{"id":"32a990d4-015c-4a57-ab2f-7621b76a9e60","arxiv_id":"2510.00811","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For unbounded domains, spectral minimal k-partitions exist below a threshold set by the essential spectrum; at the threshold they may fail for p<∞, always exist for p=∞, but need not be equipartitions.","lead":"This paper determines when an infinite domain can be optimally split into k pieces that minimize the lowest energy of a Schrödinger operator, and when no such split exists. It shows a sharp threshold controls everything: below it, minimizers always exist; at it, existence can fail or the pieces can have unequal energies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's hypothesis (3.1) is stronger than Theorem 1.4's (1.10); the missing equality eL_{k-1}=L_{k-1} creates a circular gap in the central existence proof.","rationale":"The reader's weakest assumption was the nonnegativity of V, which is a real limitation but is explicitly assumed (Assumption 1.1) and partially relaxed in Remark 1.11. The concern I identify is more load-bearing because it strikes at the logical validity of the main existence proof. Theorem 1.4 is the central claim, and its proof via Theorem 3.1 uses a hypothesis involving eL_{k-1,p} instead of L_{k-1,p}. Since eL_{k-1,p} ≤ L_{k-1,p} is all that is available at that stage, the threshold condition in Theorem 1.4 is weaker than what Theorem 3.1 requires. The later proof of equality eL=L in Proposition 1.7 appears to use Theorem 1.4, which creates a circular dependency unless an induction is made explicit. This is a fixable gap, not a counterexample, but it means the manuscript as written does not fully establish the central existence result. I therefore recommend a conditional acceptance pending an explicit induction argument or a rephrasing of Theorem 3.1's hypothesis.","tokens_in":34372,"tokens_out":20326,"duration_ms":137243,"concrete_test":"Verify the logical dependency: attempt to prove Theorem 1.4 from Theorem 3.1 by induction on k, making explicit where eL_{k-1,p}=L_{k-1,p} is used. Concretely, check whether Proposition 1.7's proof for k-1 uses Theorem 1.4 for k-1; if yes, the induction can be ordered so that equality for k-1 is established before Theorem 1.4 for k, and the gap closes. If no such ordering exists without circularity, Theorem 1.4 is unproved as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central existence theorem is not directly proved as stated. Theorem 1.4 assumes eL_{k,p}(Ω) < T_{k,p}(Ω) = (L_{k-1,p}(Ω)^p + Σ(Ω)^p)^{1/p}, but the proof given in Theorem 3.1 (Section 3) requires the stronger condition (3.1): eL_{k,p}(Ω)^p < eL_{k-1,p}(Ω)^p + Σ(Ω)^p. Only eL_{k-1,p} ≤ L_{k-1,p} is known at that point (1.6), so the two conditions need not coincide. Step 4's contradiction uses eL_{k-1,p} explicitly (the inequality after (3.6)); replacing it by L_{k-1,p} would be unjustified because the functions u_{2,n},...,u_{k,n} are a test tuple for the relaxed problem, not a partition. The equality eL_{k-1,p}=L_{k-1,p} is proved later in Proposition 1.7, whose proof in the strict-threshold case invokes Theorem 1.4 and Theorem 1.6, creating an apparent circularity. The gap is repairable by an induction on k: base eL_{1,p}=L_{1,p}, and if eL_{k-1,p}=L_{k-1,p} is assumed, then Theorem 1.4's hypothesis becomes (3.1). But this induction is not stated in the manuscript. As written, the proof of Theorem 1.4 is incomplete.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a theory of spectral minimal partitions for Schrödinger operators on unbounded, possibly infinite-volume domains. For open sets ω⊂Ω it replaces Dirichlet first eigenvalues by the bottom λ(ω) of the spectrum, defines partition functionals Λ_{k,p} and their relaxed counterparts eΛ_{k,p} on k-tuples of mutually orthogonal functions, and studies the infima L_{k,p}(Ω) and eL_{k,p}(Ω). The main results are: an energy threshold theorem (Theorem 1.3) giving L_{k,p}(Ω) ≤ T_{k,p}(Ω), where T involves L_{k−1,p}(Ω) and the bottom Σ(Ω) of the essential spectrum; an existence theorem below the threshold (Theorem 1.4); a regularity theorem (Theorem 1.6) showing that minimizers of the relaxed problem give regular, possibly nodal-type partitions; and structural results about equality of the two formulations, equipartitions, and monotonicity in p and k. A series of examples illustrates new phenomena: existence without ground states, non-attainment, non-equipartition of p=∞ minimizers, and mixed behavior for connected/disconnected cells.","tokens_in":34691,"tokens_out":7776,"duration_ms":344785,"significance":"The topic is timely and the paper is the first systematic treatment of spectral minimal partitions on genuinely unbounded domains. If the central existence theorem is established, the results are substantial: they generalize the bounded-domain theory, uncover a natural threshold phenomenon linked to the essential spectrum, and provide a catalog of counterexamples that sharply delineate the range of validity of the classical behavior. The reliance on a relaxed functional and on existing regularity theory is methodologically appropriate. However, the proof of the main existence theorem contains a hypothesis mismatch that is load-bearing; it is repairable, but the repair must be made explicit. The examples in Section 6 are rich and well chosen, but a few of them depend on sketched arguments that should be completed or clearly flagged.","major_comments":[{"comment":"Theorem 1.4 assumes (1.10): eL_{k,p}(Ω) < (L_{k−1,p}(Ω)^p + Σ(Ω)^p)^{1/p}. Theorem 3.1 proves convergence of minimizing sequences under the stronger condition (3.1): eL_{k,p}^p < eL_{k−1,p}^p + Σ^p. At that point only eL_{k−1,p} ≤ L_{k−1,p} is known, so (1.10) does not imply (3.1). Step 4 of the proof of Theorem 3.1 explicitly uses (u_{2,n},…,u_{k,n}) as a test tuple for eL_{k−1,p}, so the lower bound in the contradiction is eL_{k−1,p}, not L_{k−1,p}. The later