{"id":"6b8239a1-405e-483b-8f66-526196444b5d","arxiv_id":"2412.15937","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"On discrete graphs with purely discrete spectrum, the summed eigenvalue difference between the Laplacian with and without a potential equals the weighted sum of the potential.","lead":"This paper proves an exact formula for how much the eigenvalues of a graph Laplacian shift when a nonnegative potential is added: the total shift over all eigenvalues equals the sum of the potential values divided by the vertex weights. The result extends a known finite-graph trace identity to infinite graphs and also yields an inverse-spectral uniqueness statement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Local Weyl law (Prop. 8) is proved via Mercer's theorem without verifying trace class; since Theorem 5 invokes Prop. 8, the main proof has a rigor gap, fixable by a direct Parseval argument.","rationale":"The central formula is a trace-type identity for the multiplication operator V = c/m, and the proof strategy is sound: under Assumption 2, D(Q0) compactly embeds, giving discrete spectrum; L_c has discrete spectrum by Lemma 3; the local Weyl identity is a Parseval fact; the Hadamard formula is standard for form-bounded perturbations; the dominated/monotone convergence step is valid. The weakest point in the written proof is Prop. 8's reliance on Mercer's theorem without checking that e^{-tL_c} is trace class; this is a genuine rigor gap in a lemma the main proof invokes, but it is easily closed by the elementary expansion. Example 7's pi^2/6 is inconsistent with the theorem as stated, supporting a conditional verdict. These issues do not affect the truth of Theorem 5, so the reader's CONDITIONAL verdict remains unchanged.","tokens_in":7221,"tokens_out":23299,"duration_ms":201471,"concrete_test":"Replace the Mercer step in Prop. 8 by a direct Parseval computation: fix x, normalize delta_x, and compute its ell^2(X,m) norm in the basis {f_n^c} to confirm sum_n |f_n^c(x)|^2 = 1/m(x). If this elementary derivation is valid, Prop. 8 holds for every c and the main proof is repaired. Separately, recompute c(n)/m(n) in Example 7: with m(n)=n^{-4} and c(n)=n^2, c/m=n^6 and the sum diverges, so either Example 7's intended c or m must be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 8 states sum_n |f_n^c(x)|^2 = 1/m(x). The proof invokes Mercer's theorem for e^{-tL_c}, but under Assumption 2 alone e^{-tL_c} need not be trace class (eigenvalues can grow arbitrarily slowly), so Mercer's hypotheses are not verified. This matters because Theorem 5's proof uses Prop. 8 for every L_{tau c}. The identity itself is elementary: expand delta_x / sqrt(m(x)) in the orthonormal eigenbasis to get ||delta_x / sqrt(m(x))||^2 = 1 = m(x) sum_n |f_n^c(x)|^2. Thus the gap is repairable, but as written the main proof is not fully justified. Additionally, Example 7's claimed value pi^2/6 is inconsistent with Theorem 5: with m(n) = n^{-4} and c(n) = n^2, c/m = n^6, whose sum over n diverges, so the RHS should be +infinity. This suggests a typo in m or c, not in the theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a spectral comparison result for self-adjoint Laplacians on discrete graphs. Under Assumption 2, which requires the form domain of the unperturbed realization to be compactly embedded in ℓ²(X,m), the induced realization with a potential c ≥ 0 has purely discrete spectrum. The main theorem (Theorem 5) states that the sum over all eigenvalue differences λ_n(c) - λ_n(0) equals the weighted sum ∑_{x∈X} c(x)/m(x), with value +∞ when c/m ∉ ℓ¹(X,1). The proof uses a local Weyl law (Proposition 8), a Hadamard-type formula (Proposition 10), and a truncation argument for the non-ℓ¹ case. A corollary gives an Ambarzumian-type theorem, and an explicit infinite path graph example is provided.","tokens_in":7408,"tokens_out":10314,"duration_ms":86443,"significance":"If the proof issues are repaired, this is a clean and striking result: unlike the continuous and quantum-graph settings, where spectral comparison yields averaged or mean eigenvalue distances, here the full sum of all eigenvalue differences has an exact, simple formula. The local Weyl law identity is also presented without Tauberian arguments. The paper correctly recovers earlier finite-graph results and yields a null-sequence asymptotic isospectrality statement. The main theorem is falsifiable and the argument is largely self-contained. However, the current proof of Proposition 8 relies on an unverified trace-class condition, and Example 7 