.\n\nWhere it falls apart. In Section 4, the authors take v supported in B_a(0), d > 2a, and define U by reflection about the plane x1 = d/2. They then assume U v = v. But Uv is supported in B_a(d e1), which is disjoint from B_a(0). Hence Uv = v forces v = 0. There is no nonzero admissible v, so Theorem 4.1 and Corollary 5.2 are vacuous as stated. The intended symmetry is almost certainly evenness of v about the origin, v(-x) = v(x), which is what the 1D warmup actually uses. That correction would make the argument plausible, but it is not what the paper says.\n\nThere is a second gap. Theorem 5.1, which converts lower bounds on |rho| into splitting lower bounds, is only sketched; most of it is delegated to FSW22, and the energy-estimates section contains an unresolved 'assumption??' placeholder in the proof of Lemma 6.5 before applying the disjoint-energy estimate. So the connection between rho and Delta is not fully demonstrated here.\n\nWho should read it. Spectral theorists and people working on tight-binding models will want the 1D formula. The higher-dimensional part needs a corrected symmetry hypothesis and a completed proof of the connection theorem.\n\nRecommendation. I would not accept the paper as is. I would send it back for major revision rather than desk-reject, because the fix is identifiable and the 1D part is solid. A referee who spots the symmetry error can give the authors a clear path to a salvageable paper.","headline":"The 1D hopping formula is clean and correct, but the paper's main excited-state lower bound is vacuous as written because the reflection symmetry in §4 is impossible for a compactly supported single well.","tokens_in":15492,"tokens_out":3555,"would_cite":false,"duration_ms":37427,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q10","35P15","35J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under reflection symmetry of the single-well potential, every excited-state tunneling splitting is bounded below by an explicit exponential in the well separation.","keywords":["quantum tunneling","double-well potential","excited states","tunneling splitting","hopping coefficient","exponential decay","reflection symmetry","Agmon estimates"],"falsifier":"Test the hypothesis by choosing any nonzero $v$ with support in $B_a(0)$ and $d>2a$ and asking whether $v(x)=v(d-x)$ for every $x$; the identity forces $v$ to vanish on both half-spaces split by $x_1=d/2$, so no such potential exists. If the symmetry is instead taken as evenness about the origin, compute the hopping coefficient for a smooth even bump in one dimension and compare the exact $\\rho_j=-2A_+A_-\\kappa e^{-\\kappa d}$ with the asserted lower bound; the ratio should approach 1 if the theorem's mechanism is correct.","tokens_in":14385,"feed_emoji":"⚛️","tokens_out":16829,"duration_ms":181373,"temperature":0.7,"pith_summary":"Tunneling between two wells splits each single-well energy level into a doublet, and for the ground state the splitting is known to decay exponentially with the separation between the wells. This paper asks whether the same is true for excited levels, whose wavefunctions change sign and therefore escape the positivity arguments used for the ground state. Its central claim is that, under a reflection-symmetry hypothesis on the single-well potential, every excited-level splitting $\\Delta_j(\\lambda,d)$ obeys the explicit lower bound $\\Delta_j(\\lambda,d)\\ge (1-\\varepsilon)C_\\varepsilon(\\lambda)\\sqrt{-e_j}\\,\\exp(-d\\sqrt{-e_j+\\varepsilon})$ for all sufficiently large $d$, and that $\\Delta_j/(2|\\rho_j|)\\to 1$ as $d\\to\\infty$, where $\\rho_j$ is the hopping coefficient between the two wells. If correct, this promotes a qualitative statement about exponential smallness into a quantitative, level-by-level estimate that also fixes the exponential rate at the single-well binding energy.","feed_headline":"Excited-state tunneling splits get explicit exponential lower bounds","feed_subtitle":"If true, every excited level's tunneling splitting decays no faster than an explicit exponential in the well separation.","key_machinery":"The engine of the proof is the hopping coefficient $\\rho_j$, defined as the matrix element of the translated potential between a single-well eigenfunction and its $d$-shift; it is what the splitting $\\Delta_j$ is asymptotically twice as large as. The argument chains four mechanisms: the surface-integral identity $\\rho_j=\\int_{\\{x_1=d/2\\}}[(\\partial_1\\varphi_j)R_d\\varphi_j-\\varphi_j\\partial_1 R_d\\varphi_j]$, which moves all information to the gap between the wells; the reflection-symmetry reduction of that integral to a derivative of the profile $\\int_{\\mathbb{R}^{\\nu-1}}|\\varphi_j(x_1,x_\\perp)|^2 dx_\\perp$ at $x_1=d/2$; a nonvanishing theorem for solutions of elliptic equations together with an exponential decay estimate, which turn sign-definiteness into the explicit $d$-dependence; and resolvent energy estimates in the style of a Schur-complement reduction, which connect $\\rho_j$ to the actual eigenvalue splitting.","core_discovery":"The discovery is that the hopping coefficient $\\rho_j=\\langle\\varphi_j,\\lambda^2 v(\\cdot-d)\\varphi_j\\rangle$, rather than any property of the nodal set, controls the tunneling splitting of level $e_j$ in the fixed-$\\lambda$, large-$d$ regime. The paper proves an exact surface-integral representation for $\\rho_j$ over the bisecting hyperplane $x_1=d/2$, and shows that when the single-well potential is reflection symmetric this integral reduces to the derivative of