proof of Proposition 1.7 proves equality eL=L using Theorem 1.4 and Theorem 1.6, making the argument circular as written. The gap is repairable by a simultaneous induction on k: for k=1 the equality eL_{1,p}=L_{1,p}=λ(Ω) is immediate; assuming equality for k−1, condition (1.10) becomes (3.1), and Theorem 3.1 produces a minimizer, after which Theorem 1.6 yields equality for k; when eL_{k,p}=T_{k,p}, the inequalities eL≤L≤T give equ","section":"§3, Theorem 3.1; §5, Proposition 1.7"}],"minor_comments":[{"comment":"The assertion that for d≥3 one can always make each ω_i connected uses 'a cutoff argument (which we omit)'. Since this claim supports the description of minimizing partitions for L_{k,∞}, the omitted argument should either be supplied or the statement explicitly labelled as conditional/sketched.","section":"Remark 6.5"},{"comment":"The connected-domain variant is introduced with 'we will not go into full details'. It is used to support the discussion of ground states and equipartitions; as it stands, the connected version is only sketched. Please provide the missing details or mark the claim as provisional.","section":"Example 6.7"},{"comment":"The final part of Step 8 contains typographical errors: 'for all i=1,…,n' should be 'k', and the displayed inequality mixes a scalar ∥u_1∥₂ with the coordinate vector. Rewriting this coordinatewise would remove ambiguity.","section":"§3, Step 8"},{"comment":"The proof of the 'Otherwise' case states eL_{k,p}=T_{k,p} without spelling out that this follows from eL≤L≤T together with the non-strict threshold assumption. This is correct once the induction in the main comment is in place, but the argument should be made explicit.","section":"§5, Proposition 1.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for math.SP and the results are likely to be of considerable interest. The main obstacle is the gap between Theorem 1.4 and Theorem 3.1; the repair by induction is straightforward but must be written into the paper and the order of proofs adjusted accordingly. The examples are valuable but some should be cleaned up. I would be willing to support acceptance after these revisions."},"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J10","35B65","35J20","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For unbounded domains, spectral minimal partitions exist when the optimal energy is strictly below a threshold determined by the essential spectrum, and then each cell has a simple isolated eigenvalue (a ground state).","keywords":["spectral minimal partitions","unbounded domains","Schrödinger operator","essential spectrum","ground states","concentration-compactness","equipartition","threshold"],"falsifier":"Find a domain Ω and a nonnegative potential V for which eL_{k,p}(Ω) < T_{k,p}(Ω) but eL_{k,p} has no minimizer. The paper's Theorem 1.4 asserts this cannot happen; any such pair is a counterexample. A concrete candidate family is the half-strip of Example 6.9 with a step potential tuned so that the strict inequality (1.10) holds but numerically the minimizing sequence escapes to infinity without converging in L².","tokens_in":34191,"feed_emoji":"🧩","tokens_out":6580,"duration_ms":56817,"temperature":0.7,"pith_summary":"This paper extends the theory of spectral minimal partitions—dividing a domain into k pieces to minimize the p-norm of the pieces' lowest eigenvalues—to unbounded domains, where classical existence proofs fail because the spectrum is no longer discrete. The central result is a threshold: the infimum partition energy is always at most a value T_{k,p} built from the bottom of the essential spectrum and the (k−1)-partition energy, and if the energy is strictly below this threshold, an optimal partition always exists and every cell has a genuine ground state. The proof works through a relaxed variational problem over L²-orthogonal function tuples and a concentration-compactness argument that separates each function into a part near the origin and a part escaping to infinity. The paper also shows that at the threshold, behavior becomes delicate and p-dependent: for p=∞, minimizers always exist (but may lack ground states and need not be equipartitions), while for p<∞, minimizers may cease to exist entirely.","feed_headline":"Optimal partitions exist below a sharp spectral threshold","feed_subtitle":"For unbounded domains, a threshold set by the essential spectrum decides when minimizers exist—and when they are classical.","key_machinery":"The central object is the relaxed functional eΛ_{k,p} defined on k-tuples of L²-orthogonal functions in H^1_{0,V}(Ω), whose infimum eL_{k,p} equals the partition infimum L_{k,p}. The threshold T_{k,p} uses Σ(Ω), the bottom of the essential spectrum (via Persson's characterization), and the (k−1)-partition energy. The proof of existence below the threshold uses IMS localization: cutoffs φ_n, ψ_n with φ_n²+ψ_n²=1 split each function into a part supported near the origin and a part escaping to infinity; the strict threshold inequality forces every component of the limit to be nonzero and yields strong H¹ convergence of the minimizing sequence.","core_discovery":"The paper's core claim is Theorem 1.4: if the relaxed infimum eL_{k,p}(Ω) is strictly smaller than the threshold T_{k,p}(Ω) = (L_{k−1,p}(Ω)^p + Σ(Ω)^p)^{1/p} for p<∞, or than Σ(Ω) for p=∞, then eL_{k,p} is attained by k nonzero, L²-orthogonal functions. Combined with the regularity Theorem 1.6, the supports form a minimizing partition whose cells each carry a simple isolated eigenvalue (a ground state). The paper further establishes that L_{k,p}=eL_{k,p} always, and that for p=∞ a minimizer always exists, but at the threshold it may be a non-equipartition and its cells may lack ground states; for p<∞, neither formulation need admit a minimizer when the threshold is reached.","pith_inferences":["A natural extension suggested by the continuity results in p: the threshold gap eL_{k,p}