as stated contradicts the theorem; both issues are repairable.","major_comments":[{"comment":"The proof of Proposition 8 invokes Mercer's theorem to write p_t^c(x,x) = ∑_{n} e^{-tλ_n(c)} |f_n^c(x)|², but it does not verify that e^{-tL_c} is trace class. Under Assumption 2 the eigenvalues of L_c need only tend to infinity with no prescribed rate (for instance, λ_n = log n is compatible with purely discrete spectrum), so e^{-tλ_n} need not be summable and Mercer's theorem is not applicable. Since Theorem 5 uses Proposition 8 for every L_{τc}, this is a gap in the main proof. The identity is elementary and can be proved by expanding δ_x/√m(x) in the orthonormal eigenbasis, which gives m(x)∑_n |f_n^c(x)|² = 1 without any trace-class assumption; please replace or supplement the Mercer argument with this direct Parseval argument.","section":"§3, Proposition 8"},{"comment":"Example 7 is inconsistent with Theorem 5: with m(n) = n^{-4} and c(n) = n², the ratio c(n)/m(n) equals n⁶, whose sum over n∈N diverges, so Theorem 5 predicts +∞, not π²/6. If the intent was to illustrate a finite value, a different potential or weight is needed (e.g., c(n) = n^{-6} would give ∑ n^{-2} = π²/6). Please correct the example.","section":"§2, Example 7"}],"minor_comments":[{"comment":"The phrase 'where the right-hand side equals +∞' should be read as referring to the sum ∑_{x∈X} c(x)/m(x); consider rewording for clarity.","section":"Theorem 5"},{"comment":"In the reference list, the entry [KL W21] contains a spurious space; it should be [KLW21].","section":"References"},{"comment":"Using dim ℓ²(X,m) as the upper summation limit is unconventional; it would help to define it explicitly as ∞ when the space is infinite-dimensional.","section":"Notation"},{"comment":"After the eigenfunction expansion is established, the argument using Fatou's lemma is unnecessarily indirect; a monotone-convergence or direct Parseval argument would be clearer.","section":"§3, Proposition 8"}],"recommendation":"major_revision","confidential_remarks":"The paper is short and the main theorem is very likely correct. The two flagged issues are local and fixable: the Mercer trace-class gap in Proposition 8 and the arithmetic error in Example 7. I would encourage the authors to replace the Mercer-based proof with the direct eigenbasis expansion, and to correct or reinterpret the example. No concerns about novelty or scope; the result is a worthwhile contribution to the spectral comparison literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read of arXiv:2412.15937. The headline: the main spectral comparison result for infinite discrete graphs is correct, and the proof strategy is cleaner than the continuous analogues, but Proposition 8 is proved via Mercer's theorem without checking trace class, and Example 7 has a numerical inconsistency. Both are fixable.\n\nWhat's actually new: Theorem 5 gives the exact identity sum_n (lambda_n(c) - lambda_n(0)) = sum_x c(x)/m(x) for a broad class of infinite discrete graph Laplacians, with divergence when c/m is not summable. The finite-graph version was already in [BK23]; the infinite extension is new. Corollary 11's Ambarzumian-type statement follows immediately and is a nice touch. The paper does not rely on Tauberian arguments or Weyl asymptotics, and the proof is mostly self-contained.\n\nWhat it does well: the Hadamard-formula-plus-partial-sums argument is genuinely simple. The normalization by m(x) is the right one, and the statement that the eigenvalue differences form a null sequence when c/m is summable is a clean asymptotic-isospectrality remark.\n\nSoft spots, in proportion: First, Proposition 8's proof invokes Mercer's theorem for e^{-tL_c}. Under Assumption 2 the semigroup need not be trace class, so Mercer's hypotheses are not verified. The identity itself is just Parseval: expanding the normalized delta function gives sum_n |f_n^c(x)|^2 = 1/m(x). So the gap is real but repairable with a two-line proof. Second, Example 7 says the RHS is pi^2/6, but with m(n)=n^{-4} and c(n)=n^2, c/m = n^6, which diverges; the theorem would give +infinity. That's a typo in the example data, not in the theorem. Third, the label \"local Weyl law\" is generous — it is an elementary complete-basis identity — but they do correctly flag that no Tauberian theorem is involved.\n\nThe citation pattern is fine: the authors' prior work is the natural antecedent, and the finite case is explicitly recovered rather than assumed.