the transverse profile of $|\\varphi_j|^2$ at that plane. Because a classical nonvanishing theorem for elliptic equations prevents that profile from being identically zero, and an exponential decay estimate gives the decay rate, $\\rho_j$ has the lower bound $|\\rho_j(d)|\\ge C_\\varepsilon(\\lambda)\\sqrt{-e_j}\\exp(-d\\sqrt{-e_j+\\varepsilon})$. Resolvent estimates then show $\\Delta_j/(2|\\rho_j|)\\to1$, yielding the explicit lower bound on $\\Delta_j$ stated in Corollary 5.2.","pith_inferences":["As printed, the reflection hypothesis $U v=v$ about $x_1=d/2$ is incompatible with $\\operatorname{supp}v\\subseteq B_a(0)$ and $d>2a$: the reflected support lies in a disjoint ball, so every such $v$ is identically zero, leaving Theorem 4.1 and Corollary 5.2 without a single nontrivial instance. If the intended hypothesis is evenness about the origin, $v(x)=v(-x)$, the Fourier-transform computation","A direct check of the one-dimensional formula $\\rho_j=-2A_+A_-\\sqrt{-e_j}\\,e^{-\\sqrt{-e_j}d}$ for a smooth even, compactly supported potential would test whether the prefactor and rate of the lower bound are sharp, and at what separation the asymptotic $\\Delta_j/(2|\\rho_j|)\\to1$ becomes visible.","The lower bound only requires $|\\rho_j|$, not its sign, so the same machinery might yield lower bounds for asymmetric double wells if one can control the phase of the surface integral or prove a modulus lower bound directly."],"forward_implications":["Every excited-state level, not just the ground state, is guaranteed to have a tunneling splitting no smaller than an explicit exponential scale set by $\\sqrt{-e_j}$.","The ratio $\\Delta_j/(2|\\rho_j|)$ tends to 1 as $d\\to\\infty$, so computing the hopping coefficient from the surface integral gives the asymptotic splitting for each level.","The bound holds with rate $\\sqrt{-e_j+\\varepsilon}$ for every $\\varepsilon>0$, showing the exponential rate is pinned to the single-well binding energy in the limit $\\varepsilon\\downarrow0$.","The paper formulates its resolvent energy estimates abstractly for translation-invariant kinetic terms and countable sets of wells, so the splitting-to-hopping mechanism is set up for use in periodic or multi-well settings beyond the two-well problem.","The surface-integral representation extends to magnetic translations when the vector potential satisfies $[P_i,A_i(X)]=0$, so the same route is available for magnetized tunneling problems."],"supporting_citations":[{"why":"Supplies the theorem that a solution of an elliptic equation cannot vanish on an open set, used to show the hopping coefficient is not identically zero on the bisecting hyperplane.","marker":"[Aro57]"},{"why":"Supplies the exponential decay estimate for bound states, used to control eigenfunctions and the partial Fourier transform outside the well support.","marker":"[Agm82]"},{"why":"Supplies the resolvent and Schur-complement machinery plus the relation between splitting and hopping coefficient that Theorem 5.1 extends to excited states.","marker":"[FSW22]"},{"why":"Supplies the energy-estimate framework, generalized in Section 6 to excited states, degeneracies, and more general translations.","marker":"[SW22]"},{"why":"Establishes the ground-state lower bound in the same setting and the resolvent-estimate strategy that the present paper adapts to excited states.","marker":"[FLW18]"},{"why":"Provides the earlier surface-integral expression for a two-level splitting that Proposition 3.1 generalizes to higher dimensions and excited states.","marker":"[LL77]"}],"fun_headline_variants":["Explicit exponential lower bounds for excited-state tunneling","Excited-state tunneling splits get explicit exponential decay","Reflection symmetry yields explicit tunneling lower bounds","Hopping coefficient controls tunneling splitting bound","New bound on tunneling splitting for excited states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a single-well potential supported in one small ball is reflection-symmetric about the far-away bisecting plane; because the two support regions are disjoint, that premise forces the potential to vanish identically, so the main lower bound as stated has no nonzero example.","fun_headline_variants_meta":{"raw":{"variants":["Explicit exponential lower bounds for excited-state tunneling","Excited-state tunneling splits get explicit exponential decay","Reflection symmetry yields explicit tunneling lower bounds","Hopping coefficient controls tunneling splitting bound","New bound on tunneling splitting for excited states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1357,"prompt_tokens":780,"completion_tokens":577,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":396,"completion_tokens_details":{"reasoning_tokens":510}},"tokens_in":396,"tokens_out":577,"duration_ms":6885,"temperature":1.0,"reasoning_tokens":510,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:45:54.409446+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the hypothesis by choosing any nonzero $v$ with support in $B_a(0)$ and $d>2a$ and asking whether $v(x)=v(d-x)$ for every $x$; the identity forces $v$ to vanish on both half-spaces split by $x_1=d/2$, so no such potential exists. If the symmetry is instead taken as evenness about the origin, compute the hopping coefficient for a smooth even bump in one dimension and compare the exact $\\rho_j=-2A_+A_-\\kappa e^{-\\kappa d}$ with the asserted lower bound; the ratio should approach 1 if the theorem's mechanism is correct.","supporting_citations":[],"review_version":1}