\n\nNet: the mathematics is sound once Prop 8 is patched. This deserves a serious referee, and with minor revision it should be publishable. I'd bring it to a spectral-graph-theory reading group as a neat example of exact comparison identities; I'd cite it if I worked on discrete Laplacians.","headline":"Correct main theorem for infinite discrete graph Laplacians with a repairable gap in the local Weyl law proof and a typo in Example 7 — worth refereeing.","tokens_in":7980,"tokens_out":3399,"would_cite":true,"duration_ms":31935,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B25","05C50","34L20","81Q10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For discrete graphs, the summed eigenvalue shift from adding a potential equals the weighted total of the potential, and the formula stays exact on infinite graphs.","keywords":["discrete graph Laplacian","spectral comparison","local Weyl law","eigenvalue differences","Ambarzumian theorem","compact embedding","heat kernel","Schrödinger operator on graphs"],"falsifier":"Take the infinite path graph of Example 4, with $m(n)=n^{-4}$, edge weights $b(n,n+1)=n^2$, and $c(n)=n^2$. The theorem predicts $\\sum_n(\\lambda_n(c)-\\lambda_n(0))=\\sum_n 1/n^2=\\pi^2/6$; computing the first $N$ eigenvalue differences numerically and checking whether their partial sums converge to $\\pi^2/6$ while the summands go to zero would settle it. On any finite graph, exact diagonalization can test the same identity in closed form: the sum of all eigenvalue shifts must equal $\\sum_x c(x)/m(x)$.","tokens_in":6964,"feed_emoji":"🧮","tokens_out":8180,"duration_ms":67381,"temperature":0.7,"pith_summary":"This paper proves a spectral comparison formula for Laplacians on discrete graphs: for any realization $L_0$ whose form domain is compactly embedded in $\\ell^2(X,m)$, and any nonnegative potential $c$, the induced operator $L_c$ satisfies $\\sum_n (\\lambda_n(c)-\\lambda_n(0)) = \\sum_x c(x)/m(x)$, with the right-hand side understood as $+\\infty$ when $c/m$ is not summable. Because the right-hand side is just a weighted total of the potential, the result turns a spectral question into an explicit arithmetic sum, even on infinite graphs. The proof passes through a discrete local Weyl law, $\\sum_n |f_n^c(x)|^2 = 1/m(x)$ at every vertex, which is derived directly from the heat kernel and Mercer's theorem without any Tauberian or Weyl-law input. As a corollary, if adding $c$ changes no eigenvalue, then $c$ must be zero.","feed_headline":"Adding a potential shifts graph eigenvalues by exactly its total weight","feed_subtitle":"On discrete graphs, the summed eigenvalue change equals the weighted total of the potential, even for infinite graphs.","key_machinery":"The load-bearing identity is the discrete local Weyl law, $\\sum_{n=1}^{\\dim \\ell^2(X,m)} |f_n^c(x)|^2 = 1/m(x)$ for each vertex $x$. It is obtained from Mercer's theorem applied to the heat kernel $p_t^c(x,x)$, together with the strong continuity of the semigroup and Fatou's lemma. A second mechanism is the Hadamard-type formula $\\frac{d}{d\\tau}\\lambda_n(\\tau c) = \\sum_x c(x)|f_n^{\\tau c}(x)|^2$, which lets the spectral shift be written as an integral over $\\tau$. Assumption 2 — compact embedding of the form domain — guarantees the purely discrete spectrum needed for both identities; the truncation argument $\\{c_M\\}$ then extends the formula to potentials with $c/m\\notin\\ell^1$.","core_discovery":"The paper's central claim is that the spectral shift induced by a nonnegative potential $c$ is exactly additive over the full spectrum: writing $\\lambda_n(c)$ and $\\lambda_n(0)$ for the eigenvalues of the induced and base realizations, one has $\\sum_n (\\lambda_n(c)-\\lambda_n(0)) = \\sum_x c(x)/m(x)$, where the right-hand side is $+\\infty$ precisely when $c/m\\notin\\ell^1(X,1)$. The same statement holds for finite and infinite graphs, provided the base form domain is compactly embedded in $\\ell^2(X,m)$ (Assumption 2). The key intermediate is the local Weyl law $\\sum_n |f_n^c(x)|^2 = 1/m(x)$, valid at every vertex; combined with a Hadamard-type formula for the derivative of $\\lambda_n(\\tau c)$ along the linear path $\\tau\\mapsto \\tau c$, it yields the formula by integrating over $\\tau$ and applying dominated convergence. The non-summable case follows by truncating $c$ and using the min-max principle.","pith_inferences":["The formula ties the full eigenvalue sequence to only one scalar per graph, the weighted total of the potential; recovering the potential pointwise from spectra would therefore require additional data such as eigenfunction intensities or a family of perturbations.","Because the local Weyl law does not depend on the edge weights $b$, the summed spectral shift is insensitive to the graph's connectivity; this suggests using the identity as a normalization constraint when matching graph models to spectral measurements.","A natural testable extension is to signed potentials $c$: the same heat-kernel and Hadamard mechanism should yield the identity whenever $c/m$ is summable, although the truncation argument would need modification since the min-max monotonicity is lost.","One could probe the necessity of Assumption 2 by looking for realizations without compact embedding where the pointwise identity $\\sum_n |f_n^c(x)|^2 = 1/m(x)$ still holds; if any exist, the spectral comparison formula may extend beyond the stated class."],"forward_implications":["If $c/m\\in\\ell^1(X,1)$, the eigenvalue differences $\\lambda_n(c)-\\lambda_n(0)$ form a null sequence, so the two operators are asymptotically isospectral.","If $c/m\\notin\\ell^1(X,1)$, the summed eigenvalue shift is $+\\infty$; the spectrum is still purely discrete, but there is no finite comparison.","Adding a potential that leaves every eigenvalue unchanged forces $c=0$; this is the Ambarzumian-type theorem for discrete graphs.","The discrete local Weyl law holds for every realization satisfying Assumption 2, giving a pointwise completeness relation for the eigenfunction system at each vertex.","For finite graphs the formula is an exact trace identity, since both sides are finite and the eigenvalue shift sums to $\\sum_x c(x)/m(x)$."],"supporting_citations":[{"why":"Benchmark comparison of Robin and Neumann eigenvalue sums on bounded domains; the discrete result is the graph analogue of this program.","marker":"[RWY21]"},{"why":"Establishes spectral comparison results for Schrödinger operators on compact quantum graphs, the continuous setting this paper transfers to discrete graphs.","marker":"[BK24c]"},{"why":"Provides the truncation argument and modified local Weyl law for infinite quantum graphs used in the non-summable case.","marker":"[BK24d]"},{"why":"Its Lemma 2.6 is invoked to verify the compact embedding (Assumption 2) for the infinite path example.","marker":"[BM25]"},{"why":"Standard result that compact embedding of the form domain implies purely discrete spectrum.","marker":"[Sch12]"},{"why":"First-order asymptotic perturbation theory that yields the Hadamard-type formula for eigenvalue derivatives.","marker":"[LS24]"},{"why":"Classical perturbation theory reference for the Hadamard formula used along the path $\\tau c$.","marker":"[Kat66]"},{"why":"Heat-kernel diagonal expansion for compact metric graphs, the counterpart of the local Weyl law proved here.","marker":"[BEJ22]"}],"fun_headline_variants":["Eigenvalue shifts sum to total potential weight on graphs","Sum of eigenvalue shifts equals weighted potential sum on graphs","On infinite graphs, weighted potential equals summed spectral shift","Local Weyl law gives exact eigenvalue shift sum on graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument stands on a compactness assumption: the energy space of the base operator must be squeezed inside the square-summable functions tightly enough that the spectrum is purely discrete, which is automatic on finite graphs but a genuine restriction on infinite ones.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvalue shifts sum to total potential weight on graphs","Sum of eigenvalue shifts equals weighted potential sum on graphs","On infinite graphs, weighted potential equals summed spectral shift","Local Weyl law gives exact eigenvalue shift sum on graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000593,"raw_usage":{"total_tokens":2721,"prompt_tokens":831,"completion_tokens":1890,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":1837}},"tokens_in":447,"tokens_out":1890,"duration_ms":12163,"temperature":1.0,"reasoning_tokens":1837,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:57:23.006965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the infinite path graph of Example 4, with $m(n)=n^{-4}$, edge weights $b(n,n+1)=n^2$, and $c(n)=n^2$. The theorem predicts $\\sum_n(\\lambda_n(c)-\\lambda_n(0))=\\sum_n 1/n^2=\\pi^2/6$; computing the first $N$ eigenvalue differences numerically and checking whether their partial sums converge to $\\pi^2/6$ while the summands go to zero would settle it. On any finite graph, exact diagonalization can test the same identity in closed form: the sum of all eigenvalue shifts must equal $\\sum_x c(x)/m(x)$.","supporting_citations":[],"review_version":1}